research article faster multistep iterations for the ...fixed points applied to zamfirescu operators...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 464593 4 pageshttpdxdoiorg1011552013464593

Research ArticleFaster Multistep Iterations for the Approximation ofFixed Points Applied to Zamfirescu Operators

Shin Min Kang1 Ljubomir B SiriT2 Arif Rafiq3 Faisal Ali4 and Young Chel Kwun5

1 Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea2 Faculty of Mechanical Engineering University of Belgrade Kraljice Marije 16 Belgrade 11000 Serbia3 Department of Mathematics Lahore Leads University Lahore 54810 Pakistan4Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University Multan 54000 Pakistan5 Department of Mathematics Dong-A University Pusan 614-714 Republic of Korea

Correspondence should be addressed to Young Chel Kwun yckwundauackr

Received 13 July 2013 Accepted 7 September 2013

Academic Editor Salvador Romaguera

Copyright copy 2013 Shin Min Kang et alThis 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

By taking a counterexample we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processesfor Zamfirescu operators

1 Introduction

Let 119862 be a nonempty convex subset of a normed space 119864 andlet 119879 119862 rarr 119862 be a mapping

(a) For arbitrary 1199090isin 119862 the sequence 119909

119899 defined by

119909119899+1

= (1 minus 119887119899) 119909119899+ 119887119899119879119909119899 119899 ge 0 (119872

119899)

where 119887119899 is a sequence in [0 1] is known as the Mann

iteration process [1](b) For arbitrary 119909

0isin 119862 the sequence 119909

119899 defined by

119909119899+1

= (1 minus 119887119899) 119909119899+ 119887119899119879119910119899

119910119899= (1 minus 119887

1015840

119899) 119909119899+ 1198871015840

119899119879119909119899 119899 ge 0

(119868119899)

where 119887119899 and 119887

1015840

119899 are sequences in [0 1] is known as the

Ishikawa iteration process [2](c) For arbitrary 119909

0isin 119862 the sequence 119909

119899 defined by

119909119899+1

= (1 minus 119887119899) 119909119899+ 1198871198991198791199101

119899

119910119894

119899= (1 minus 119887

119894

119899) 119909119899+ 119887119894

119899119879119910119894+1

119899

119910119901minus1

119899= (1 minus 119887

119901minus1

119899) 119909119899+ 119887119901minus1

119899119879119899119909119899 119899 ge 0

(119877119878119899)

where 119887119899 and 119887119894

119899 119894 = 1 2 119901 minus 2 (119901 ge 2) are sequences

in [0 1] and denoted by (119877119878119899) is known as the multistep

iteration process [3]

Definition 1 (see [4]) Suppose that 119886119899 and 119887

119899 are two real

convergent sequences with limits 119886 and 119887 respectively Then119886119899 is said to converge faster than 119887

119899 if

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119886119899minus 119886

119887119899minus 119887

10038161003816100381610038161003816100381610038161003816

= 0 (1)

Definition 2 (see [4]) Let 119906119899 and V

119899 be two fixed-point

iteration procedures which both converge to the same fixedpoint 119901 say with error estimates

1003817100381710038171003817119906119899 minus 1199011003817100381710038171003817 le 119886119899

1003817100381710038171003817V119899 minus 1199011003817100381710038171003817 le 119887119899 119899 ge 0 (2)

where lim 119886119899= 0 = lim 119887

119899 If 119886119899 converges faster than 119887

119899

then 119906119899 is said to converge faster than V

119899

Theorem 3 (see [5]) Let (119883 119889) be a complete metric spaceand let 119879 119883 rarr 119883 be a mapping for which there exist realnumbers 119886 119887 and 119888 satisfying 119886 isin (0 1) and 119887 119888 isin (0 12) suchthat for each pair 119909 119910 isin 119883 at least one of the following is true

(1199111) 119889(119879119909 119879119910) le 119886119889(119909 119910)

2 Abstract and Applied Analysis

(1199112) 119889(119879119909 119879119910) le 119887[119889(119909 119879119909) + 119889(119910 119879119910)](1199113) 119889(119879119909 119879119910) le 119888[119889(119909 119879119910) + 119889(119910 119879119909)]

Then119879 has a unique fixed point119901 and the Picard iteration119909119899 defined by

119909119899+1

= 119879119909119899 119899 ge 0 (3)

converges to 119901 for any 1199090isin 119883

Remark 4 An operator 119879 which satisfies the contractionconditions (1199111)ndash(1199113) of Theorem 3 will be called a Zam-firescu operator [4 6 7]

In [6 7] Berinde introduced a new class of operators ona normed space 119864 satisfying

1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 120575

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 119871 119879119909 minus 119909 (4)

for any 119909 119910 isin 119864 0 le 120575 lt 1 and 119871 ge 0He proved that this class is wider than the class of

Zamfirescu operatorsThe following results are proved in [6 7]

Theorem5 (see [7]) Let119862 be a nonempty closed convex subsetof a normed space 119864 Let 119879 119862 rarr 119862 be an operator satisfying(4) Let 119909

119899 be defined through the iterative process (119872

119899) and

1199090isin 119862 If 119865(119879) = 0 and sum

infin

119899=0119887119899= infin then 119909

119899 converges

strongly to the unique fixed point of 119879

Theorem6 (see [6]) Let119862 be a nonempty closed convex subsetof an arbitrary Banach space 119864 and let 119879 119862 rarr 119862 be anoperator satisfying (4) Let 119909

119899 be defined through the iterative

process (119868119899) and 119909

0isin 119862 where 119887

119899 and 1198871015840

119899 are sequences of

positive numbers in [0 1] with 119887119899 satisfying suminfin

119899=0119887119899= infin

Then 119909119899 converges strongly to the fixed point of 119879

The following result can be found in [8]

Theorem 7 Let 119862 be a closed convex subset of an arbitraryBanach space 119864 Let theMann and Ishikawa iteration processeswith real sequences 119887

119899 and 1198871015840

119899 satisfy 0 le 119887

119899 1198871015840119899le 1 and

suminfin

119899=0119887119899= infin Then (119872

119899) and (119868

119899) converge strongly to the

unique fixed point of 119879 Let 119879 119862 rarr 119862 be a Zamfirescuoperator and moreover the Mann iteration process convergesfaster than the Ishikawa iteration process to the fixed point of119879

In [4] Berinde proved the following result

Theorem 8 Let 119862 be a closed convex subset of an arbitraryBanach space 119864 and let 119879 119862 rarr 119862 be a Zamfirescu operatorLet 119910

119899 be defined by (119872

119899) and 119910

0isin 119862 with a sequence 119887

119899

in [0 1] satisfying suminfin119899=0

119887119899= infin Then 119910

119899 converges strongly

to the fixed point of 119879 and moreover the Picard iteration 119909119899

converges faster than the Mann iteration

Remark 9 In [9] Qing and Rhoades by taking a counterex-ample showed that the Mann iteration process convergesmore slowly than the Ishikawa iteration process for Zam-firescu operators

In this paper we establish a general theorem to approxi-mate fixed points of quasi-contractive operators in a Banachspace through themultistep iteration process Our result gen-eralizes and improves upon amongothers the correspondingresults of Babu and Vara Prasad [8] and Berinde [4 6 7]

We also prove that the Mann iteration process and theIshikawa iteration process converge more slowly than themultistep iteration process for Zamfirescu operators

2 Main Results

We now prove our main results

Theorem 10 Let 119862 be a nonempty closed convex subset of anarbitrary Banach space 119864 and let 119879 119862 rarr 119862 be an operatorsatisfying (4) Let 119909

119899 be defined through the iterative process

(119877119878119899) and 119909

0isin 119862 where 119887

119899 and 119887119894

119899 119894 = 1 2 119901minus2 (119901 ge

2) are sequences in [0 1] with suminfin119899=0

119887119899= infin If 119865(119879) = 0 then

119865(119879) is a singleton and the sequence 119909119899 converges strongly to

the fixed point of 119879

Proof Assume that 119865(119879) = 0 and 119908 isin 119865(119879) Then using(119877119878119899) we have

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 =

10038171003817100381710038171003817(1 minus 119887

119899) 119909119899+ 1198871198991198791199101

119899minus 119908

10038171003817100381710038171003817

=10038171003817100381710038171003817(1 minus 119887

119899) (119909119899minus 119908) + 119887

119899(1198791199101

119899minus 119908)

10038171003817100381710038171003817

le (1 minus 119887119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 119887119899

100381710038171003817100381710038171198791199101

119899minus 119908

10038171003817100381710038171003817

(5)

Now for 119909 = 119908 and 119910 = 1199101

119899 (4) gives

100381710038171003817100381710038171198791199101

119899minus 119908

10038171003817100381710038171003817le 120575

100381710038171003817100381710038171199101

119899minus 119908

10038171003817100381710038171003817 (6)

By substituting (6) in (5) we obtain

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 le (1 minus 119887

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119899

100381710038171003817100381710038171199101

119899minus 119908

10038171003817100381710038171003817 (7)

In a similar fashion again by using (119877119878119899) we can get

10038171003817100381710038171003817119910119894

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

119894

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119894

119899

10038171003817100381710038171003817119910119894+1

119899minus 119908

10038171003817100381710038171003817 (8)

where 119894 = 1 2 119901 minus 2 (119901 ge 2) and

10038171003817100381710038171003817119910119901minus1

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus1

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (9)

It can be easily seen that for 119894 = 1 2 119901 minus 2 (119901 ge 2) wehave

100381710038171003817100381710038171199101

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

1

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 1205751198871

119899

100381710038171003817100381710038171199102

119899minus 119908

10038171003817100381710038171003817

10038171003817100381710038171003817119910119901minus3

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

119901minus3

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119901minus3

119899

10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817

10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

119901minus2

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119901minus2

119899

10038171003817100381710038171003817119910119901minus1

119899minus 119908

10038171003817100381710038171003817

(10)

Abstract and Applied Analysis 3

Substituting (9) in (10) gives us

10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus2

119899(1 + 120575119887

119901minus1

119899))1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (11)

It may be noted that for 120575 isin [0 1) and 120578119899 isin [0 1] the

following inequality is always true

1 le 1 + 120575120578119899le 1 + 120575 (12)

From (11) and (12) we get

10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus2

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (13)

By repeating the same procedure finally from (7) and (10) weyield

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 le [1 minus (1 minus 120575) 119887

119899]1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (14)

By (14) we inductively obtain

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 le

119899

prod

119896=0

[1 minus (1 minus 120575) 119887119896]10038171003817100381710038171199090 minus 119908

1003817100381710038171003817 119899 ge 0 (15)

Using the fact that 0 le 120575 lt 1 0 le 119887119899le 1 andsuminfin

119899=0119887119899= infin it

results that

lim119899rarrinfin

119899

prod

119896=0

[1 minus (1 minus 120575) 119887119896] = 0 (16)

which by (15) implies that

lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 = 0 (17)

Consequently 119909119899rarr 119908 isin 119865 and this completes the proof

Now by a counterexample we prove that the multistepiteration process is faster than the Mann and Ishikawaiteration processes for Zamfirescu operators

Example 11 Suppose that 119879 [0 1] rarr [0 1] is defined by119879119909 = (12)119909 119887

119899= 0 = 119887

119901minus1

119899= 119887119894

119899 119894 = 1 2 119901 minus 1 (119901 ge

2) and 119899 = 1 2 15 119887119899= 4radic119899 = 119887

119901minus1

119899= 119887119894

119899 119894 =

1 2 119901 minus 1 (119901 ge 2) and 119899 ge 16 It is clear that 119879 is aZamfirescu operator with a unique fixed point 0 and that allof the conditions ofTheorem 10 are satisfied Also119872

119899= 1199090=

119868119899= 119877119878119899 119899 = 1 2 15 Suppose that 119909

0= 0 For the Mann

and Ishikawa iteration processes we have

119872119899= (1 minus 119887

119899) 119909119899+ 119887119899119879119909119899

= (1 minus4

radic119899)119909119899+

4

radic119899

1

2119909119899

= (1 minus2

radic119899)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus2

radic119894) 1199090

119868119899= (1 minus 119887

119899) 119909119899+ 119887119899119879 ((1 minus 119887

1015840

119899) 119909119899+ 1198871015840

119899119879119909119899)

= (1 minus4

radic119899)119909119899+

4

radic119899

1

2(1 minus

2

radic119899)119909119899

= (1 minus2

radic119899minus4

119899)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus2

radic119894minus4

119894) 1199090

119877119878119899= (1 minus 119887

119899) 119909119899+ 1198871198991198791199101

119899

119910119894

119899= (1 minus 119887

119894

119899) 119909119899+ 119887119894

119899119879119910119894+1

119899

119910119901minus1

119899= (1 minus 119887

119901minus1

119899) 119909119899+ 119887119901minus1

119899119879119909119899 119899 ge 0

(18)where 119894 = 1 2 119901 minus 2 (119901 ge 2) implies that

119877119878119899= (1 minus

119901

sum

119895=1

(2

radic119899)

119895

)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus

119901

sum

119895=1

(2

radic119894)

119895

)1199090

(19)

Now consider

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119872119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899

119894=16(1 minus (2radic119894)) 119909

0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899

119894=16(1 minus (2radic119894))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

1 minus (2radic119894)

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

radic119894

radic119894 minus 2)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(20)

It is easy to see that

0 le lim119899rarrinfin

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

radic119894


le lim119899rarrinfin

119899

prod

119894=16

(1 minus1

119894)

= lim119899rarrinfin

15

119899

= 0

(21)

Hence

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119872119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (22)


Thus theMann iteration process converges more slowly thanthe multistep iteration process to the fixed point 0 of 119879

Similarly

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899

119894=16(1 minus (2radic119894) minus (4119894)) 119909

0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899

119894=16(1 minus (2radic119894) minus (4119894))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

1 minus (2radic119894) minus (4119894)

)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894

119894radic119894 minus 2119894 minus 4radic119894)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

with


119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894



119899

prod

119894=16

(1 minus1

119894)


15

119899

= 0

(24)

implies that

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (25)

Thus the Ishikawa iteration process converges more slowlythan the multistep iteration process to the fixed point 0 of 119879

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the editor and the refereesfor their useful comments and suggestions This study wassupported by research funds from Dong-A University

References

[1] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953

[2] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974

[3] B E Rhoades and S M Soltuz ldquoThe equivalence betweenMann-Ishikawa iterations and multistep iterationrdquo NonlinearAnalysis Theory Methods amp Applications vol 58 no 1-2 pp219ndash228 2004

[4] V Berinde ldquoPicard iteration converges faster than Mann iter-ation for a class of quasi-contractive operatorsrdquo Fixed PointTheory and Applications no 2 pp 97ndash105 2004

[5] T Zamfirescu ldquoFix point theorems inmetric spacesrdquoArchiv derMathematik vol 23 pp 292ndash298 1972

[6] V Berinde ldquoOn the convergence of the Ishikawa iteration inthe class of quasi contractive operatorsrdquoActaMathematica Uni-versitatis Comenianae vol 73 no 1 pp 119ndash126 2004

[7] V Berinde ldquoA convergence theorem for some mean value fixedpoint iteration proceduresrdquoDemonstratio Mathematica vol 38no 1 pp 177ndash184 2005

[8] G V R Babu and K N V V Vara Prasad ldquoMann iteration con-verges faster than Ishikawa iteration for the class of Zamfirescuoperatorsrdquo Fixed Point Theory and Applications vol 2006Article ID 49615 6 pages 2006

[9] Y Qing and B E Rhoades ldquoComments on the rate of con-vergence between Mann and Ishikawa iterations applied toZamfirescu operatorsrdquo Fixed PointTheory and Applications vol2008 Article ID 387504 3 pages 2008

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Stochastic AnalysisInternational Journal of


(1199112) 119889(119879119909 119879119910) le 119887[119889(119909 119879119909) + 119889(119910 119879119910)](1199113) 119889(119879119909 119879119910) le 119888[119889(119909 119879119910) + 119889(119910 119879119909)]

Then119879 has a unique fixed point119901 and the Picard iteration119909119899 defined by

119909119899+1

= 119879119909119899 119899 ge 0 (3)

converges to 119901 for any 1199090isin 119883

Remark 4 An operator 119879 which satisfies the contractionconditions (1199111)ndash(1199113) of Theorem 3 will be called a Zam-firescu operator [4 6 7]

In [6 7] Berinde introduced a new class of operators ona normed space 119864 satisfying

1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 120575

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + 119871 119879119909 minus 119909 (4)

for any 119909 119910 isin 119864 0 le 120575 lt 1 and 119871 ge 0He proved that this class is wider than the class of

Zamfirescu operatorsThe following results are proved in [6 7]

Theorem5 (see [7]) Let119862 be a nonempty closed convex subsetof a normed space 119864 Let 119879 119862 rarr 119862 be an operator satisfying(4) Let 119909

119899 be defined through the iterative process (119872

119899) and

1199090isin 119862 If 119865(119879) = 0 and sum

infin

119899=0119887119899= infin then 119909

119899 converges

strongly to the unique fixed point of 119879

Theorem6 (see [6]) Let119862 be a nonempty closed convex subsetof an arbitrary Banach space 119864 and let 119879 119862 rarr 119862 be anoperator satisfying (4) Let 119909

119899 be defined through the iterative

process (119868119899) and 119909

0isin 119862 where 119887

119899 and 1198871015840

119899 are sequences of

positive numbers in [0 1] with 119887119899 satisfying suminfin

119899=0119887119899= infin

Then 119909119899 converges strongly to the fixed point of 119879

The following result can be found in [8]

Theorem 7 Let 119862 be a closed convex subset of an arbitraryBanach space 119864 Let theMann and Ishikawa iteration processeswith real sequences 119887

119899 and 1198871015840

119899 satisfy 0 le 119887

119899 1198871015840119899le 1 and

suminfin

119899=0119887119899= infin Then (119872

119899) and (119868

119899) converge strongly to the

unique fixed point of 119879 Let 119879 119862 rarr 119862 be a Zamfirescuoperator and moreover the Mann iteration process convergesfaster than the Ishikawa iteration process to the fixed point of119879

In [4] Berinde proved the following result

Theorem 8 Let 119862 be a closed convex subset of an arbitraryBanach space 119864 and let 119879 119862 rarr 119862 be a Zamfirescu operatorLet 119910

119899 be defined by (119872

119899) and 119910

0isin 119862 with a sequence 119887

119899

in [0 1] satisfying suminfin119899=0

119887119899= infin Then 119910

119899 converges strongly

to the fixed point of 119879 and moreover the Picard iteration 119909119899

converges faster than the Mann iteration

Remark 9 In [9] Qing and Rhoades by taking a counterex-ample showed that the Mann iteration process convergesmore slowly than the Ishikawa iteration process for Zam-firescu operators

In this paper we establish a general theorem to approxi-mate fixed points of quasi-contractive operators in a Banachspace through themultistep iteration process Our result gen-eralizes and improves upon amongothers the correspondingresults of Babu and Vara Prasad [8] and Berinde [4 6 7]

We also prove that the Mann iteration process and theIshikawa iteration process converge more slowly than themultistep iteration process for Zamfirescu operators

2 Main Results

We now prove our main results

Theorem 10 Let 119862 be a nonempty closed convex subset of anarbitrary Banach space 119864 and let 119879 119862 rarr 119862 be an operatorsatisfying (4) Let 119909

119899 be defined through the iterative process

(119877119878119899) and 119909

0isin 119862 where 119887

119899 and 119887119894

119899 119894 = 1 2 119901minus2 (119901 ge

2) are sequences in [0 1] with suminfin119899=0

119887119899= infin If 119865(119879) = 0 then

119865(119879) is a singleton and the sequence 119909119899 converges strongly to

the fixed point of 119879

Proof Assume that 119865(119879) = 0 and 119908 isin 119865(119879) Then using(119877119878119899) we have

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 =

10038171003817100381710038171003817(1 minus 119887

119899) 119909119899+ 1198871198991198791199101

119899minus 119908

10038171003817100381710038171003817

=10038171003817100381710038171003817(1 minus 119887

119899) (119909119899minus 119908) + 119887

119899(1198791199101

119899minus 119908)

10038171003817100381710038171003817

le (1 minus 119887119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 119887119899

100381710038171003817100381710038171198791199101

119899minus 119908

10038171003817100381710038171003817

(5)

Now for 119909 = 119908 and 119910 = 1199101

119899 (4) gives

100381710038171003817100381710038171198791199101

119899minus 119908

10038171003817100381710038171003817le 120575

100381710038171003817100381710038171199101

119899minus 119908

10038171003817100381710038171003817 (6)

By substituting (6) in (5) we obtain

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 le (1 minus 119887

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119899

100381710038171003817100381710038171199101

119899minus 119908

10038171003817100381710038171003817 (7)

In a similar fashion again by using (119877119878119899) we can get

10038171003817100381710038171003817119910119894

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

119894

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119894

119899

10038171003817100381710038171003817119910119894+1

119899minus 119908

10038171003817100381710038171003817 (8)

where 119894 = 1 2 119901 minus 2 (119901 ge 2) and

10038171003817100381710038171003817119910119901minus1

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus1

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (9)

It can be easily seen that for 119894 = 1 2 119901 minus 2 (119901 ge 2) wehave

100381710038171003817100381710038171199101

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

1

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 1205751198871

119899

100381710038171003817100381710038171199102

119899minus 119908

10038171003817100381710038171003817

10038171003817100381710038171003817119910119901minus3

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

119901minus3

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119901minus3

119899

10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817

10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus 119887

119901minus2

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 + 120575119887119901minus2

119899

10038171003817100381710038171003817119910119901minus1

119899minus 119908

10038171003817100381710038171003817

(10)



10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus2

119899(1 + 120575119887

119901minus1

119899))1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (11)



1 le 1 + 120575120578119899le 1 + 120575 (12)


10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus2

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (13)



119899]1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (14)


1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 le

119899

prod

119896=0


1003817100381710038171003817 119899 ge 0 (15)


119899=0119887119899= infin it

results that

lim119899rarrinfin

119899

prod

119896=0

[1 minus (1 minus 120575) 119887119896] = 0 (16)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 = 0 (17)




119899= 0 = 119887

119901minus1

119899= 119887119894

119899 119894 = 1 2 119901 minus 1 (119901 ge

2) and 119899 = 1 2 15 119887119899= 4radic119899 = 119887

119901minus1

119899= 119887119894

119899 119894 =


119899= 1199090=

119868119899= 119877119878119899 119899 = 1 2 15 Suppose that 119909

0= 0 For the Mann


119872119899= (1 minus 119887

119899) 119909119899+ 119887119899119879119909119899

= (1 minus4

radic119899)119909119899+

4

radic119899

1

2119909119899

= (1 minus2

radic119899)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus2

radic119894) 1199090

119868119899= (1 minus 119887

119899) 119909119899+ 119887119899119879 ((1 minus 119887

1015840

119899) 119909119899+ 1198871015840

119899119879119909119899)

= (1 minus4

radic119899)119909119899+

4

radic119899

1

2(1 minus

2

radic119899)119909119899

= (1 minus2

radic119899minus4

119899)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus2

radic119894minus4

119894) 1199090

119877119878119899= (1 minus 119887

119899) 119909119899+ 1198871198991198791199101

119899

119910119894

119899= (1 minus 119887

119894

119899) 119909119899+ 119887119894

119899119879119910119894+1

119899

119910119901minus1

119899= (1 minus 119887

119901minus1

119899) 119909119899+ 119887119901minus1

119899119879119909119899 119899 ge 0


119877119878119899= (1 minus

119901

sum

119895=1

(2

radic119899)

119895

)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus

119901

sum

119895=1

(2

radic119894)

119895

)1199090

(19)

Now consider

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119872119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899

119894=16(1 minus (2radic119894)) 119909

0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899

119894=16(1 minus (2radic119894))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896


)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

radic119894


100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(20)



119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

radic119894



119899

prod

119894=16

(1 minus1

119894)


15

119899

= 0

(21)

Hence

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119872119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (22)



Similarly

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899


0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896


)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894


100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

with


119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894



119899

prod

119894=16

(1 minus1

119894)


15

119899

= 0

(24)

implies that

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (25)




Acknowledgments


References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus2

119899(1 + 120575119887

119901minus1

119899))1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (11)



1 le 1 + 120575120578119899le 1 + 120575 (12)


10038171003817100381710038171003817119910119901minus2

119899minus 119908

10038171003817100381710038171003817le (1 minus (1 minus 120575) 119887

119901minus2

119899)1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (13)



119899]1003817100381710038171003817119909119899 minus 119908

1003817100381710038171003817 (14)


1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 le

119899

prod

119896=0


1003817100381710038171003817 119899 ge 0 (15)


119899=0119887119899= infin it

results that

lim119899rarrinfin

119899

prod

119896=0

[1 minus (1 minus 120575) 119887119896] = 0 (16)


lim119899rarrinfin

1003817100381710038171003817119909119899+1 minus 1199081003817100381710038171003817 = 0 (17)




119899= 0 = 119887

119901minus1

119899= 119887119894

119899 119894 = 1 2 119901 minus 1 (119901 ge

2) and 119899 = 1 2 15 119887119899= 4radic119899 = 119887

119901minus1

119899= 119887119894

119899 119894 =


119899= 1199090=

119868119899= 119877119878119899 119899 = 1 2 15 Suppose that 119909

0= 0 For the Mann


119872119899= (1 minus 119887

119899) 119909119899+ 119887119899119879119909119899

= (1 minus4

radic119899)119909119899+

4

radic119899

1

2119909119899

= (1 minus2

radic119899)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus2

radic119894) 1199090

119868119899= (1 minus 119887

119899) 119909119899+ 119887119899119879 ((1 minus 119887

1015840

119899) 119909119899+ 1198871015840

119899119879119909119899)

= (1 minus4

radic119899)119909119899+

4

radic119899

1

2(1 minus

2

radic119899)119909119899

= (1 minus2

radic119899minus4

119899)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus2

radic119894minus4

119894) 1199090

119877119878119899= (1 minus 119887

119899) 119909119899+ 1198871198991198791199101

119899

119910119894

119899= (1 minus 119887

119894

119899) 119909119899+ 119887119894

119899119879119910119894+1

119899

119910119901minus1

119899= (1 minus 119887

119901minus1

119899) 119909119899+ 119887119901minus1

119899119879119909119899 119899 ge 0


119877119878119899= (1 minus

119901

sum

119895=1

(2

radic119899)

119895

)119909119899

= sdot sdot sdot =

119899

prod

119894=16

(1 minus

119901

sum

119895=1

(2

radic119894)

119895

)1199090

(19)

Now consider

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119872119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899

119894=16(1 minus (2radic119894)) 119909

0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899

119894=16(1 minus (2radic119894))

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896


)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

radic119894


100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(20)



119899

prod

119894=16

(1 minus

sum119901

119896=2(2radic119894)

119896

radic119894



119899

prod

119894=16

(1 minus1

119894)


15

119899

= 0

(21)

Hence

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119872119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (22)



Similarly

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899


0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896


)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894


100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

with


119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894



119899

prod

119894=16

(1 minus1

119894)


15

119899

= 0

(24)

implies that

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (25)




Acknowledgments


References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Similarly

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

) 1199090

prod119899


0

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

prod119899

119894=16(1 minus sum

119901

119895=1(2radic119894)

119895

)

prod119899


1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896


)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894


100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(23)

with


119899

prod

119894=16

(1 minus

sum119901

119896=3(2radic119894)

119896

119894radic119894



119899

prod

119894=16

(1 minus1

119894)


15

119899

= 0

(24)

implies that

lim119899rarrinfin

10038161003816100381610038161003816100381610038161003816

119877119878119899minus 0

119868119899minus 0

10038161003816100381610038161003816100381610038161003816

= 0 (25)




Acknowledgments


References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article faster multistep iterations for the ...fixed points applied to zamfirescu operators...

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