Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 549518 11 pageshttpdxdoiorg1011552013549518

Research ArticleCommon Fixed Point Results for Mappings withRational Expressions

M A Kutbi1 Jamshaid Ahmad2 Nawab Hussain1 and Muhammad Arshad3

1 Department of Mathematics King Abdulaziz University Jeddah Saudi Arabia2Department of Mathematics COMSATS Institute of Information Technology Chak Shahzad Islamabad 44000 Pakistan3 Department of Mathematics International Islamic University Sector H-10 Islamabad 44000 Pakistan

Correspondence should be addressed to Jamshaid Ahmad jamshaid jasimyahoocom

Received 2 March 2013 Accepted 14 May 2013

Academic Editor Irena Rachunkova

Copyright copy 2013 M A Kutbi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We obtain some common fixed point results for single as well as set valued mappings involving certain rational expressions incomplete partial metric spaces In the process we generalize various results of the literature Two examples are also included toillustrate the fact that our results cannot be obtained from the corresponding results in metric spaces

1 Introduction and Preliminaries

In 1994 Matthews [1] introduced the concept of a partialmetric space and obtained a Banach type fixed point theoremon a complete partial metric space Later on several authors(see eg [1ndash28]) proved fixed point theorems in partialmetric spaces After the definition of the Partial Hausdorffmetric Aydi et al [9] proved Banach type fixed point resultfor set valued mappings in complete partial metric spaceHere we prove some common fixed point results for singleas well as set valued mappings involving certain rationalexpressions in complete partial metric spaces and show byexamples that the results proved in this paper cannot bededuced from the corresponding results inmetric spaces (seeExample 10 Remark 13)

We start with recalling some basic definitions and lemmason partial metric space The definition of a partial metricspace is given by Matthews (see [1]) as follows

Definition 1 A partial metric on a nonempty set 119883 is afunction 119901 119883 times 119883 rarr [0infin) such that for all 119909 119910 119911 isin 119883

(P1) 119901(119909 119909) = 119901(119910 119910) = 119901(119909 119910) if and only if 119909 = 119910

(P2) 119901(119909 119909) le 119901(119909 119910)

(P3) 119901(119909 119910) = 119901(119910 119909)

(P4) 119901(119909 119911) le 119901(119909 119910) + 119901(119910 119911) minus 119901(119910 119910)

The pair (119883 119901) is then called a partial metric spaceIf (119883 119901) is a partial metric space then the function 119901

119904

119883times119883 rarr R+ given by 119901119904(119909 119910) = 2119901(119909 119910)minus119901(119909 119909)minus119901(119910 119910)

119909 119910 isin 119883 is a metric on119883A basic example of a partial metric space is the pair

(119877+ 119901) where 119901(119909 119910) = max119909 119910 for all 119909 119910 isin 119877

+

Lemma 2 (see [1]) Let (119883 119901) be a partial metric space thenone has the following

(1) A sequence 119909119899 in a partial metric space (119883 119901)

converges to a point 119909 isin 119883 if and only iflim119899rarrinfin

119901(119909 119909119899) = 119901(119909 119909)

(2) A sequence 119909119899 in a partialmetric space (119883 119901) is called

a Cauchy sequence if the lim119899119898rarrinfin

119901(119909119899 119909119898) exists

and is finite

(3) A partial metric space (119883 119901) is said to be complete ifevery Cauchy sequence 119909

119899 in 119883 converges to a point

119909 isin 119883 that is 119901(119909 119909) = lim119899119898rarrinfin

119901(119909119899 119909119898)

(4) A partial metric space (119883 119901) is complete if and onlyif the metric space (119883 119901

119904) is complete Furthermore

lim119899rarrinfin

119901119904(119909119899 119911) = 0 if and only if 119901(119911 119911) =

lim119899rarrinfin

119901(119909119899 119911) = lim

119899119898rarrinfin119901(119909119899 119909119898)

2 Abstract and Applied Analysis

Remark 3 (see [1]) Let (119883 119901) be a partial metric space andlet 119860 be a nonempty set in (119883 119901) then 119886 isin 119860 if and only if

119901 (119886 119860) = 119901 (119886 119886) (1)

where 119860 denotes the closure of 119860 with respect to the partialmetric 119901 Note 119860 is closed in (119883 119901) if and only if 119860 = 119860

Definition 4 (see [24]) Two families of self-mappings 119879119894119898

1

and 119878119894119899

1are said to be pairwise commuting if

(1) 119879119894119879119895= 119879119895119879119894 119894 119895 isin 1 2 119898

(2) 119878119896119878119897= 119878119897119878119896 119896 119897 isin 1 2 119899

(3) 119879119894119878119896= 119878119896119879119894 119894 isin 1 2 119898 119896 isin 1 2 119899

Now we recall the following definitions and results from[9]

Let 119862119861119901(119883) be the collection of all nonempty closed and

bounded subsets of119883with respect to the partial metric119901 For119860 isin 119862119861

119901(119883) we define

119901 (119886 119860) = inf 119901 (119886 119909) 119909 isin 119860 (2)

For 119860 119861 isin 119862119861119901(119883)

120575119901(119860 119861) = sup 119901 (119886 119861) 119886 isin 119860

120575119901(119861 119860) = sup 119901 (119887 119860) 119887 isin 119861

(3)

For 119860 119861 isin 119862119861119901(119883)

119867119901(119860 119861) = max 120575

119901(119860 119861) 120575

119901(119861 119860) (4)

Proposition 5 (see [9]) Let (119883 119901) be a partial metric spaceFor any 119860 119861 119862 isin 119862119861

119901(119883) one has

(i) 120575119901(119860 119860) = sup119901(119886 119886) 119886 isin 119860

(ii) 120575119901(119860 119860) le 120575

119901(119860 119861)

(iii) 120575119901(119860 119861) = 0 implies that 119860 sube 119861

(iv) 120575119901(119860 119861) le 120575

119901(119860 119862) + 120575

119901(119862 119861) minus inf

119888isin119862119901(119888 119888)

Proposition 6 (see [9]) Let (119883 119901) be a partial metric spaceFor any 119860 119861 119862 isin 119862119861

119901(119883) one has

(h1) 119867119901(119860 119860) le 119867

119901(119860 119861)

(h2) 119867119901(119860 119861) = 119867

119901(119861 119860)

(h3) 119867119901(119860 119861) le 119867

119901(119860 119862) + 119867

119901(119862 119861) minus inf

119888isin119862119901(119888 119888)

Lemma 7 (see [9]) Let 119860 and 119861 be nonempty closed andbounded subsets of a partial metric space (119883 119901) and ℎ gt 1Then for every 119886 isin 119860 there exists 119887 isin 119861 such that 119901(119886 119887) le

ℎ119867119901(119860 119861)

Lemma 8 (see [10]) Let 119860 and 119861 be nonempty closed andbounded subsets of a partialmetric space (119883 119901) and 0 lt ℎ isin RThen for every 119886 isin 119860 there exists 119887 isin 119861 such that 119901(119886 119887) le

119867119901(119860 119861) + ℎ

2 Results for Single Valued Mappings

The following result regarding the existence of the commonfixed point of themappings satisfying a contractive conditionon the closed ball is very useful in the sense that it requiresthe contractiveness of the mappings only on the closed ballinstead of the whole space

Theorem 9 Let 119878 119879 119883 rarr 119883 be mappings on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(5)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198781199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (6)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861119901(1199090 119903) such that 119906 = 119878119906 = 119879119906 Also 119901(119906 119906) = 0

Proof Let 1199090be an arbitrary point in119883 and define

1199092119896+1

= 1198781199092119896 1199092119896+2

= 1198791199092119896+1

where 119896 = 0 1 2

(7)

We will prove that 119909119899

isin (119861(1199090 119903)) for all 119899 isin N by

mathematical induction Using inequality (6) and the factthat 120582 = (120572 + 120574)(1 minus 120573 minus 120574) lt 1 we have

119901 (1199090 1198781199090) le 119903 + 119901 (119909

0 1199090) (8)

It implies that 1199091

isin (119861(1199090 119903)) Let 119909

2 119909

119895isin 119861(119909

0 119903) for

some 119895 isin 119873 If 119895 = 2119896 + 1 where 119896 = 0 1 2 (119895 minus 1)2 sousing inequality (5) we obtain

119901 (1199092119896+1

1199092119896+2

)

= 119901 (1198781199092119896 1198791199092119896+1

)

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1198781199092119896) 119901 (119909

2119896+1 1198791199092119896+1

)

+120574119901 (1199092119896+1

1198781199092119896) 119901 (119909

2119896 1198791199092119896+1

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1199092119896+1

) 119901 (1199092119896+1

1199092119896+2

)

+120574119901 (1199092119896+1

1199092119896+1

) 119901 (1199092119896 1199092119896+2

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

Abstract and Applied Analysis 3

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1199092119896+1

) 119901 (1199092119896+1

1199092119896+2

)

+120574119901 (1199092119896 1199092119896+1

) 119901 (1199092119896 1199092119896+2

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

le 120572119901 (1199092119896 1199092119896+1

) + 120573119901 (1199092119896+1

1199092119896+2

) + 120574119901 (1199092119896 1199092119896+2

)

(9)

as 1 + 119901(1199092119896 1199092119896+1

) gt 119901(1199092119896 1199092119896+1

) and so

119901 (1199092119896+1

1199092119896+2

) le (120572 + 120574

1 minus 120573 minus 120574)119901 (119909

2119896 1199092119896+1

) (10)

which implies that

119901 (1199092119896+1

1199092119896+2

) le 120582119901 (1199092119896 1199092119896+1

) le sdot sdot sdot le 1205822119896+1

119901 (1199090 1199091)

(11)

If 119895 = 2119896 + 2 where 119896 = 0 1 2 (119895 minus 2)2 one can easilyprove that

119901 (1199092119896+2

1199092119896+3

) le 1205822119896+2

119901 (1199090 1199091) (12)

Thus from inequality (11) and (12) we have

119901 (119909119895 119909119895+1

) le 120582119895119901 (1199090 1199091) for some 119895 isin 119873 (13)

Now

119901 (1199090 119909119895+1

) le 119901 (1199090 1199091) + sdot sdot sdot + 119901 (119909

119895 119909119895+1

)

minus [119901 (1199091 1199091) + sdot sdot sdot + 119901 (119909

119895 119909119895)]

le 119901 (1199090 1199091) + sdot sdot sdot + 120582

119895119901 (1199090 1199091)

= 119901 (1199090 1199091) [1 + sdot sdot sdot + 120582

119895minus1+ 120582119895]

le (1 minus 120582) [119903 + 119901 (1199090 1199090)]

(1 minus 120582119895+1

)

1 minus 120582

le 119903 + 119901 (1199090 1199090)

(14)

gives 119909119895+1

isin 119861(1199090 119903) Hence 119909

119899isin 119861(119909

0 119903) for all 119899 isin N One

can easily prove that

119901 (119909119899 119909119899+1

) le 120582119899119901 (1199090 1199091) (13

lowast)

for all 119899 isin N We now show that 119909119899 is a Cauchy sequence

Without loss of generality assume that 119898 gt 119899 Then using(13lowast) and the triangle inequality for partial metrics (P

4) we

have119901 (119909119899 119909119898) le 119901 (119909

119899 119909119899+1

) + 119901 (119909119899+1

119909119898) minus 119901 (119909

119899+1 119909119899+1

)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + 119901 (119909119899+2

119909119898)

minus 119901 (119909119899+2

119909119899+2

)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + 119901 (119909119899+2

119909119898)

(15)

Inductively we have

0 le 119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 120582119899119901 (1199090 1199091) + 120582119899+1

119901 (1199090 1199091) + sdot sdot sdot + 120582

119898minus1119901 (1199090 1199091)

le (120582119899+ 120582119899+1

+ sdot sdot sdot + 120582119898minus1

) 119901 (1199090 1199091)

le (120582119899

1 minus 120582)119901 (119909

0 1199091)

997888rarr 0 as 119899 997888rarr infin (since 0 lt 120582 lt 1)

(16)

Thus

lim119899rarr+infin

119901 (119909119899 119909119898) = 0 (17)

By the definition of 119901119904 we get for any119898 isin Nlowast

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (18)

Hence the sequence 119909119899 is a Cauchy sequence in

(119861(1199090 119903) 119901119904) By Lemma 2(4) 119909

119899 is a Cauchy sequence in

(119861(1199090 119903) 119901) Therefore there exists a point 119906 isin 119861(119909

0 119903) with

lim119899rarrinfin

119909119899

= 119906 Also lim119899rarrinfin

119901119904(119909119899 119906) = 0 Again from

Lemma 2(4) we have

119901 (119906 119906) = lim119899rarr+infin

119901 (119909119899 119906) = lim

119899rarr+infin119901 (119909119899 119909119898) = 0

(19)

By the triangle inequality (P4) we have

119901 (119906 119878119906) le 119901 (119906 1199092119899+2

) + 119901 (1199092119899+2

119878119906) minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119906 1199092119899+2

) + 119901 (1198791199092119899+1

119878119906)

le 119901 (119906 1199092119899+2

) + 120572119901 (1199092119899+1

119906)

+ (120573119901 (1199092119899+1

1198791199092119899+1

) 119901 (119906 119878119906)

+120574119901 (119906 1198791199092119899+1

) 119901 (1199092119899+1

119878119906))

times (1 + 119901 (1199092119899+1

119906))minus1

le 119901 (119906 1199092119899+2

) + 120572119901 (1199092119899+1

119906)

+ (120573119901 (1199092119899+1

1199092119899+2

) 119901 (119906 119878119906)

+120574119901 (119906 1199092119899+2

) 119901 (1199092119899+1

119878119906))

times (1 + 119901 (1199092119899+1

119906))minus1

(20)

Letting 119899 rarr +infin and using (19) we obtain

119901 (119906 119878119906) = 0 (21)

By (P1) we concluded that 119906 = 119878119906 It follows similarly that

119906 = 119879119906 To prove the uniqueness of common fixed point let

4 Abstract and Applied Analysis

119906lowastisin 119861(119909

0 119903) be another common fixed point of 119878 and119879 that

is 119906lowast = 119878119906lowast= 119879119906lowast Then

119901 (119906 119906lowast) = 119901 (119878119906 119879119906

lowast)

le 120572119901 (119906 119906lowast)

+120573119901 (119906 119878119906) 119901 (119906

lowast 119879119906lowast)+120574119901 (119906

lowast 119878119906) 119901 (119906 119879119906

lowast)

1+119901 (119906 119906lowast)

= 120572119901 (119906 119906lowast) +

120574119901 (119906lowast 119906) 119901 (119906 119906

lowast)

1 + 119901 (119906 119906lowast)

(22)

so that119901(119906 119906lowast) le 120572119901(119906 119906lowast)+120574119901(119906 119906

lowast) because 1+119901(119906 119906

lowast) gt

119901(119906 119906lowast) Therefore 119901(119906 119906

lowast) le (120572 + 120574)119901(119906 119906

lowast) which is a

contradiction so that 119906 = 119906lowast (as 120572 + 120574 lt 1) Hence 119878 and

119879 have a unique common fixed point in 119861(1199090 119903)

Example 10 Let119883 = [0 +infin) endowed with the usual partialmetric 119901 defined by 119901 119883 times 119883 rarr R+ with 119901(119909 119910) =

max119909 119910 Clearly (119883 119901) is a partial metric space Now wedefine 119878 119879 119883 rarr 119883 as

119878 (119909) =

119909

16if 0 le 119909 le 1

119909 minus1

6if 119909 gt 1

119879 (119909) =

5119909

17if 0 le 119909 le 1

119909 minus1

7if 119909 gt 1

(23)

for all 119909 isin 119883 Taking 120572 = 15 120573 = 16 120574 = 18 1199090= 12

and 119903 = 12 then 119861119901(1199090 119903) = [0 1] Also we have 119901(119909

0 1199090) =

max12 12 = 12 120582 = (120572 + 120574)(1 minus 120573 minus 120574) = 3985 with

(1 minus 120582) (119903 + 119901 (1199090 1199090)) =

46

85

119901 (1199090 1198781199090) = 119901 (

1

21

32) =

1

2lt (1 minus 120582) (119903 + 119901 (119909

0 1199090))

(24)

Also if 119909 119910 isin (1 +infin) then

119901 (119878119909 119879119910) = max 119909 minus1

6 119909 minus

1

7 ge

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

= 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(25)

So the contractive condition does not hold on whole of 119883Now if 119909 119910 isin 119861

119901(1199090 119903) then

119901 (119878119909 119879119910) = max 119909

165119910

17 le

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

(26)

Therefore all the conditions of Theorem 9 are satisfied Thus0 is the common fixed point of 119878 and 119879 and 119901(0 0) = 0Moreover note that for any metric 119889 on119883

119889 (1198781 1198791) = 119889 (1

165

17) gt

1

5119889 (1 1)

+ (1

6119889 (1

1

16) 119889 (1

5

17)

+1

8119889 (1

1

16) 119889 (1

5

17))

times (1 + 119889 (1 1))minus1

(27)

Therefore commonfixed points of 119878 and119879 cannot be obtainedfrom a metric fixed point theorem

Corollary 11 Let 119878 119879 119883 rarr 119883 be mappings on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(28)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119878119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119878 and 119879 have nofixed point other than 119906

By choosing 120573 = 120574 = 0 in Corollary 11 we get thefollowing corollary

Corollary 12 Let 119878 119879 119883 rarr 119883 be a mappings on completePMS (119883 119901) If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910) (29)

for all 119909 119910 isin 119883 120572 lt 1 (120572 is a nonnegative real) Then 119878 and 119879

have a common fixed point 119906 isin 119883 and 119901(119906 119906) = 0

Remark 13 If we impose Banach type contractive conditionfor a pair 119878 119879 119883 rarr 119883 of mappings on a metric space(119883 119889) that is 119889(119878119909 119879119910) le 120572119889(119909 119910) for all 119909 119910 isin 119883 andthen it follows that 119878119909 = 119879119909 for all 119909 isin 119883 (ie 119878 and 119879 are

Abstract and Applied Analysis 5

equal) Therefore the above condition fails to find commonfixed points of 119878 and 119879 This can be seen as

119889 (1198781 1198791) = 119889 ((1

16) (

5

17)) gt 120572119889 (1 1) = 0 (30)

However the same condition in partial metric space does notassert that 119878 = 119879 This can be seen as by taking the partialmetric same as in Example 10

119901 (1198781 1198791) = 119901 ((1

16) (

5

17)) =

5

17le 120572119901 (1 1) = 120572 (1)

(31)

for any 120572 ge 13 Hence Corollary 12 cannot be obtained froma metric fixed point theorem

Remark 14 By equating 120572 120573 120574 to 0 in all possible combi-nations one can derive a host of corollaries which includeMatthews theorem for mappings defined on a completepartial metric space

By taking 119878 = 119879 in the Theorem 9 we get the followingcorollary

Corollary 15 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(32)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198791199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (33)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861

119901(1199090 119903) such that 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879

has no fixed point other than 119906

By taking 119878 = 119879 in Corollary 11 we get the followingcorollary

Corollary 16 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(34)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879 has no fixed pointother than 119906

Now we give an example in favour of Corollary 16

Example 17 Let 119883 = [0 4] endowed with the usual partialmetric 119901 defined by 119901(119909 119910) = max119909 119910 Clearly (119883 119901) is acomplete partial metric space Now we define 119865 119883 rarr 119883 asfollows

119865 (119909) =

119909

3if 0 le 119909 lt 2

119909

1 + 119909if 2 le 119909 le 4

(35)

for all 119909 isin 119883 Now let 119910 le 119909 If 119909 isin [0 2) (and so 119910 isin [0 2))Then 119901(119865119909 119865119910) = 1199093 119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) =

119910 119901(119910 119865119909) = 1199093 119901(119909 119865119910) = 119909 Taking 120572 = 13 120573 = 115120574 = 215 we can prove that all the conditions of Corollary 16are satisfied Now if 119909 isin [2 4] then 119901(119865119909 119865119910) = 119909(1 + 119909)119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) = 119910 119901(119910 119865119909) = 119909(1+119909)119901(119909 119865119910) = 119909 and taking 120572 = 13 120573 = 115 120574 = 215 onecan verify the condition of the above corollary Thus all theconditions of Corollary 16 are satisfied and 119906 = 0 is a fixedpoint of the mapping 119865

As an application of Theorem 9 we prove the followingtheorem for two finite families of mappings

Theorem 18 If 119879119894119898

1and 119878

119894119899

1are two pairwise commuting

finite families of self-mapping defined on a complete partialmetric space (119883 119901) such that the mappings 119878 and 119879 (with119879 = 119879

11198792sdot sdot sdot 119879119898and 119878 = 119878

11198782sdot sdot sdot 119878119899) satisfy the contractive

condition (5) then the component maps of the two families119879119894119898

1and 119878

119894119899

1have a unique common fixed point

Proof FromTheorem 9 we can say that the mappings 119879 and119878 have a unique common fixed point 119911 that is 119879119911 = 119878119911 = 119911Now our requirement is to show that 119911 is a common fixedpoint of all the component mappings of both the families Inviewof pairwise commutativity of the families 119879

119894119898

1and 119878

119894119899

1

(for every 1 le 119896 le 119898) we can write 119879119896119911 = 119879

119896119879119911 = 119879119879

119896119911 and

119879119896119911 = 119879119896119878119911 = 119878119879

119896119911 which show that 119879

119896119911 (for every 119896) is also

a common fixed point of 119879 and 119878 By using the uniqueness ofcommonfixed point we canwrite119879

119896119911 = 119911 (for every 119896) which

shows that 119911 is a common fixed point of the family 119879119894119898

1

Using the same argument one can also show that (for every1 le 119896 le 119899) 119878

119896119911 = 119911Thus componentmaps of the two families

119879119894119898

1and 119878

119894119899

1have a unique common fixed point

By setting 1198791= 1198792= sdot sdot sdot = 119879

119898= 119865 and 119878

1= 1198782= sdot sdot sdot =

119878119899= 119866 in Theorem 18 we get the following corollary

Corollary 19 Let 119865 119866 119883 rarr 119883 be two commuting self-mappings defined on a complete PMS (119883 119901) satisfying thecondition

119901 (119865119898119909 119866119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119865

119898119909) 119901 (119910 119866

119899119910) + 120574119901 (119910 119865

119898119909) 119901 (119909 119866

119899119910)

1 + 119901 (119909 119910)

(36)

6 Abstract and Applied Analysis

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F and G have a unique common fixed point

By setting 119898 = 119899 and 119865 = 119866 = 119879 in Corollary 19 wededuce the following corollary

Corollary 20 Let 119879 119883 rarr 119883 be a mapping defined on acomplete PMS (119883 119901) satisfying the condition

119901 (119879119899119909 119879119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119879

119899119909) 119901 (119910 119879

119899119910) + 120574119901 (119910 119879

119899119909) 119901 (119909 119879

119899119910)

1 + 119901 (119909 119910)

(37)

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F has a unique fixed point

By setting 120573 = 120574 = 0 we draw following corollary whichcan be viewed as an extension of Bryantrsquos theorem [15] for amapping on a complete PMS (119883 119901)

Corollary 21 Let 119865 119883 rarr 119883 be a mapping on a completePMS (119883 119901) If 119865 satisfies

119901 (119865119899119909 119865119899119910) le 120572119901 (119909 119910) (38)

for all 119909 119910 isin 119883 120572 lt 1 Then 119865 has a unique fixed point

The following example demonstrates the superiority ofBryantrsquos theorem overMatthews theorem on complete partialmetric space

Example 22 Let119883 = [0 4] Define the partial metric 119901 119883 times

119883 rarr R by

119901 (119909 119910) = max 119909 119910 (39)

Then (119883 119901) is a complete partial metric space Let 119865 119883 rarr

119883 be defined as follows

119865 (119909) =

1199092 if 119909 isin [0 1 [

2 if 119909 isin [1 2 [

0 if 119909 isin [2 4]

(40)

Then for 119909 = 0 and 119910 = 1 we get

119901 (119865 (0) 119865 (1)) = 119901 (0 2) = 2 gt 120572119901 (0 1) = 120572 (1) (41)

because 0 le 120572 lt 1 However 1198652 satisfies the requirement ofBryantrsquos theorem and 119911 = 0 is the unique fixed point of 119865

3 Results for Set Valued Mappings

Theorem23 Let (119883 119901) be a complete partial metric space andlet 119878 119879 119883 rarr 119862119861

119901(119883) be mappings such that

119867119901(119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(42)

for all 119909 119910 isin 119883 0 le 120572 120573 120574 with 120572 + 120573 + 2120574 lt 1 Then 119878 and 119879

have a common fixed point

Proof Assume that119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) Let 1199090isin 119883 be

arbitrary but fixed element of 119883 and choose 1199091

isin 119878(1199090) By

Lemma 8 we can choose 1199092isin 119879(119909

1) such that

119901 (1199091 1199092) le 119867

119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1198781199090) 119901 (119909

1 1198791199091)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1198781199090) 119901 (119909

0 1198791199091)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199090 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573119901 (119909

1 1199092) + 120574119901 (119909

0 1199092)

+ (120572 + 120574)

(43)

So we get

119901 (1199091 1199092) le (

120572 + 120574

1 minus 120573 minus 120574)119901 (119909

0 1199091) + (

120572 + 120574

1 minus 120573 minus 120574) (44)

Since119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) so it further implies that119901 (1199091 1199092) le 119872119901 (119909

0 1199091) + 119872 (45)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093) le 119867

119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120574)2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1198791199091) 119901 (119909

2 1198781199092)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1198791199091) 119901 (119909

1 1198781199092)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Remark 3 (see [1]) Let (119883 119901) be a partial metric space andlet 119860 be a nonempty set in (119883 119901) then 119886 isin 119860 if and only if

119901 (119886 119860) = 119901 (119886 119886) (1)

where 119860 denotes the closure of 119860 with respect to the partialmetric 119901 Note 119860 is closed in (119883 119901) if and only if 119860 = 119860

Definition 4 (see [24]) Two families of self-mappings 119879119894119898

1

and 119878119894119899

1are said to be pairwise commuting if

(1) 119879119894119879119895= 119879119895119879119894 119894 119895 isin 1 2 119898

(2) 119878119896119878119897= 119878119897119878119896 119896 119897 isin 1 2 119899

(3) 119879119894119878119896= 119878119896119879119894 119894 isin 1 2 119898 119896 isin 1 2 119899

Now we recall the following definitions and results from[9]

Let 119862119861119901(119883) be the collection of all nonempty closed and

bounded subsets of119883with respect to the partial metric119901 For119860 isin 119862119861

119901(119883) we define

119901 (119886 119860) = inf 119901 (119886 119909) 119909 isin 119860 (2)

For 119860 119861 isin 119862119861119901(119883)

120575119901(119860 119861) = sup 119901 (119886 119861) 119886 isin 119860

120575119901(119861 119860) = sup 119901 (119887 119860) 119887 isin 119861

(3)

For 119860 119861 isin 119862119861119901(119883)

119867119901(119860 119861) = max 120575

119901(119860 119861) 120575

119901(119861 119860) (4)

Proposition 5 (see [9]) Let (119883 119901) be a partial metric spaceFor any 119860 119861 119862 isin 119862119861

119901(119883) one has

(i) 120575119901(119860 119860) = sup119901(119886 119886) 119886 isin 119860

(ii) 120575119901(119860 119860) le 120575

119901(119860 119861)

(iii) 120575119901(119860 119861) = 0 implies that 119860 sube 119861

(iv) 120575119901(119860 119861) le 120575

119901(119860 119862) + 120575

119901(119862 119861) minus inf

119888isin119862119901(119888 119888)

Proposition 6 (see [9]) Let (119883 119901) be a partial metric spaceFor any 119860 119861 119862 isin 119862119861

119901(119883) one has

(h1) 119867119901(119860 119860) le 119867

119901(119860 119861)

(h2) 119867119901(119860 119861) = 119867

119901(119861 119860)

(h3) 119867119901(119860 119861) le 119867

119901(119860 119862) + 119867

119901(119862 119861) minus inf

119888isin119862119901(119888 119888)

Lemma 7 (see [9]) Let 119860 and 119861 be nonempty closed andbounded subsets of a partial metric space (119883 119901) and ℎ gt 1Then for every 119886 isin 119860 there exists 119887 isin 119861 such that 119901(119886 119887) le

ℎ119867119901(119860 119861)

Lemma 8 (see [10]) Let 119860 and 119861 be nonempty closed andbounded subsets of a partialmetric space (119883 119901) and 0 lt ℎ isin RThen for every 119886 isin 119860 there exists 119887 isin 119861 such that 119901(119886 119887) le

119867119901(119860 119861) + ℎ

2 Results for Single Valued Mappings

The following result regarding the existence of the commonfixed point of themappings satisfying a contractive conditionon the closed ball is very useful in the sense that it requiresthe contractiveness of the mappings only on the closed ballinstead of the whole space

Theorem 9 Let 119878 119879 119883 rarr 119883 be mappings on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(5)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198781199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (6)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861119901(1199090 119903) such that 119906 = 119878119906 = 119879119906 Also 119901(119906 119906) = 0

Proof Let 1199090be an arbitrary point in119883 and define

1199092119896+1

= 1198781199092119896 1199092119896+2

= 1198791199092119896+1

where 119896 = 0 1 2

(7)

We will prove that 119909119899

isin (119861(1199090 119903)) for all 119899 isin N by

mathematical induction Using inequality (6) and the factthat 120582 = (120572 + 120574)(1 minus 120573 minus 120574) lt 1 we have

119901 (1199090 1198781199090) le 119903 + 119901 (119909

0 1199090) (8)

It implies that 1199091

isin (119861(1199090 119903)) Let 119909

2 119909

119895isin 119861(119909

0 119903) for

some 119895 isin 119873 If 119895 = 2119896 + 1 where 119896 = 0 1 2 (119895 minus 1)2 sousing inequality (5) we obtain

119901 (1199092119896+1

1199092119896+2

)

= 119901 (1198781199092119896 1198791199092119896+1

)

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1198781199092119896) 119901 (119909

2119896+1 1198791199092119896+1

)

+120574119901 (1199092119896+1

1198781199092119896) 119901 (119909

2119896 1198791199092119896+1

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1199092119896+1

) 119901 (1199092119896+1

1199092119896+2

)

+120574119901 (1199092119896+1

1199092119896+1

) 119901 (1199092119896 1199092119896+2

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

Abstract and Applied Analysis 3

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1199092119896+1

) 119901 (1199092119896+1

1199092119896+2

)

+120574119901 (1199092119896 1199092119896+1

) 119901 (1199092119896 1199092119896+2

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

le 120572119901 (1199092119896 1199092119896+1

) + 120573119901 (1199092119896+1

1199092119896+2

) + 120574119901 (1199092119896 1199092119896+2

)

(9)

as 1 + 119901(1199092119896 1199092119896+1

) gt 119901(1199092119896 1199092119896+1

) and so

119901 (1199092119896+1

1199092119896+2

) le (120572 + 120574

1 minus 120573 minus 120574)119901 (119909

2119896 1199092119896+1

) (10)

which implies that

119901 (1199092119896+1

1199092119896+2

) le 120582119901 (1199092119896 1199092119896+1

) le sdot sdot sdot le 1205822119896+1

119901 (1199090 1199091)

(11)

If 119895 = 2119896 + 2 where 119896 = 0 1 2 (119895 minus 2)2 one can easilyprove that

119901 (1199092119896+2

1199092119896+3

) le 1205822119896+2

119901 (1199090 1199091) (12)

Thus from inequality (11) and (12) we have

119901 (119909119895 119909119895+1

) le 120582119895119901 (1199090 1199091) for some 119895 isin 119873 (13)

Now

119901 (1199090 119909119895+1

) le 119901 (1199090 1199091) + sdot sdot sdot + 119901 (119909

119895 119909119895+1

)

minus [119901 (1199091 1199091) + sdot sdot sdot + 119901 (119909

119895 119909119895)]

le 119901 (1199090 1199091) + sdot sdot sdot + 120582

119895119901 (1199090 1199091)

= 119901 (1199090 1199091) [1 + sdot sdot sdot + 120582

119895minus1+ 120582119895]

le (1 minus 120582) [119903 + 119901 (1199090 1199090)]

(1 minus 120582119895+1

)

1 minus 120582

le 119903 + 119901 (1199090 1199090)

(14)

gives 119909119895+1

isin 119861(1199090 119903) Hence 119909

119899isin 119861(119909

0 119903) for all 119899 isin N One

can easily prove that

119901 (119909119899 119909119899+1

) le 120582119899119901 (1199090 1199091) (13

lowast)

for all 119899 isin N We now show that 119909119899 is a Cauchy sequence

Without loss of generality assume that 119898 gt 119899 Then using(13lowast) and the triangle inequality for partial metrics (P

4) we

have119901 (119909119899 119909119898) le 119901 (119909

119899 119909119899+1

) + 119901 (119909119899+1

119909119898) minus 119901 (119909

119899+1 119909119899+1

)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + 119901 (119909119899+2

119909119898)

minus 119901 (119909119899+2

119909119899+2

)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + 119901 (119909119899+2

119909119898)

(15)

Inductively we have

0 le 119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 120582119899119901 (1199090 1199091) + 120582119899+1

119901 (1199090 1199091) + sdot sdot sdot + 120582

119898minus1119901 (1199090 1199091)

le (120582119899+ 120582119899+1

+ sdot sdot sdot + 120582119898minus1

) 119901 (1199090 1199091)

le (120582119899

1 minus 120582)119901 (119909

0 1199091)

997888rarr 0 as 119899 997888rarr infin (since 0 lt 120582 lt 1)

(16)

Thus

lim119899rarr+infin

119901 (119909119899 119909119898) = 0 (17)

By the definition of 119901119904 we get for any119898 isin Nlowast

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (18)

Hence the sequence 119909119899 is a Cauchy sequence in

(119861(1199090 119903) 119901119904) By Lemma 2(4) 119909

119899 is a Cauchy sequence in

(119861(1199090 119903) 119901) Therefore there exists a point 119906 isin 119861(119909

0 119903) with

lim119899rarrinfin

119909119899

= 119906 Also lim119899rarrinfin

119901119904(119909119899 119906) = 0 Again from

Lemma 2(4) we have

119901 (119906 119906) = lim119899rarr+infin

119901 (119909119899 119906) = lim

119899rarr+infin119901 (119909119899 119909119898) = 0

(19)

By the triangle inequality (P4) we have

119901 (119906 119878119906) le 119901 (119906 1199092119899+2

) + 119901 (1199092119899+2

119878119906) minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119906 1199092119899+2

) + 119901 (1198791199092119899+1

119878119906)

le 119901 (119906 1199092119899+2

) + 120572119901 (1199092119899+1

119906)

+ (120573119901 (1199092119899+1

1198791199092119899+1

) 119901 (119906 119878119906)

+120574119901 (119906 1198791199092119899+1

) 119901 (1199092119899+1

119878119906))

times (1 + 119901 (1199092119899+1

119906))minus1

le 119901 (119906 1199092119899+2

) + 120572119901 (1199092119899+1

119906)

+ (120573119901 (1199092119899+1

1199092119899+2

) 119901 (119906 119878119906)

+120574119901 (119906 1199092119899+2

) 119901 (1199092119899+1

119878119906))

times (1 + 119901 (1199092119899+1

119906))minus1

(20)

Letting 119899 rarr +infin and using (19) we obtain

119901 (119906 119878119906) = 0 (21)

By (P1) we concluded that 119906 = 119878119906 It follows similarly that

119906 = 119879119906 To prove the uniqueness of common fixed point let

4 Abstract and Applied Analysis

119906lowastisin 119861(119909

0 119903) be another common fixed point of 119878 and119879 that

is 119906lowast = 119878119906lowast= 119879119906lowast Then

119901 (119906 119906lowast) = 119901 (119878119906 119879119906

lowast)

le 120572119901 (119906 119906lowast)

+120573119901 (119906 119878119906) 119901 (119906

lowast 119879119906lowast)+120574119901 (119906

lowast 119878119906) 119901 (119906 119879119906

lowast)

1+119901 (119906 119906lowast)

= 120572119901 (119906 119906lowast) +

120574119901 (119906lowast 119906) 119901 (119906 119906

lowast)

1 + 119901 (119906 119906lowast)

(22)

so that119901(119906 119906lowast) le 120572119901(119906 119906lowast)+120574119901(119906 119906

lowast) because 1+119901(119906 119906

lowast) gt

119901(119906 119906lowast) Therefore 119901(119906 119906

lowast) le (120572 + 120574)119901(119906 119906

lowast) which is a

contradiction so that 119906 = 119906lowast (as 120572 + 120574 lt 1) Hence 119878 and

119879 have a unique common fixed point in 119861(1199090 119903)

Example 10 Let119883 = [0 +infin) endowed with the usual partialmetric 119901 defined by 119901 119883 times 119883 rarr R+ with 119901(119909 119910) =

max119909 119910 Clearly (119883 119901) is a partial metric space Now wedefine 119878 119879 119883 rarr 119883 as

119878 (119909) =

119909

16if 0 le 119909 le 1

119909 minus1

6if 119909 gt 1

119879 (119909) =

5119909

17if 0 le 119909 le 1

119909 minus1

7if 119909 gt 1

(23)

for all 119909 isin 119883 Taking 120572 = 15 120573 = 16 120574 = 18 1199090= 12

and 119903 = 12 then 119861119901(1199090 119903) = [0 1] Also we have 119901(119909

0 1199090) =

max12 12 = 12 120582 = (120572 + 120574)(1 minus 120573 minus 120574) = 3985 with

(1 minus 120582) (119903 + 119901 (1199090 1199090)) =

46

85

119901 (1199090 1198781199090) = 119901 (

1

21

32) =

1

2lt (1 minus 120582) (119903 + 119901 (119909

0 1199090))

(24)

Also if 119909 119910 isin (1 +infin) then

119901 (119878119909 119879119910) = max 119909 minus1

6 119909 minus

1

7 ge

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

= 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(25)

So the contractive condition does not hold on whole of 119883Now if 119909 119910 isin 119861

119901(1199090 119903) then

119901 (119878119909 119879119910) = max 119909

165119910

17 le

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

(26)

Therefore all the conditions of Theorem 9 are satisfied Thus0 is the common fixed point of 119878 and 119879 and 119901(0 0) = 0Moreover note that for any metric 119889 on119883

119889 (1198781 1198791) = 119889 (1

165

17) gt

1

5119889 (1 1)

+ (1

6119889 (1

1

16) 119889 (1

5

17)

+1

8119889 (1

1

16) 119889 (1

5

17))

times (1 + 119889 (1 1))minus1

(27)

Therefore commonfixed points of 119878 and119879 cannot be obtainedfrom a metric fixed point theorem

Corollary 11 Let 119878 119879 119883 rarr 119883 be mappings on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(28)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119878119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119878 and 119879 have nofixed point other than 119906

By choosing 120573 = 120574 = 0 in Corollary 11 we get thefollowing corollary

Corollary 12 Let 119878 119879 119883 rarr 119883 be a mappings on completePMS (119883 119901) If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910) (29)

for all 119909 119910 isin 119883 120572 lt 1 (120572 is a nonnegative real) Then 119878 and 119879

have a common fixed point 119906 isin 119883 and 119901(119906 119906) = 0

Remark 13 If we impose Banach type contractive conditionfor a pair 119878 119879 119883 rarr 119883 of mappings on a metric space(119883 119889) that is 119889(119878119909 119879119910) le 120572119889(119909 119910) for all 119909 119910 isin 119883 andthen it follows that 119878119909 = 119879119909 for all 119909 isin 119883 (ie 119878 and 119879 are

Abstract and Applied Analysis 5

equal) Therefore the above condition fails to find commonfixed points of 119878 and 119879 This can be seen as

119889 (1198781 1198791) = 119889 ((1

16) (

5

17)) gt 120572119889 (1 1) = 0 (30)

However the same condition in partial metric space does notassert that 119878 = 119879 This can be seen as by taking the partialmetric same as in Example 10

119901 (1198781 1198791) = 119901 ((1

16) (

5

17)) =

5

17le 120572119901 (1 1) = 120572 (1)

(31)

for any 120572 ge 13 Hence Corollary 12 cannot be obtained froma metric fixed point theorem

Remark 14 By equating 120572 120573 120574 to 0 in all possible combi-nations one can derive a host of corollaries which includeMatthews theorem for mappings defined on a completepartial metric space

By taking 119878 = 119879 in the Theorem 9 we get the followingcorollary

Corollary 15 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(32)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198791199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (33)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861

119901(1199090 119903) such that 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879

has no fixed point other than 119906

By taking 119878 = 119879 in Corollary 11 we get the followingcorollary

Corollary 16 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(34)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879 has no fixed pointother than 119906

Now we give an example in favour of Corollary 16

Example 17 Let 119883 = [0 4] endowed with the usual partialmetric 119901 defined by 119901(119909 119910) = max119909 119910 Clearly (119883 119901) is acomplete partial metric space Now we define 119865 119883 rarr 119883 asfollows

119865 (119909) =

119909

3if 0 le 119909 lt 2

119909

1 + 119909if 2 le 119909 le 4

(35)

for all 119909 isin 119883 Now let 119910 le 119909 If 119909 isin [0 2) (and so 119910 isin [0 2))Then 119901(119865119909 119865119910) = 1199093 119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) =

119910 119901(119910 119865119909) = 1199093 119901(119909 119865119910) = 119909 Taking 120572 = 13 120573 = 115120574 = 215 we can prove that all the conditions of Corollary 16are satisfied Now if 119909 isin [2 4] then 119901(119865119909 119865119910) = 119909(1 + 119909)119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) = 119910 119901(119910 119865119909) = 119909(1+119909)119901(119909 119865119910) = 119909 and taking 120572 = 13 120573 = 115 120574 = 215 onecan verify the condition of the above corollary Thus all theconditions of Corollary 16 are satisfied and 119906 = 0 is a fixedpoint of the mapping 119865

As an application of Theorem 9 we prove the followingtheorem for two finite families of mappings

Theorem 18 If 119879119894119898

1and 119878

119894119899

1are two pairwise commuting

finite families of self-mapping defined on a complete partialmetric space (119883 119901) such that the mappings 119878 and 119879 (with119879 = 119879

11198792sdot sdot sdot 119879119898and 119878 = 119878

11198782sdot sdot sdot 119878119899) satisfy the contractive

condition (5) then the component maps of the two families119879119894119898

1and 119878

119894119899

1have a unique common fixed point

Proof FromTheorem 9 we can say that the mappings 119879 and119878 have a unique common fixed point 119911 that is 119879119911 = 119878119911 = 119911Now our requirement is to show that 119911 is a common fixedpoint of all the component mappings of both the families Inviewof pairwise commutativity of the families 119879

119894119898

1and 119878

119894119899

1

(for every 1 le 119896 le 119898) we can write 119879119896119911 = 119879

119896119879119911 = 119879119879

119896119911 and

119879119896119911 = 119879119896119878119911 = 119878119879

119896119911 which show that 119879

119896119911 (for every 119896) is also

a common fixed point of 119879 and 119878 By using the uniqueness ofcommonfixed point we canwrite119879

119896119911 = 119911 (for every 119896) which

shows that 119911 is a common fixed point of the family 119879119894119898

1

Using the same argument one can also show that (for every1 le 119896 le 119899) 119878

119896119911 = 119911Thus componentmaps of the two families

119879119894119898

1and 119878

119894119899

1have a unique common fixed point

By setting 1198791= 1198792= sdot sdot sdot = 119879

119898= 119865 and 119878

1= 1198782= sdot sdot sdot =

119878119899= 119866 in Theorem 18 we get the following corollary

Corollary 19 Let 119865 119866 119883 rarr 119883 be two commuting self-mappings defined on a complete PMS (119883 119901) satisfying thecondition

119901 (119865119898119909 119866119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119865

119898119909) 119901 (119910 119866

119899119910) + 120574119901 (119910 119865

119898119909) 119901 (119909 119866

119899119910)

1 + 119901 (119909 119910)

(36)

6 Abstract and Applied Analysis

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F and G have a unique common fixed point

By setting 119898 = 119899 and 119865 = 119866 = 119879 in Corollary 19 wededuce the following corollary

Corollary 20 Let 119879 119883 rarr 119883 be a mapping defined on acomplete PMS (119883 119901) satisfying the condition

119901 (119879119899119909 119879119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119879

119899119909) 119901 (119910 119879

119899119910) + 120574119901 (119910 119879

119899119909) 119901 (119909 119879

119899119910)

1 + 119901 (119909 119910)

(37)

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F has a unique fixed point

By setting 120573 = 120574 = 0 we draw following corollary whichcan be viewed as an extension of Bryantrsquos theorem [15] for amapping on a complete PMS (119883 119901)

Corollary 21 Let 119865 119883 rarr 119883 be a mapping on a completePMS (119883 119901) If 119865 satisfies

119901 (119865119899119909 119865119899119910) le 120572119901 (119909 119910) (38)

for all 119909 119910 isin 119883 120572 lt 1 Then 119865 has a unique fixed point

The following example demonstrates the superiority ofBryantrsquos theorem overMatthews theorem on complete partialmetric space

Example 22 Let119883 = [0 4] Define the partial metric 119901 119883 times

119883 rarr R by

119901 (119909 119910) = max 119909 119910 (39)

Then (119883 119901) is a complete partial metric space Let 119865 119883 rarr

119883 be defined as follows

119865 (119909) =

1199092 if 119909 isin [0 1 [

2 if 119909 isin [1 2 [

0 if 119909 isin [2 4]

(40)

Then for 119909 = 0 and 119910 = 1 we get

119901 (119865 (0) 119865 (1)) = 119901 (0 2) = 2 gt 120572119901 (0 1) = 120572 (1) (41)

because 0 le 120572 lt 1 However 1198652 satisfies the requirement ofBryantrsquos theorem and 119911 = 0 is the unique fixed point of 119865

3 Results for Set Valued Mappings

Theorem23 Let (119883 119901) be a complete partial metric space andlet 119878 119879 119883 rarr 119862119861

119901(119883) be mappings such that

119867119901(119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(42)

for all 119909 119910 isin 119883 0 le 120572 120573 120574 with 120572 + 120573 + 2120574 lt 1 Then 119878 and 119879

have a common fixed point

Proof Assume that119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) Let 1199090isin 119883 be

arbitrary but fixed element of 119883 and choose 1199091

isin 119878(1199090) By

Lemma 8 we can choose 1199092isin 119879(119909

1) such that

119901 (1199091 1199092) le 119867

119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1198781199090) 119901 (119909

1 1198791199091)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1198781199090) 119901 (119909

0 1198791199091)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199090 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573119901 (119909

1 1199092) + 120574119901 (119909

0 1199092)

+ (120572 + 120574)

(43)

So we get

119901 (1199091 1199092) le (

120572 + 120574

1 minus 120573 minus 120574)119901 (119909

0 1199091) + (

120572 + 120574

1 minus 120573 minus 120574) (44)

Since119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) so it further implies that119901 (1199091 1199092) le 119872119901 (119909

0 1199091) + 119872 (45)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093) le 119867

119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120574)2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1198791199091) 119901 (119909

2 1198781199092)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1198791199091) 119901 (119909

1 1198781199092)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

le 120572119901 (1199092119896 1199092119896+1

)

+ (120573119901 (1199092119896 1199092119896+1

) 119901 (1199092119896+1

1199092119896+2

)

+120574119901 (1199092119896 1199092119896+1

) 119901 (1199092119896 1199092119896+2

))

times (1 + 119901 (1199092119896 1199092119896+1

))minus1

le 120572119901 (1199092119896 1199092119896+1

) + 120573119901 (1199092119896+1

1199092119896+2

) + 120574119901 (1199092119896 1199092119896+2

)

(9)

as 1 + 119901(1199092119896 1199092119896+1

) gt 119901(1199092119896 1199092119896+1

) and so

119901 (1199092119896+1

1199092119896+2

) le (120572 + 120574

1 minus 120573 minus 120574)119901 (119909

2119896 1199092119896+1

) (10)

which implies that

119901 (1199092119896+1

1199092119896+2

) le 120582119901 (1199092119896 1199092119896+1

) le sdot sdot sdot le 1205822119896+1

119901 (1199090 1199091)

(11)

If 119895 = 2119896 + 2 where 119896 = 0 1 2 (119895 minus 2)2 one can easilyprove that

119901 (1199092119896+2

1199092119896+3

) le 1205822119896+2

119901 (1199090 1199091) (12)

Thus from inequality (11) and (12) we have

119901 (119909119895 119909119895+1

) le 120582119895119901 (1199090 1199091) for some 119895 isin 119873 (13)

Now

119901 (1199090 119909119895+1

) le 119901 (1199090 1199091) + sdot sdot sdot + 119901 (119909

119895 119909119895+1

)

minus [119901 (1199091 1199091) + sdot sdot sdot + 119901 (119909

119895 119909119895)]

le 119901 (1199090 1199091) + sdot sdot sdot + 120582

119895119901 (1199090 1199091)

= 119901 (1199090 1199091) [1 + sdot sdot sdot + 120582

119895minus1+ 120582119895]

le (1 minus 120582) [119903 + 119901 (1199090 1199090)]

(1 minus 120582119895+1

)

1 minus 120582

le 119903 + 119901 (1199090 1199090)

(14)

gives 119909119895+1

isin 119861(1199090 119903) Hence 119909

119899isin 119861(119909

0 119903) for all 119899 isin N One

can easily prove that

119901 (119909119899 119909119899+1

) le 120582119899119901 (1199090 1199091) (13

lowast)

for all 119899 isin N We now show that 119909119899 is a Cauchy sequence

Without loss of generality assume that 119898 gt 119899 Then using(13lowast) and the triangle inequality for partial metrics (P

4) we

have119901 (119909119899 119909119898) le 119901 (119909

119899 119909119899+1

) + 119901 (119909119899+1

119909119898) minus 119901 (119909

119899+1 119909119899+1

)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + 119901 (119909119899+2

119909119898)

minus 119901 (119909119899+2

119909119899+2

)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + 119901 (119909119899+2

119909119898)

(15)

Inductively we have

0 le 119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 120582119899119901 (1199090 1199091) + 120582119899+1

119901 (1199090 1199091) + sdot sdot sdot + 120582

119898minus1119901 (1199090 1199091)

le (120582119899+ 120582119899+1

+ sdot sdot sdot + 120582119898minus1

) 119901 (1199090 1199091)

le (120582119899

1 minus 120582)119901 (119909

0 1199091)

997888rarr 0 as 119899 997888rarr infin (since 0 lt 120582 lt 1)

(16)

Thus

lim119899rarr+infin

119901 (119909119899 119909119898) = 0 (17)

By the definition of 119901119904 we get for any119898 isin Nlowast

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (18)

Hence the sequence 119909119899 is a Cauchy sequence in

(119861(1199090 119903) 119901119904) By Lemma 2(4) 119909

119899 is a Cauchy sequence in

(119861(1199090 119903) 119901) Therefore there exists a point 119906 isin 119861(119909

0 119903) with

lim119899rarrinfin

119909119899

= 119906 Also lim119899rarrinfin

119901119904(119909119899 119906) = 0 Again from

Lemma 2(4) we have

119901 (119906 119906) = lim119899rarr+infin

119901 (119909119899 119906) = lim

119899rarr+infin119901 (119909119899 119909119898) = 0

(19)

By the triangle inequality (P4) we have

119901 (119906 119878119906) le 119901 (119906 1199092119899+2

) + 119901 (1199092119899+2

119878119906) minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119906 1199092119899+2

) + 119901 (1198791199092119899+1

119878119906)

le 119901 (119906 1199092119899+2

) + 120572119901 (1199092119899+1

119906)

+ (120573119901 (1199092119899+1

1198791199092119899+1

) 119901 (119906 119878119906)

+120574119901 (119906 1198791199092119899+1

) 119901 (1199092119899+1

119878119906))

times (1 + 119901 (1199092119899+1

119906))minus1

le 119901 (119906 1199092119899+2

) + 120572119901 (1199092119899+1

119906)

+ (120573119901 (1199092119899+1

1199092119899+2

) 119901 (119906 119878119906)

+120574119901 (119906 1199092119899+2

) 119901 (1199092119899+1

119878119906))

times (1 + 119901 (1199092119899+1

119906))minus1

(20)

Letting 119899 rarr +infin and using (19) we obtain

119901 (119906 119878119906) = 0 (21)

By (P1) we concluded that 119906 = 119878119906 It follows similarly that

119906 = 119879119906 To prove the uniqueness of common fixed point let

4 Abstract and Applied Analysis

119906lowastisin 119861(119909

0 119903) be another common fixed point of 119878 and119879 that

is 119906lowast = 119878119906lowast= 119879119906lowast Then

119901 (119906 119906lowast) = 119901 (119878119906 119879119906

lowast)

le 120572119901 (119906 119906lowast)

+120573119901 (119906 119878119906) 119901 (119906

lowast 119879119906lowast)+120574119901 (119906

lowast 119878119906) 119901 (119906 119879119906

lowast)

1+119901 (119906 119906lowast)

= 120572119901 (119906 119906lowast) +

120574119901 (119906lowast 119906) 119901 (119906 119906

lowast)

1 + 119901 (119906 119906lowast)

(22)

so that119901(119906 119906lowast) le 120572119901(119906 119906lowast)+120574119901(119906 119906

lowast) because 1+119901(119906 119906

lowast) gt

119901(119906 119906lowast) Therefore 119901(119906 119906

lowast) le (120572 + 120574)119901(119906 119906

lowast) which is a

contradiction so that 119906 = 119906lowast (as 120572 + 120574 lt 1) Hence 119878 and

119879 have a unique common fixed point in 119861(1199090 119903)

Example 10 Let119883 = [0 +infin) endowed with the usual partialmetric 119901 defined by 119901 119883 times 119883 rarr R+ with 119901(119909 119910) =

max119909 119910 Clearly (119883 119901) is a partial metric space Now wedefine 119878 119879 119883 rarr 119883 as

119878 (119909) =

119909

16if 0 le 119909 le 1

119909 minus1

6if 119909 gt 1

119879 (119909) =

5119909

17if 0 le 119909 le 1

119909 minus1

7if 119909 gt 1

(23)

for all 119909 isin 119883 Taking 120572 = 15 120573 = 16 120574 = 18 1199090= 12

and 119903 = 12 then 119861119901(1199090 119903) = [0 1] Also we have 119901(119909

0 1199090) =

max12 12 = 12 120582 = (120572 + 120574)(1 minus 120573 minus 120574) = 3985 with

(1 minus 120582) (119903 + 119901 (1199090 1199090)) =

46

85

119901 (1199090 1198781199090) = 119901 (

1

21

32) =

1

2lt (1 minus 120582) (119903 + 119901 (119909

0 1199090))

(24)

Also if 119909 119910 isin (1 +infin) then

119901 (119878119909 119879119910) = max 119909 minus1

6 119909 minus

1

7 ge

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

= 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(25)

So the contractive condition does not hold on whole of 119883Now if 119909 119910 isin 119861

119901(1199090 119903) then

119901 (119878119909 119879119910) = max 119909

165119910

17 le

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

(26)

Therefore all the conditions of Theorem 9 are satisfied Thus0 is the common fixed point of 119878 and 119879 and 119901(0 0) = 0Moreover note that for any metric 119889 on119883

119889 (1198781 1198791) = 119889 (1

165

17) gt

1

5119889 (1 1)

+ (1

6119889 (1

1

16) 119889 (1

5

17)

+1

8119889 (1

1

16) 119889 (1

5

17))

times (1 + 119889 (1 1))minus1

(27)

Therefore commonfixed points of 119878 and119879 cannot be obtainedfrom a metric fixed point theorem

Corollary 11 Let 119878 119879 119883 rarr 119883 be mappings on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(28)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119878119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119878 and 119879 have nofixed point other than 119906

By choosing 120573 = 120574 = 0 in Corollary 11 we get thefollowing corollary

Corollary 12 Let 119878 119879 119883 rarr 119883 be a mappings on completePMS (119883 119901) If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910) (29)

for all 119909 119910 isin 119883 120572 lt 1 (120572 is a nonnegative real) Then 119878 and 119879

have a common fixed point 119906 isin 119883 and 119901(119906 119906) = 0

Remark 13 If we impose Banach type contractive conditionfor a pair 119878 119879 119883 rarr 119883 of mappings on a metric space(119883 119889) that is 119889(119878119909 119879119910) le 120572119889(119909 119910) for all 119909 119910 isin 119883 andthen it follows that 119878119909 = 119879119909 for all 119909 isin 119883 (ie 119878 and 119879 are

Abstract and Applied Analysis 5

equal) Therefore the above condition fails to find commonfixed points of 119878 and 119879 This can be seen as

119889 (1198781 1198791) = 119889 ((1

16) (

5

17)) gt 120572119889 (1 1) = 0 (30)

However the same condition in partial metric space does notassert that 119878 = 119879 This can be seen as by taking the partialmetric same as in Example 10

119901 (1198781 1198791) = 119901 ((1

16) (

5

17)) =

5

17le 120572119901 (1 1) = 120572 (1)

(31)

for any 120572 ge 13 Hence Corollary 12 cannot be obtained froma metric fixed point theorem

Remark 14 By equating 120572 120573 120574 to 0 in all possible combi-nations one can derive a host of corollaries which includeMatthews theorem for mappings defined on a completepartial metric space

By taking 119878 = 119879 in the Theorem 9 we get the followingcorollary

Corollary 15 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(32)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198791199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (33)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861

119901(1199090 119903) such that 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879

has no fixed point other than 119906

By taking 119878 = 119879 in Corollary 11 we get the followingcorollary

Corollary 16 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(34)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879 has no fixed pointother than 119906

Now we give an example in favour of Corollary 16

Example 17 Let 119883 = [0 4] endowed with the usual partialmetric 119901 defined by 119901(119909 119910) = max119909 119910 Clearly (119883 119901) is acomplete partial metric space Now we define 119865 119883 rarr 119883 asfollows

119865 (119909) =

119909

3if 0 le 119909 lt 2

119909

1 + 119909if 2 le 119909 le 4

(35)

for all 119909 isin 119883 Now let 119910 le 119909 If 119909 isin [0 2) (and so 119910 isin [0 2))Then 119901(119865119909 119865119910) = 1199093 119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) =

119910 119901(119910 119865119909) = 1199093 119901(119909 119865119910) = 119909 Taking 120572 = 13 120573 = 115120574 = 215 we can prove that all the conditions of Corollary 16are satisfied Now if 119909 isin [2 4] then 119901(119865119909 119865119910) = 119909(1 + 119909)119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) = 119910 119901(119910 119865119909) = 119909(1+119909)119901(119909 119865119910) = 119909 and taking 120572 = 13 120573 = 115 120574 = 215 onecan verify the condition of the above corollary Thus all theconditions of Corollary 16 are satisfied and 119906 = 0 is a fixedpoint of the mapping 119865

As an application of Theorem 9 we prove the followingtheorem for two finite families of mappings

Theorem 18 If 119879119894119898

1and 119878

119894119899

1are two pairwise commuting

finite families of self-mapping defined on a complete partialmetric space (119883 119901) such that the mappings 119878 and 119879 (with119879 = 119879

11198792sdot sdot sdot 119879119898and 119878 = 119878

11198782sdot sdot sdot 119878119899) satisfy the contractive

condition (5) then the component maps of the two families119879119894119898

1and 119878

119894119899

1have a unique common fixed point

Proof FromTheorem 9 we can say that the mappings 119879 and119878 have a unique common fixed point 119911 that is 119879119911 = 119878119911 = 119911Now our requirement is to show that 119911 is a common fixedpoint of all the component mappings of both the families Inviewof pairwise commutativity of the families 119879

119894119898

1and 119878

119894119899

1

(for every 1 le 119896 le 119898) we can write 119879119896119911 = 119879

119896119879119911 = 119879119879

119896119911 and

119879119896119911 = 119879119896119878119911 = 119878119879

119896119911 which show that 119879

119896119911 (for every 119896) is also

a common fixed point of 119879 and 119878 By using the uniqueness ofcommonfixed point we canwrite119879

119896119911 = 119911 (for every 119896) which

shows that 119911 is a common fixed point of the family 119879119894119898

1

Using the same argument one can also show that (for every1 le 119896 le 119899) 119878

119896119911 = 119911Thus componentmaps of the two families

119879119894119898

1and 119878

119894119899

1have a unique common fixed point

By setting 1198791= 1198792= sdot sdot sdot = 119879

119898= 119865 and 119878

1= 1198782= sdot sdot sdot =

119878119899= 119866 in Theorem 18 we get the following corollary

Corollary 19 Let 119865 119866 119883 rarr 119883 be two commuting self-mappings defined on a complete PMS (119883 119901) satisfying thecondition

119901 (119865119898119909 119866119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119865

119898119909) 119901 (119910 119866

119899119910) + 120574119901 (119910 119865

119898119909) 119901 (119909 119866

119899119910)

1 + 119901 (119909 119910)

(36)

6 Abstract and Applied Analysis

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F and G have a unique common fixed point

By setting 119898 = 119899 and 119865 = 119866 = 119879 in Corollary 19 wededuce the following corollary

Corollary 20 Let 119879 119883 rarr 119883 be a mapping defined on acomplete PMS (119883 119901) satisfying the condition

119901 (119879119899119909 119879119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119879

119899119909) 119901 (119910 119879

119899119910) + 120574119901 (119910 119879

119899119909) 119901 (119909 119879

119899119910)

1 + 119901 (119909 119910)

(37)

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F has a unique fixed point

By setting 120573 = 120574 = 0 we draw following corollary whichcan be viewed as an extension of Bryantrsquos theorem [15] for amapping on a complete PMS (119883 119901)

Corollary 21 Let 119865 119883 rarr 119883 be a mapping on a completePMS (119883 119901) If 119865 satisfies

119901 (119865119899119909 119865119899119910) le 120572119901 (119909 119910) (38)

for all 119909 119910 isin 119883 120572 lt 1 Then 119865 has a unique fixed point

The following example demonstrates the superiority ofBryantrsquos theorem overMatthews theorem on complete partialmetric space

Example 22 Let119883 = [0 4] Define the partial metric 119901 119883 times

119883 rarr R by

119901 (119909 119910) = max 119909 119910 (39)

Then (119883 119901) is a complete partial metric space Let 119865 119883 rarr

119883 be defined as follows

119865 (119909) =

1199092 if 119909 isin [0 1 [

2 if 119909 isin [1 2 [

0 if 119909 isin [2 4]

(40)

Then for 119909 = 0 and 119910 = 1 we get

119901 (119865 (0) 119865 (1)) = 119901 (0 2) = 2 gt 120572119901 (0 1) = 120572 (1) (41)

because 0 le 120572 lt 1 However 1198652 satisfies the requirement ofBryantrsquos theorem and 119911 = 0 is the unique fixed point of 119865

3 Results for Set Valued Mappings

Theorem23 Let (119883 119901) be a complete partial metric space andlet 119878 119879 119883 rarr 119862119861

119901(119883) be mappings such that

119867119901(119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(42)

for all 119909 119910 isin 119883 0 le 120572 120573 120574 with 120572 + 120573 + 2120574 lt 1 Then 119878 and 119879

have a common fixed point

Proof Assume that119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) Let 1199090isin 119883 be

arbitrary but fixed element of 119883 and choose 1199091

isin 119878(1199090) By

Lemma 8 we can choose 1199092isin 119879(119909

1) such that

119901 (1199091 1199092) le 119867

119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1198781199090) 119901 (119909

1 1198791199091)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1198781199090) 119901 (119909

0 1198791199091)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199090 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573119901 (119909

1 1199092) + 120574119901 (119909

0 1199092)

+ (120572 + 120574)

(43)

So we get

119901 (1199091 1199092) le (

120572 + 120574

1 minus 120573 minus 120574)119901 (119909

0 1199091) + (

120572 + 120574

1 minus 120573 minus 120574) (44)

Since119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) so it further implies that119901 (1199091 1199092) le 119872119901 (119909

0 1199091) + 119872 (45)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093) le 119867

119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120574)2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1198791199091) 119901 (119909

2 1198781199092)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1198791199091) 119901 (119909

1 1198781199092)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

119906lowastisin 119861(119909

0 119903) be another common fixed point of 119878 and119879 that

is 119906lowast = 119878119906lowast= 119879119906lowast Then

119901 (119906 119906lowast) = 119901 (119878119906 119879119906

lowast)

le 120572119901 (119906 119906lowast)

+120573119901 (119906 119878119906) 119901 (119906

lowast 119879119906lowast)+120574119901 (119906

lowast 119878119906) 119901 (119906 119879119906

lowast)

1+119901 (119906 119906lowast)

= 120572119901 (119906 119906lowast) +

120574119901 (119906lowast 119906) 119901 (119906 119906

lowast)

1 + 119901 (119906 119906lowast)

(22)

so that119901(119906 119906lowast) le 120572119901(119906 119906lowast)+120574119901(119906 119906

lowast) because 1+119901(119906 119906

lowast) gt

119901(119906 119906lowast) Therefore 119901(119906 119906

lowast) le (120572 + 120574)119901(119906 119906

lowast) which is a

contradiction so that 119906 = 119906lowast (as 120572 + 120574 lt 1) Hence 119878 and

119879 have a unique common fixed point in 119861(1199090 119903)

Example 10 Let119883 = [0 +infin) endowed with the usual partialmetric 119901 defined by 119901 119883 times 119883 rarr R+ with 119901(119909 119910) =

max119909 119910 Clearly (119883 119901) is a partial metric space Now wedefine 119878 119879 119883 rarr 119883 as

119878 (119909) =

119909

16if 0 le 119909 le 1

119909 minus1

6if 119909 gt 1

119879 (119909) =

5119909

17if 0 le 119909 le 1

119909 minus1

7if 119909 gt 1

(23)

for all 119909 isin 119883 Taking 120572 = 15 120573 = 16 120574 = 18 1199090= 12

and 119903 = 12 then 119861119901(1199090 119903) = [0 1] Also we have 119901(119909

0 1199090) =

max12 12 = 12 120582 = (120572 + 120574)(1 minus 120573 minus 120574) = 3985 with

(1 minus 120582) (119903 + 119901 (1199090 1199090)) =

46

85

119901 (1199090 1198781199090) = 119901 (

1

21

32) =

1

2lt (1 minus 120582) (119903 + 119901 (119909

0 1199090))

(24)

Also if 119909 119910 isin (1 +infin) then

119901 (119878119909 119879119910) = max 119909 minus1

6 119909 minus

1

7 ge

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

= 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(25)

So the contractive condition does not hold on whole of 119883Now if 119909 119910 isin 119861

119901(1199090 119903) then

119901 (119878119909 119879119910) = max 119909

165119910

17 le

1

5max 119909 119910

+ (1

6max 119909 119909 minus

1

6max 119910 119910 minus

1

7

+1

8max 119910 119909 minus

1

6max 119909 119910 minus

1

7)

times (1 +max 119909 119910)minus1

(26)

Therefore all the conditions of Theorem 9 are satisfied Thus0 is the common fixed point of 119878 and 119879 and 119901(0 0) = 0Moreover note that for any metric 119889 on119883

119889 (1198781 1198791) = 119889 (1

165

17) gt

1

5119889 (1 1)

+ (1

6119889 (1

1

16) 119889 (1

5

17)

+1

8119889 (1

1

16) 119889 (1

5

17))

times (1 + 119889 (1 1))minus1

(27)

Therefore commonfixed points of 119878 and119879 cannot be obtainedfrom a metric fixed point theorem

Corollary 11 Let 119878 119879 119883 rarr 119883 be mappings on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(28)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119878119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119878 and 119879 have nofixed point other than 119906

By choosing 120573 = 120574 = 0 in Corollary 11 we get thefollowing corollary

Corollary 12 Let 119878 119879 119883 rarr 119883 be a mappings on completePMS (119883 119901) If 119878 and 119879 satisfy

119901 (119878119909 119879119910) le 120572119901 (119909 119910) (29)

for all 119909 119910 isin 119883 120572 lt 1 (120572 is a nonnegative real) Then 119878 and 119879

have a common fixed point 119906 isin 119883 and 119901(119906 119906) = 0

Remark 13 If we impose Banach type contractive conditionfor a pair 119878 119879 119883 rarr 119883 of mappings on a metric space(119883 119889) that is 119889(119878119909 119879119910) le 120572119889(119909 119910) for all 119909 119910 isin 119883 andthen it follows that 119878119909 = 119879119909 for all 119909 isin 119883 (ie 119878 and 119879 are

Abstract and Applied Analysis 5

equal) Therefore the above condition fails to find commonfixed points of 119878 and 119879 This can be seen as

119889 (1198781 1198791) = 119889 ((1

16) (

5

17)) gt 120572119889 (1 1) = 0 (30)

However the same condition in partial metric space does notassert that 119878 = 119879 This can be seen as by taking the partialmetric same as in Example 10

119901 (1198781 1198791) = 119901 ((1

16) (

5

17)) =

5

17le 120572119901 (1 1) = 120572 (1)

(31)

for any 120572 ge 13 Hence Corollary 12 cannot be obtained froma metric fixed point theorem

Remark 14 By equating 120572 120573 120574 to 0 in all possible combi-nations one can derive a host of corollaries which includeMatthews theorem for mappings defined on a completepartial metric space

By taking 119878 = 119879 in the Theorem 9 we get the followingcorollary

Corollary 15 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(32)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198791199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (33)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861

119901(1199090 119903) such that 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879

has no fixed point other than 119906

By taking 119878 = 119879 in Corollary 11 we get the followingcorollary

Corollary 16 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(34)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879 has no fixed pointother than 119906

Now we give an example in favour of Corollary 16

Example 17 Let 119883 = [0 4] endowed with the usual partialmetric 119901 defined by 119901(119909 119910) = max119909 119910 Clearly (119883 119901) is acomplete partial metric space Now we define 119865 119883 rarr 119883 asfollows

119865 (119909) =

119909

3if 0 le 119909 lt 2

119909

1 + 119909if 2 le 119909 le 4

(35)

for all 119909 isin 119883 Now let 119910 le 119909 If 119909 isin [0 2) (and so 119910 isin [0 2))Then 119901(119865119909 119865119910) = 1199093 119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) =

119910 119901(119910 119865119909) = 1199093 119901(119909 119865119910) = 119909 Taking 120572 = 13 120573 = 115120574 = 215 we can prove that all the conditions of Corollary 16are satisfied Now if 119909 isin [2 4] then 119901(119865119909 119865119910) = 119909(1 + 119909)119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) = 119910 119901(119910 119865119909) = 119909(1+119909)119901(119909 119865119910) = 119909 and taking 120572 = 13 120573 = 115 120574 = 215 onecan verify the condition of the above corollary Thus all theconditions of Corollary 16 are satisfied and 119906 = 0 is a fixedpoint of the mapping 119865

As an application of Theorem 9 we prove the followingtheorem for two finite families of mappings

Theorem 18 If 119879119894119898

1and 119878

119894119899

1are two pairwise commuting

finite families of self-mapping defined on a complete partialmetric space (119883 119901) such that the mappings 119878 and 119879 (with119879 = 119879

11198792sdot sdot sdot 119879119898and 119878 = 119878

11198782sdot sdot sdot 119878119899) satisfy the contractive

condition (5) then the component maps of the two families119879119894119898

1and 119878

119894119899

1have a unique common fixed point

Proof FromTheorem 9 we can say that the mappings 119879 and119878 have a unique common fixed point 119911 that is 119879119911 = 119878119911 = 119911Now our requirement is to show that 119911 is a common fixedpoint of all the component mappings of both the families Inviewof pairwise commutativity of the families 119879

119894119898

1and 119878

119894119899

1

(for every 1 le 119896 le 119898) we can write 119879119896119911 = 119879

119896119879119911 = 119879119879

119896119911 and

119879119896119911 = 119879119896119878119911 = 119878119879

119896119911 which show that 119879

119896119911 (for every 119896) is also

a common fixed point of 119879 and 119878 By using the uniqueness ofcommonfixed point we canwrite119879

119896119911 = 119911 (for every 119896) which

shows that 119911 is a common fixed point of the family 119879119894119898

1

Using the same argument one can also show that (for every1 le 119896 le 119899) 119878

119896119911 = 119911Thus componentmaps of the two families

119879119894119898

1and 119878

119894119899

1have a unique common fixed point

By setting 1198791= 1198792= sdot sdot sdot = 119879

119898= 119865 and 119878

1= 1198782= sdot sdot sdot =

119878119899= 119866 in Theorem 18 we get the following corollary

Corollary 19 Let 119865 119866 119883 rarr 119883 be two commuting self-mappings defined on a complete PMS (119883 119901) satisfying thecondition

119901 (119865119898119909 119866119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119865

119898119909) 119901 (119910 119866

119899119910) + 120574119901 (119910 119865

119898119909) 119901 (119909 119866

119899119910)

1 + 119901 (119909 119910)

(36)

6 Abstract and Applied Analysis

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F and G have a unique common fixed point

By setting 119898 = 119899 and 119865 = 119866 = 119879 in Corollary 19 wededuce the following corollary

Corollary 20 Let 119879 119883 rarr 119883 be a mapping defined on acomplete PMS (119883 119901) satisfying the condition

119901 (119879119899119909 119879119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119879

119899119909) 119901 (119910 119879

119899119910) + 120574119901 (119910 119879

119899119909) 119901 (119909 119879

119899119910)

1 + 119901 (119909 119910)

(37)

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F has a unique fixed point

By setting 120573 = 120574 = 0 we draw following corollary whichcan be viewed as an extension of Bryantrsquos theorem [15] for amapping on a complete PMS (119883 119901)

Corollary 21 Let 119865 119883 rarr 119883 be a mapping on a completePMS (119883 119901) If 119865 satisfies

119901 (119865119899119909 119865119899119910) le 120572119901 (119909 119910) (38)

for all 119909 119910 isin 119883 120572 lt 1 Then 119865 has a unique fixed point

The following example demonstrates the superiority ofBryantrsquos theorem overMatthews theorem on complete partialmetric space

Example 22 Let119883 = [0 4] Define the partial metric 119901 119883 times

119883 rarr R by

119901 (119909 119910) = max 119909 119910 (39)

Then (119883 119901) is a complete partial metric space Let 119865 119883 rarr

119883 be defined as follows

119865 (119909) =

1199092 if 119909 isin [0 1 [

2 if 119909 isin [1 2 [

0 if 119909 isin [2 4]

(40)

Then for 119909 = 0 and 119910 = 1 we get

119901 (119865 (0) 119865 (1)) = 119901 (0 2) = 2 gt 120572119901 (0 1) = 120572 (1) (41)

because 0 le 120572 lt 1 However 1198652 satisfies the requirement ofBryantrsquos theorem and 119911 = 0 is the unique fixed point of 119865

3 Results for Set Valued Mappings

Theorem23 Let (119883 119901) be a complete partial metric space andlet 119878 119879 119883 rarr 119862119861

119901(119883) be mappings such that

119867119901(119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(42)

for all 119909 119910 isin 119883 0 le 120572 120573 120574 with 120572 + 120573 + 2120574 lt 1 Then 119878 and 119879

have a common fixed point

Proof Assume that119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) Let 1199090isin 119883 be

arbitrary but fixed element of 119883 and choose 1199091

isin 119878(1199090) By

Lemma 8 we can choose 1199092isin 119879(119909

1) such that

119901 (1199091 1199092) le 119867

119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1198781199090) 119901 (119909

1 1198791199091)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1198781199090) 119901 (119909

0 1198791199091)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199090 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573119901 (119909

1 1199092) + 120574119901 (119909

0 1199092)

+ (120572 + 120574)

(43)

So we get

119901 (1199091 1199092) le (

120572 + 120574

1 minus 120573 minus 120574)119901 (119909

0 1199091) + (

120572 + 120574

1 minus 120573 minus 120574) (44)

Since119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) so it further implies that119901 (1199091 1199092) le 119872119901 (119909

0 1199091) + 119872 (45)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093) le 119867

119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120574)2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1198791199091) 119901 (119909

2 1198781199092)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1198791199091) 119901 (119909

1 1198781199092)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

equal) Therefore the above condition fails to find commonfixed points of 119878 and 119879 This can be seen as

119889 (1198781 1198791) = 119889 ((1

16) (

5

17)) gt 120572119889 (1 1) = 0 (30)

However the same condition in partial metric space does notassert that 119878 = 119879 This can be seen as by taking the partialmetric same as in Example 10

119901 (1198781 1198791) = 119901 ((1

16) (

5

17)) =

5

17le 120572119901 (1 1) = 120572 (1)

(31)

for any 120572 ge 13 Hence Corollary 12 cannot be obtained froma metric fixed point theorem

Remark 14 By equating 120572 120573 120574 to 0 in all possible combi-nations one can derive a host of corollaries which includeMatthews theorem for mappings defined on a completepartial metric space

By taking 119878 = 119879 in the Theorem 9 we get the followingcorollary

Corollary 15 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) and 119909

0 119909 119910 isin 119883 and 119903 gt 0 Suppose that there

exist nonnegative reals 120572 120573 and 120574 such that 120572 + 120573 + 2120574 lt 1 If119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(32)

for all 119909 119910 isin 119861119901(1199090 119903)

119901 (1199090 1198791199090) le (1 minus 120582) (119903 + 119901 (119909

0 1199090)) (33)

where 120582 = (120572+ 120574)(1 minus120573minus 120574) Then there exists a unique point119906 isin 119861

119901(1199090 119903) such that 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879

has no fixed point other than 119906

By taking 119878 = 119879 in Corollary 11 we get the followingcorollary

Corollary 16 Let 119879 119883 rarr 119883 be a mapping on a completePMS (119883 119901) Suppose that there exist nonnegative reals 120572 120573and 120574 such that 120572 + 120573 + 2120574 lt 1 If 119879 satisfies

119901 (119879119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119879119909) 119901 (119910 119879119910) + 120574119901 (119910 119879119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(34)

for all 119909 119910 isin 119883 Then there exists a unique point 119906 isin 119883 suchthat 119906 = 119879119906 Also 119901(119906 119906) = 0 Further 119879 has no fixed pointother than 119906

Now we give an example in favour of Corollary 16

Example 17 Let 119883 = [0 4] endowed with the usual partialmetric 119901 defined by 119901(119909 119910) = max119909 119910 Clearly (119883 119901) is acomplete partial metric space Now we define 119865 119883 rarr 119883 asfollows

119865 (119909) =

119909

3if 0 le 119909 lt 2

119909

1 + 119909if 2 le 119909 le 4

(35)

for all 119909 isin 119883 Now let 119910 le 119909 If 119909 isin [0 2) (and so 119910 isin [0 2))Then 119901(119865119909 119865119910) = 1199093 119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) =

119910 119901(119910 119865119909) = 1199093 119901(119909 119865119910) = 119909 Taking 120572 = 13 120573 = 115120574 = 215 we can prove that all the conditions of Corollary 16are satisfied Now if 119909 isin [2 4] then 119901(119865119909 119865119910) = 119909(1 + 119909)119901(119909 119910) = 119909 119901(119909 119865119909) = 119909 119901(119910 119865119910) = 119910 119901(119910 119865119909) = 119909(1+119909)119901(119909 119865119910) = 119909 and taking 120572 = 13 120573 = 115 120574 = 215 onecan verify the condition of the above corollary Thus all theconditions of Corollary 16 are satisfied and 119906 = 0 is a fixedpoint of the mapping 119865

As an application of Theorem 9 we prove the followingtheorem for two finite families of mappings

Theorem 18 If 119879119894119898

1and 119878

119894119899

1are two pairwise commuting

finite families of self-mapping defined on a complete partialmetric space (119883 119901) such that the mappings 119878 and 119879 (with119879 = 119879

11198792sdot sdot sdot 119879119898and 119878 = 119878

11198782sdot sdot sdot 119878119899) satisfy the contractive

condition (5) then the component maps of the two families119879119894119898

1and 119878

119894119899

1have a unique common fixed point

Proof FromTheorem 9 we can say that the mappings 119879 and119878 have a unique common fixed point 119911 that is 119879119911 = 119878119911 = 119911Now our requirement is to show that 119911 is a common fixedpoint of all the component mappings of both the families Inviewof pairwise commutativity of the families 119879

119894119898

1and 119878

119894119899

1

(for every 1 le 119896 le 119898) we can write 119879119896119911 = 119879

119896119879119911 = 119879119879

119896119911 and

119879119896119911 = 119879119896119878119911 = 119878119879

119896119911 which show that 119879

119896119911 (for every 119896) is also

a common fixed point of 119879 and 119878 By using the uniqueness ofcommonfixed point we canwrite119879

119896119911 = 119911 (for every 119896) which

shows that 119911 is a common fixed point of the family 119879119894119898

1

Using the same argument one can also show that (for every1 le 119896 le 119899) 119878

119896119911 = 119911Thus componentmaps of the two families

119879119894119898

1and 119878

119894119899

1have a unique common fixed point

By setting 1198791= 1198792= sdot sdot sdot = 119879

119898= 119865 and 119878

1= 1198782= sdot sdot sdot =

119878119899= 119866 in Theorem 18 we get the following corollary

Corollary 19 Let 119865 119866 119883 rarr 119883 be two commuting self-mappings defined on a complete PMS (119883 119901) satisfying thecondition

119901 (119865119898119909 119866119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119865

119898119909) 119901 (119910 119866

119899119910) + 120574119901 (119910 119865

119898119909) 119901 (119909 119866

119899119910)

1 + 119901 (119909 119910)

(36)

6 Abstract and Applied Analysis

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F and G have a unique common fixed point

By setting 119898 = 119899 and 119865 = 119866 = 119879 in Corollary 19 wededuce the following corollary

Corollary 20 Let 119879 119883 rarr 119883 be a mapping defined on acomplete PMS (119883 119901) satisfying the condition

119901 (119879119899119909 119879119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119879

119899119909) 119901 (119910 119879

119899119910) + 120574119901 (119910 119879

119899119909) 119901 (119909 119879

119899119910)

1 + 119901 (119909 119910)

(37)

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F has a unique fixed point

By setting 120573 = 120574 = 0 we draw following corollary whichcan be viewed as an extension of Bryantrsquos theorem [15] for amapping on a complete PMS (119883 119901)

Corollary 21 Let 119865 119883 rarr 119883 be a mapping on a completePMS (119883 119901) If 119865 satisfies

119901 (119865119899119909 119865119899119910) le 120572119901 (119909 119910) (38)

for all 119909 119910 isin 119883 120572 lt 1 Then 119865 has a unique fixed point

The following example demonstrates the superiority ofBryantrsquos theorem overMatthews theorem on complete partialmetric space

Example 22 Let119883 = [0 4] Define the partial metric 119901 119883 times

119883 rarr R by

119901 (119909 119910) = max 119909 119910 (39)

Then (119883 119901) is a complete partial metric space Let 119865 119883 rarr

119883 be defined as follows

119865 (119909) =

1199092 if 119909 isin [0 1 [

2 if 119909 isin [1 2 [

0 if 119909 isin [2 4]

(40)

Then for 119909 = 0 and 119910 = 1 we get

119901 (119865 (0) 119865 (1)) = 119901 (0 2) = 2 gt 120572119901 (0 1) = 120572 (1) (41)

because 0 le 120572 lt 1 However 1198652 satisfies the requirement ofBryantrsquos theorem and 119911 = 0 is the unique fixed point of 119865

3 Results for Set Valued Mappings

Theorem23 Let (119883 119901) be a complete partial metric space andlet 119878 119879 119883 rarr 119862119861

119901(119883) be mappings such that

119867119901(119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(42)

for all 119909 119910 isin 119883 0 le 120572 120573 120574 with 120572 + 120573 + 2120574 lt 1 Then 119878 and 119879

have a common fixed point

Proof Assume that119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) Let 1199090isin 119883 be

arbitrary but fixed element of 119883 and choose 1199091

isin 119878(1199090) By

Lemma 8 we can choose 1199092isin 119879(119909

1) such that

119901 (1199091 1199092) le 119867

119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1198781199090) 119901 (119909

1 1198791199091)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1198781199090) 119901 (119909

0 1198791199091)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199090 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573119901 (119909

1 1199092) + 120574119901 (119909

0 1199092)

+ (120572 + 120574)

(43)

So we get

119901 (1199091 1199092) le (

120572 + 120574

1 minus 120573 minus 120574)119901 (119909

0 1199091) + (

120572 + 120574

1 minus 120573 minus 120574) (44)

Since119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) so it further implies that119901 (1199091 1199092) le 119872119901 (119909

0 1199091) + 119872 (45)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093) le 119867

119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120574)2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1198791199091) 119901 (119909

2 1198781199092)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1198791199091) 119901 (119909

1 1198781199092)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F and G have a unique common fixed point

By setting 119898 = 119899 and 119865 = 119866 = 119879 in Corollary 19 wededuce the following corollary

Corollary 20 Let 119879 119883 rarr 119883 be a mapping defined on acomplete PMS (119883 119901) satisfying the condition

119901 (119879119899119909 119879119899119910)

le 120572119901 (119909 119910)

+120573119901 (119909 119879

119899119909) 119901 (119910 119879

119899119910) + 120574119901 (119910 119879

119899119909) 119901 (119909 119879

119899119910)

1 + 119901 (119909 119910)

(37)

for all 119909 119910 isin 119883 120572 + 120573 + 2120574 lt 1 (120572 120573 and 120574 are nonnegativereals) Then F has a unique fixed point

By setting 120573 = 120574 = 0 we draw following corollary whichcan be viewed as an extension of Bryantrsquos theorem [15] for amapping on a complete PMS (119883 119901)

Corollary 21 Let 119865 119883 rarr 119883 be a mapping on a completePMS (119883 119901) If 119865 satisfies

119901 (119865119899119909 119865119899119910) le 120572119901 (119909 119910) (38)

for all 119909 119910 isin 119883 120572 lt 1 Then 119865 has a unique fixed point

The following example demonstrates the superiority ofBryantrsquos theorem overMatthews theorem on complete partialmetric space

Example 22 Let119883 = [0 4] Define the partial metric 119901 119883 times

119883 rarr R by

119901 (119909 119910) = max 119909 119910 (39)

Then (119883 119901) is a complete partial metric space Let 119865 119883 rarr

119883 be defined as follows

119865 (119909) =

1199092 if 119909 isin [0 1 [

2 if 119909 isin [1 2 [

0 if 119909 isin [2 4]

(40)

Then for 119909 = 0 and 119910 = 1 we get

119901 (119865 (0) 119865 (1)) = 119901 (0 2) = 2 gt 120572119901 (0 1) = 120572 (1) (41)

because 0 le 120572 lt 1 However 1198652 satisfies the requirement ofBryantrsquos theorem and 119911 = 0 is the unique fixed point of 119865

3 Results for Set Valued Mappings

Theorem23 Let (119883 119901) be a complete partial metric space andlet 119878 119879 119883 rarr 119862119861

119901(119883) be mappings such that

119867119901(119878119909 119879119910) le 120572119901 (119909 119910)

+120573119901 (119909 119878119909) 119901 (119910 119879119910) + 120574119901 (119910 119878119909) 119901 (119909 119879119910)

1 + 119901 (119909 119910)

(42)

for all 119909 119910 isin 119883 0 le 120572 120573 120574 with 120572 + 120573 + 2120574 lt 1 Then 119878 and 119879

have a common fixed point

Proof Assume that119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) Let 1199090isin 119883 be

arbitrary but fixed element of 119883 and choose 1199091

isin 119878(1199090) By

Lemma 8 we can choose 1199092isin 119879(119909

1) such that

119901 (1199091 1199092) le 119867

119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1198781199090) 119901 (119909

1 1198791199091)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1198781199090) 119901 (119909

0 1198791199091)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199091 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573

119901 (1199090 1199091) 119901 (119909

1 1199092)

1 + 119901 (1199090 1199091)

+ 120574119901 (1199090 1199091) 119901 (119909

0 1199092)

1 + 119901 (1199090 1199091)

+ (120572 + 120574)

le 120572119901 (1199090 1199091) + 120573119901 (119909

1 1199092) + 120574119901 (119909

0 1199092)

+ (120572 + 120574)

(43)

So we get

119901 (1199091 1199092) le (

120572 + 120574

1 minus 120573 minus 120574)119901 (119909

0 1199091) + (

120572 + 120574

1 minus 120573 minus 120574) (44)

Since119872 = ((120572 + 120574)(1 minus 120573 minus 120574)) so it further implies that119901 (1199091 1199092) le 119872119901 (119909

0 1199091) + 119872 (45)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093) le 119867

119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120574)2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1198791199091) 119901 (119909

2 1198781199092)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1198791199091) 119901 (119909

1 1198781199092)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199092 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

le 120572119901 (1199091 1199092) + 120573

119901 (1199091 1199092) 119901 (119909

2 1199093)

1 + 119901 (1199091 1199092)

+ 120574119901 (1199091 1199092) 119901 (119909

1 1199093)

1 + 119901 (1199091 1199092)

+(120572 + 120574)

2

1 minus 120573 minus 120574

le 120572119901 (1199091 1199092) + 120573119901 (119909

2 1199093) + 120574119901 (119909

1 1199093)

+(120572 + 120574)

2

1 minus 120573 minus 120574

(46)

So we get

119901 (1199092 1199093) le (

120572 + 120574

1 minus 120573 minus 120574)

2

119901 (1199090 1199091) + 2(

120572 + 120574

1 minus 120573 minus 120574)

2

(47)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119872119901 (119909119899minus1

119909119899) + 119872

le sdot sdot sdot le 119872119899119901 (1199090 1199091) + 119899119872

119899

(48)

where 119872 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Without lossof generality assume that 119899 gt 119898 Then using (48) and thetriangle inequality for partial metrics (P

4) we have

119901 (119909119899 119909119898)

le 119901 (119909119899 119909119899+1

) + 119901 (119909119899+1

119909119899+2

) + sdot sdot sdot + 119901 (119909119898minus1

119909119898)

le 119872119899119901 (1199090 1199091) + 119899119872

119899+ 119872119899+1

119901 (1199090 1199091)

+ (119899 + 1)119872119899+1

+ sdot sdot sdot + 119872119898minus1

119901 (1199090 1199091)

+ (119898 minus 1)119872119898minus1

le

119898minus1

sum

119894=119899

119872119894119901 (1199090 1199091)

+

119898minus1

sum

119894=119899

119894119872119894997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119872 lt 1)

(49)

By the definition of 119901119904 we get

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (50)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 119909lowast

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 119909lowast) = 0 Again from Lemma 2(4) we get

119901 (119909lowast 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909lowast) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0 (51)

119867119901(119879 (1199092119899+2

) 119878 (119909lowast))

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

119879 (1199092119899+2

)) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 119879 (1199092119899+2

))

1 + 119901 (1199092119899+2

119909lowast)

le 120572119901 (1199092119899+2

119909lowast)

+ 120573119901 (1199092119899+2

1199092119899+3

) 119901 (119909lowast 119878 (119909lowast))

1 + 119901 (1199092119899+2

119909lowast)

+ 120574119901 (1199092119899+2

119878 (119909lowast)) 119901 (119909

lowast 1199092119899+1

)

1 + 119901 (1199092119899+2

119909lowast)

(52)

taking limit as 119899 rarr infin and using (51) we get

lim119899rarr+infin

119867119901(119879 (1199092119899+2

) 119878 (119909lowast)) = 0 (53)

Now 1199092119899+1

isin 119879(1199092119899+2

) gives that

119901 (1199092119899+1

119878 (119909lowast)) le 119867

119901(119879 (1199092119899+2

) 119878 (119909lowast)) (54)

which implies that

lim119899rarr+infin

119901 (1199092119899+1

119878 (119909lowast)) = 0 (55)

On the other hand by (P4) we have

119901 (119909lowast 119878 (119909lowast)) le 119901 (119909

lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (119909lowast 1199092119899+2

) + 119901 (1199092119899+2

119878 (119909lowast))

(56)

Taking limit as 119899 rarr +infin and using (51) and (55) we obtain119901(119909lowast 119878(119909lowast)) = 0 Therefore from (51) (119901(119909lowast 119909lowast) = 0) we

obtain

119901 (119909lowast 119878 (119909lowast)) = 119901 (119909

lowast 119909lowast) (57)

which from Remark 14 implies that 119909lowast

isin 119878(119909lowast) = 119878(119909lowast)

Similarly one can easily prove that 119909lowast isin 119879(119909lowast) Thus 119878 and

119879 have a common fixed point

Remark 24 For 120573 = 120574 = 0 and 119878 = 119879 Theorem 23 reduces tothe following result of Aydi et al [9]

Corollary 25 (see [9 Theorem 32]) Let (119883 119901) be a partialmetric space If 119879 119883 rarr 119862119861

119901(119883) is a multivalued mapping

such that for all 119909 119910 isin 119883 one has

119867119901(119879119909 119879119910) le 119896119901 (119886 119887) (58)

where 119896 isin (0 1) Then 119879 has a fixed point

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Theorem26 Let (119883 119901) be a complete partialmetric space and119878 119879 119883 rarr 119862119861

119901(119883) be multivalued mappings such that

119867119901(119878119909 119879119910) le 120572

119901 (119909 119878119909) 119901 (119909 119879119910) + 119901 (119910 119879119910) 119901 (119910 119878119909)

119901 (119909 119879119910) + 119901 (119910 119878119909)

+ 120573119889 (119909 119910)

(59)

for all 119909 119910 isin 119883 0 le 120572 120573 with 2120572 + 120573 lt 1 and 119901(119909 119879119910) +

119901(119910 119878119909) = 0 Then 119878 and 119879 have a common fixed point

Proof Assume that 119897 = (120572+120573)(1minus120572) Let 1199090isin 119883 be arbitrary

but fixed element of119883 and choose 1199091isin 119878(1199090)

When 119901(119909 119879119910)+119901(119910 119878119909) = 0 By Lemma 8we can choose1199092isin 119879(119909

1) such that

119901 (1199091 1199092)

le 119867119901(119878 (1199090) 119879 (119909

1)) + (120572 + 120573)

le 120572119901 (1199090 1198781199090) 119901 (119909

0 1198791199091) + 119901 (119909

1 1198791199091) 119901 (119909

1 1198781199090)

119901 (1199090 1198791199091) + 119901 (119909

1 1198781199090)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572119901 (1199090 1199091) 119901 (119909

0 1199092) + 119901 (119909

1 1199092) 119901 (119909

1 1199091)

119901 (1199090 1199092) + 119901 (119909

1 1199091)

+ 120573119901 (1199090 1199091) + (120572 + 120573)

le 120572 119901 (1199090 1199091) + 119901 (119909

1 1199092) + 120573119901 (119909

0 1199091) + (120572 + 120573)

(60)

because 119901(1199090 1199092) lt 119901(119909

0 1199092) + 119901(119909

1 1199091) and 119901(119909

1 1199091) lt

119901(1199091 1199091) + 119901(119909

0 1199092) Thus we get

119901 (1199091 1199092) le (

120572 + 120573

1 minus 120572)119901 (119909

0 1199091) + (

120572 + 120573

1 minus 120572) (61)

It further implies that

119901 (1199091 1199092) le 119897119901 (119909

0 1199091) + 119897 (62)

By Lemma 8 we can choose 1199093isin 119878(1199092) such that

119901 (1199092 1199093)

le 119867119901(119879 (1199091) 119878 (119909

2)) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1198791199091) 119901 (119909

1 1198781199092) + 119901 (119909

2 1198781199092) 119901 (119909

2 1198791199091)

119901 (1199091 1198781199092) + 119901 (119909

2 1198791199091)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572119901 (1199091 1199092) 119901 (119909

1 1199093) + 119901 (119909

2 1199093) 119901 (119909

2 1199092)

119901 (1199091 1199093) + 119901 (119909

2 1199092)

+ 120573119901 (1199091 1199092) +

(120572 + 120573)2

1 minus 120572

le 120572 119901 (1199091 1199092) + 119901 (119909

2 1199093) + 120573119901 (119909

1 1199092) +

(120572 + 120573)2

1 minus 120572

(63)

because 119901(1199091 1199093) lt 119901(119909

1 1199093) + 119901(119909

2 1199092) and 119901(119909

2 1199092) lt

119901(1199092 1199092) + 119901(119909

1 1199093) Thus we get

119901 (1199092 1199093) le (

120572 + 120573

1 minus 120572)119901 (119909

1 1199092) + (

120572 + 120573

1 minus 120572)

2

(64)

It further implies that

119901 (1199092 1199093) le 1198972119901 (1199090 1199090) + 21198972 (65)

Continuing in this manner one can obtain a sequence 119909119899 in

119883 as 1199092119899+1

isin 119878(1199092119899) and 119909

2119899+2isin 119879(119909

2119899+1) such that

119901 (119909119899 119909119899+1

) le 119897119901 (119909119899minus1

119909119899) + 119897 le sdot sdot sdot le 119897

119899119901 (1199090 1199091) + 119899119897119899

(66)

where 119897 = ((120572 + 120573)(1 minus 120572)) lt 1 for all 119899 ge 0 Withoutloss of generality assume that 119898 gt 119899 Then using (66) andthe triangle inequality for partial metrics (P

4) one can easily

prove that

119901 (119909119899 119909119898) le

119898minus1

sum

119894=119899

119897119894119901 (1199090 1199091) +

119898minus1

sum

119894=119899

119894119897119894

997888rarr 0 as 119899 997888rarr +infin (since 0 lt 119897 lt 1)

(67)

By the definition of 119901119904

119901119904(119909119899 119909119898) le 2119901 (119909

119899 119909119898) 997888rarr 0 as 119899 997888rarr +infin (68)

This yields that 119909119899 is a Cauchy sequence in (119883 119901

119904) Since

(119883 119901) is complete then from Lemma 2(4) (119883 119901119904) is a com-

plete metric space Therefore the sequence 119909119899 converges

to some 1199091015840

isin 119883 with respect to the metric 119901119904 that is

lim119899rarr+infin

119901119904(119909119899 1199091015840) = 0 Again from Lemma 2(4) we get

119901 (1199091015840 1199091015840) = lim119899rarr+infin

119901 (119909119899 119909) = lim119899rarr+infin

119901 (119909119899 119909119899) = 0

(69)

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

119867119901(119879 (1199092119899+1

) 119878 (1199091015840))

le 120572 (119901 (1199092119899+1

119879 (1199092119899+1

)) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 119879 (1199092119899+1

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 119879 (1199092119899+1

)))minus1

+ 120573119901 (1199092119899+1

119909)

le 120572 (119901 ((1199092119899+1

1199092119899+2

) 119901 (1199092119899+1

119878 (1199091015840))

+119901 (1199091015840 119878 (1199091015840)) 119901 (119909

1015840 1199092119899+2

)))

times(119901 (1199092119899+1

119878 (1199091015840)) + 119901 (119909

1015840 1199092119899+2

))minus1

+ 120573119901 (1199092119899+1

119909)

(70)

therefore

lim119899rarr+infin

119867119901(119879 (1199092119899+1

) 119878 (1199091015840)) = 0 (71)

Now 1199092119899+2

isin 119879(1199092119899+1

) gives that

119901 (1199092119899+2

119878 (1199091015840)) le 119867

119901(119879 (1199092119899+1

) 119878 (1199091015840)) (72)

which implies that

lim119899rarr+infin

119901 (1199092119899+2

119878 (1199091015840)) = 0 (73)

On the other hand we have

119901 (1199091015840 119878 (1199091015840)) le 119901 (119909

1015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

minus 119901 (1199092119899+2

1199092119899+2

)

le 119901 (1199091015840 1199092119899+2

) + 119901 (1199092119899+2

119878 (1199091015840))

(74)

Taking limit as 119899 rarr +infin and using (69) and (73) we obtain119901(1199091015840 119878(1199091015840)) = 0 Therefore from (69) (119901(1199091015840 1199091015840) = 0) we

obtain

119901 (1199091015840 119878 (1199091015840)) = 119901 (119909

1015840 1199091015840) (75)

which from Remark 14 implies that 1199091015840

isin 119878(1199091015840) = 119878(1199091015840)

It follows similarly that 1199091015840

isin 119879(1199091015840) Thus 119878 and 119879 have a

common fixed point

Now we give an example which illustrates our Theo-rem 26

Example 27 Let 119883 = 1 2 3 be endowed with usual orderand let 119901 be a partial metric on119883 defined as

119901 (1 1) = 119901 (2 2) = 0 119901 (3 3) =5

11

119901 (1 2) = 119901 (2 1) =3

10

119901 (1 3) = 119901 (3 1) =9

20

119901 (2 3) = 119901 (3 2) =1

2

(76)

Define the mappings 119878 119879 119883 rarr 119862119861119901(119883) by

119878119909 = 1 if 119909 119910 isin 1 2

1 2 otherwise

119879119909 = 1 if 119909 119910 isin 1 2

2 otherwise

(77)

Note that 119878119909 and 119879119909 are closed and bounded for all 119909 isin 119883

with respect to the partial metric 119901 To show that for all 119909 119910in119883 (59) is satisfied with 120572 = 111 120573 = 45 we consider thefollowing cases if 119909 119910 isin 1 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 1) = 0 (78)

and condition (59) is satisfied obviously

If 119909 = 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 2) =

3

10

119901 (119909 119878119909) = 119901 (119910 119878119909) = 119901 (3 1 2) =9

20

119901 (119909 119879119910) = 119901 (119910 119879119910) = 119901 (119910 119878119909) = 119901 (3 2) =11

24

119901 (119909 119910) = 119901 (3 3) =5

11

(79)

If 119909 = 3 119910 = 1 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (1 1) = 0 119901 (119910 119878119909) = 119901 (1 1 2) = 0

119901 (119909 119910) = 119901 (3 1) =9

20

(80)

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

If 119909 = 3 119910 = 2 then

119867119901(119878119909 119879119910) = 119867

119901(1 2 1) =

3

10

119901 (119909 119878119909)=119901 (3 1 2)=9

20 119901 (119909 119879119910)=119901 (3 1)=

9

20

119901 (119910 119879119910) = 119901 (2 1) =3

10 119901 (119910 119878119909) = 119901 (2 1 2) = 0

119901 (119909 119910) = 119901 (3 2) =1

2

(81)

If 119909 = 1 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (1 1) = 0 119901 (119909 119879119910) = 119901 (1 2) =3

10

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (1 3) =9

20

(82)

If 119909 = 2 119910 = 3 then

119867119901(119878119909 119879119910) = 119867

119901(1 2) =

3

10

119901 (119909 119878119909) = 119901 (2 1) =3

10 119901 (119909 119879119910) = 119901 (2 2) = 0

119901 (119910 119879119910) = 119901 (3 2) =1

2 119901 (119910 119878119909) = 119901 (3 1) =

9

20

119901 (119909 119910) = 119901 (2 3) =1

2

(83)

Thus all the conditions ofTheorem 26 are satisfied Here 119909 =

1 is a common fixed point of 119878 and 119879On the other hand the metric 119901

119904 induced by the partialmetric 119901 is given by

119901119904(1 1) = 119901

119904(2 2) = 119901

119904(3 3) = 0

119901119904(2 1) = 119901

119904(2 1) =

3

4

119901119904(3 2) = 119901

119904(2 3) =

4

7

119901119904(3 1) = 119901

119904(1 3) =

14

29

(84)

Note that in case of ordinary Hausdorff metric given map-ping does not satisfy the condition Indeed for 119909 = 1 and119910 = 3 we have

119867(119878119909 119879119910) = 119867 (1 2) =3

4

119901119904(119909 119878119909) = 119901

119904(1 1) = 0

119901119904(119909 119879119910) = 119901

119904(1 2) =

3

4

119901119904(119910 119879119910) = 119901

119904(3 2) =

4

7

119901119904(119910 119878119909) = 119901

119904(3 1) =

14

29

(85)

for the values of 120572 = 111 120573 = 45 By a routine calculationone can easily verify that the mapping does not satisfy thecondition which involved ordinary Hausdorff metric

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support

References

[1] S G Matthews ldquoPartial metric topologyrdquo in Proceedings of the8th Summer Conference on General Topology and Applicationsvol 728 pp 183ndash197 New York Academy of Science 1994

[2] T Abdeljawad E Karapinar andK Tas ldquoA generalized contrac-tion principle with control functions on partial metric spacesrdquoComputers amp Mathematics with Applications vol 63 no 3 pp716ndash719 2012

[3] T Abdeljawad ldquoFixed points for generalized weakly contractivemappings in partialmetric spacesrdquoMathematical andComputerModelling vol 54 no 11-12 pp 2923ndash2927 2011

[4] J Ahmad M Arshad and C Vetro ldquoOn a theorem of Khan in ageneralized metric spacerdquo International Journal of Analysis vol2013 Article ID 852727 6 pages 2013

[5] S Alsulami N Hussain and A Alotaibi ldquoCoupled fixed andcoincidence points for monotone operators in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article173 2012

[6] I Altun and A Erduran ldquoFixed point theorems for monotonemappings on partial metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 508730 2011

[7] M Arshad J Ahmad and E Karapinar ldquoSome common fixedpoint results in rectangularmetric spacesrdquo International Journalof Analysis vol 2013 Article ID 307234 7 pages 2013

[8] M Arshad E Karapinar and J Ahmad ldquoSome unique fixedpoint theorem for rational contractions in partially orderedmetric spacesrdquo Journal of Inequalities andApplications vol 2013article 248 2013

[9] H Aydi M Abbas and C Vetro ldquoPartial Hausdorff metric andNadlerrsquos fixed point theorem on partial metric spacesrdquo Topologyand Its Applications vol 159 no 14 pp 3234ndash3242 2012

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

[10] H Aydi M Abbas and C Vetro ldquoCommon Fixed points formultivalued generalized contractions on partial metric spacesrdquoRevista de la Real Academia de Ciencias Exactas Fisicas yNaturales A 2013

[11] A Azam and M Arshad ldquoFixed points of a sequence of locallycontractive multivalued mapsrdquo Computers amp Mathematics withApplications vol 57 no 1 pp 96ndash100 2009

[12] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[13] S Banach ldquoSur les operations dans les ensembles abstraitset leurs applications aux equations integralesrdquo FundamentaMathematicae vol 3 pp 133ndash181 1922

[14] I Beg and A Azam ldquoFixed points of asymptotically regularmultivalued mappingsrdquo Journal of the Australian MathematicalSociety A vol 53 no 3 pp 313ndash326 1992

[15] V W Bryant ldquoA remark on a fixed-point theorem for iteratedmappingsrdquo The American Mathematical Monthly vol 75 pp399ndash400 1968

[16] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012

[17] N Hussain H K Nashine Z Kadelburg and S M AlsulamildquoWeakly isotone increasingmappings and endpoints in partiallyordered metric spacesrdquo Journal of Inequalities and Applicationsvol 2012 article 232 2012

[18] NHussain DDJoric Z Kadelburg and S Radenovic ldquoSuzuki-type fixed point results in metric type spacesrdquo Fixed PointTheory and Applications vol 2012 article 126 2012

[19] N Hussain Z Kadelburg S Radenovic and F R Al-SolamyldquoComparison functions and fixed point results in partial metricspacesrdquo Abstract and Applied Analysis vol 2012 Article ID605781 15 pages 2012

[20] N Hussain and M Abbas ldquoCommon fixed point results fortwo new classes of hybrid pairs in symmetric spacesrdquo AppliedMathematics and Computation vol 218 no 2 pp 542ndash547 2011

[21] S Hong ldquoFixed points of multivalued operators in orderedmetric spaces with applicationsrdquo Nonlinear Analysis TheoryMethods amp Applications vol 72 no 11 pp 3929ndash3942 2010

[22] S Hong ldquoFixed points for mixed monotone multivalued oper-ators in Banach spaces with applicationsrdquo Journal of Mathemat-ical Analysis and Applications vol 337 no 1 pp 333ndash342 2008

[23] S Hong ldquoFixed points of discontinuous multivalued increasingoperators in Banach spaces with applicationsrdquo Journal of Math-ematical Analysis and Applications vol 282 no 1 pp 151ndash1622003

[24] M Imdad J Ali and M Tanveer ldquoCoincidence and commonfixed point theorems for nonlinear contractions in Menger PMspacesrdquo Chaos Solitons amp Fractals vol 42 no 5 pp 3121ndash31292009

[25] E Karapinar ldquoCiric type nonunique fixed point theorems onpartial metric spacesrdquo The Journal of Nonlinear Science andApplications vol 5 no 2 pp 74ndash83 2012

[26] E Karapnar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011

[27] E KarapinarMMarudai andV Pragadeeshwarar ldquoFixed pointtheorems for generalized weak contractions satisfying rationalexpressions on ordered Partial Metric Spacerdquo LobachevskiiJournal of Mathematics vol 34 no 1 pp 116ndash123 2013

[28] M S Khan M Swaleh and M Imdad Some Fixed PointTheorems IV Communications 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of