research article strong convergence for hybrid implicit...
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Research ArticleStrong Convergence for Hybrid Implicit S-Iteration Scheme ofNonexpansive and Strongly Pseudocontractive Mappings
Shin Min Kang1 Arif Rafiq2 Faisal Ali3 and Young Chel Kwun4
1 Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea2Department of Mathematics Lahore Leads University Lahore 54810 Pakistan3 Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University Multan 60800 Pakistan4Department of Mathematics Dong-A University Busan 604-714 Republic of Korea
Correspondence should be addressed to Young Chel Kwun yckwundauackr
Received 7 April 2014 Accepted 13 June 2014 Published 7 July 2014
Academic Editor Abdul Latif
Copyright copy 2014 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
Let 119870 be a nonempty closed convex subset of a real Banach space 119864 let 119878 119870 rarr 119870 be nonexpansive and let 119879 119870 rarr 119870
be Lipschitz strongly pseudocontractive mappings such that 119901 isin 119865 (119878) cap 119865 (119879) = 119909 isin 119870 119878119909 = 119879119909 = 119909 and 1003817100381710038171003817119909 minus 1198781199101003817100381710038171003817 le
1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 and
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 for all 119909 119910 isin 119870 Let 120573
119899 be a sequence in [0 1] satisfying (i) sum
infin
119899=1120573119899
= infin (ii) lim119899rarrinfin
120573119899
= 0
For arbitrary 1199090
isin 119870 let 119909119899 be a sequence iteratively defined by 119909
119899= 119878119910119899 119910119899
= (1 minus 120573119899) 119909119899minus1
+ 120573119899119879119909119899 119899 ge 1 Then the sequence
119909119899 converges strongly to a common fixed point p of S and T
1 Introduction and Preliminaries
Let 119864 be a real Banach space and let 119870 be a nonempty convexsubset of 119864 Let 119869 denote the normalized duality mappingfrom 119864 to 2
119864lowast
defined by
119869 (119909) = 119891lowast
isin 119864lowast
⟨119909 119891lowast
⟩ = 1199092
1003817100381710038171003817119891lowast1003817100381710038171003817 = 119909 119909 isin 119864
(1)
where 119864lowast denotes the dual space of 119864 and ⟨sdot sdot⟩ denotes the
generalized duality pairing We will denote the single-valuedduality map by 119895
Let 119879 119870 rarr 119870 be a mapping
Definition 1 Themapping 119879 is said to be Lipschitzian if thereexists 119871 gt 1 such that
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 119871
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (2)
for all 119909 119910 isin 119870
Definition 2 Themapping 119879 is said to be nonexpansive if1003817100381710038171003817119879119909 minus 119879119910
1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 (3)
for all 119909 119910 isin 119870
Definition 3 Themapping 119879 is said to be pseudocontractive if1003817100381710038171003817119909 minus 119910
1003817100381710038171003817 le1003817100381710038171003817119909 minus 119910 + 119905 ((119868 minus 119879) 119909 minus (119868 minus 119879) 119910)
1003817100381710038171003817 (4)
for all 119909 119910 isin 119870 and 119905 gt 0
Remark 4 As a consequence of a result of Kato [1] it followsfrom the inequality that 119879 is pseudocontractive if and only ifthere exists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that
⟨119879119909 minus 119879119910 119895 (119909 minus 119910)⟩ le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 (5)
for all 119909 119910 isin 119870
Definition 5 Themapping 119879 is said to be strongly pseudocon-tractive if there exists a constant 119905 gt 1 such that
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 le
1003817100381710038171003817(1 + 119903) (119909 minus 119910) minus 119903119905 (119879119909 minus 119879119910)1003817100381710038171003817 (6)
for all 119909 119910 isin 119870 and 119903 gt 0 Or equivalently (see [2]) one hasfor 0 lt 119896 lt 1
⟨119879119909 minus 119879119910 119895 (119909 minus 119910)⟩ le 1198961003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 (7)
for all 119909 119910 isin 119870
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 735673 5 pageshttpdxdoiorg1011552014735673
2 Abstract and Applied Analysis
For a nonempty convex subset 119870 of a normed space 119864119879 119870 rarr 119870 is a mapping
(I) The sequence 119909119899 defined by for arbitrary 119909
1isin 119870
119909119899+1
= (1 minus 119886119899) 119909119899
+ 119886119899119879119910119899
119910119899
= (1 minus 119887119899) 119909119899
+ 119887119899119879119909119899 119899 ge 1
(8)
where 119886119899 and 119887
119899 are sequences in [0 1] is known as the
Ishikawa iteration process [3]If 119887119899
= 0 for 119899 ge 1 then the Ishikawa iteration schemebecomes the Mann iteration process [4]
(S) The sequence 119909119899 defined by for arbitrary 119909
1isin 119870
119909119899+1
= 119879119910119899
119910119899
= (1 minus 119887119899) 119909119899
+ 119887119899119879119909119899 119899 ge 1
(9)
where 119887119899 is a sequence in [0 1] is known as the 119878-iteration
process [5 6]In the last few years or so numerous papers have been
published on the iterative approximation of fixed pointsof Lipschitz strongly pseudocontractive mappings using theIshikawa iteration scheme (see eg [3]) Results which hadbeen known only in Hilbert spaces and only for Lipschitzmappings have been extended tomore general Banach spaces(see eg [7ndash13] and the references cited therein)
In 1974 Ishikawa [3] proved the following result
Theorem 6 Let 119870 be a compact convex subset of a Hilbertspace 119867 and let 119879 119870 rarr 119870 be a Lipschitzian pseudocon-tractive mapping For arbitrary 119909
1isin 119870 let 119909
119899 be a sequence
defined iteratively by
119909119899+1
= (1 minus 120572119899) 119909119899
+ 120572119899119879119910119899
119910119899
= (1 minus 120573119899) 119909119899
+ 120573119899119879119909119899 119899 ge 1
(10)
where 120572119899 and 120573
119899 are sequences satisfying
(i) 0 le 120572119899
le 120573119899
lt 1
(ii) lim119899rarrinfin
120573119899
= 0
(iii) suminfin
119899=1120572119899120573119899
= infin
Then the sequence 119909119899 converges strongly to a fixed point of 119879
In [7] Chidume extended the results of Schu [12] fromHilbert spaces to the much more general class of real Banachspaces and approximate the fixed points of pseudocontractivemappings Also in [14] he investigated the approximation ofthe fixed points of strongly pseudocontractive mappings
In [15] Zhou and Jia gave the answer of the questionraised by Chidume [14] and proved the following
If 119883 is a real Banach space with a uniformly convex dual119883lowast 119870 is a nonempty bounded closed convex subset of 119883 and
119879 119870 rarr 119870 is a continuous strongly pseudocontractivemapping then the Ishikawa iteration scheme convergesstrongly to the unique fixed point of 119879
In [16] Liu et al introduced the following condition
Remark 7 Let 119878 119879 119870 rarr 119870 be twomappingsThemappings119878 and 119879 are said to satisfy condition (1198621) if
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119878119909 minus 1198791199101003817100381710038171003817 (1198621)
for all 119909 119910 isin 119870
In 2012 Kang et al [17] established the strong conver-gence for the implicit 119878-iterative process associated withLipschitzian hemicontractive mappings in Hilbert spaces
Theorem8 Let 119870 be a compact convex subset of a real Hilbertspace 119867 and let 119879 119870 rarr 119870 be a Lipschitzian hemicontractivemapping satisfying
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 (1198622)
for all 119909 119910 isin 119870 Let 120573119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) sum
infin
119899=11205732
119899lt infin
For arbitrary 1199090
isin 119870 let 119909119899 be a sequence iteratively defined
by
119909119899
= 119879119910119899
119910119899
= (1 minus 120573119899) 119909119899minus1
+ 120573119899119879119909119899 119899 ge 1
(11)
Then the sequence 119909119899 converges strongly to the fixed point 119909
lowast
of 119879
In 2013 Kang et al [18] proved the following result
Theorem9 Let119870 be a nonempty closed convex subset of a realBanach space 119864 let 119878 119870 rarr 119870 be a nonexpansive mappingand let 119879 119870 rarr 119870 be a Lipschitz strongly pseudocontractivemapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin 119870 119878119909 = 119879119909 = 119909
and1003817100381710038171003817119909 minus 119878119910
1003817100381710038171003817 le1003817100381710038171003817119878119909 minus 119878119910
1003817100381710038171003817 1003817100381710038171003817119909 minus 119879119910
1003817100381710038171003817 le1003817100381710038171003817119879119909 minus 119879119910
1003817100381710038171003817 (1198623)
for all 119909 119910 isin 119870 Let 120573119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) lim
119899rarrinfin120573119899
= 0
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by
119909119899+1
= 119878119910119899
119910119899
= (1 minus 120573119899) 119909119899
+ 120573119899119879119909119899 119899 ge 1
(12)
Then the sequence 119909119899 converges strongly to a common fixed
point 119901 of 119878 and 119879
Keeping in view the importance of the implicit iterationschemes (see [17]) in this paper we establish the strongconvergence theorem for the hybrid implicit 119878-iterativescheme associated with nonexpansive and Lipschitz stronglypseudocontractive mappings in real Banach spaces
Abstract and Applied Analysis 3
2 Main Results
We will need the following results
Lemma 10 (see [19 20]) Let 119869 119864 rarr 2119864lowast
be the normalizedduality mapping Then for any 119909 119910 isin 119864 one has
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2
le 1199092
+ 2 ⟨119910 119895 (119909 + 119910)⟩
forall119895 (119909 + 119910) isin 119869 (119909 + 119910)
(13)
Lemma 11 (see [13]) Let 120588119899 and 120579
119899 be nonnegative
sequences satisfying
120588119899+1
le (1 minus 120579119899) 120588119899
+ 119887119899 (14)
where 120579119899
isin [0 1) suminfin
119899=1120579119899
= infin and 119887119899
= 119900(120579119899) Then
lim119899rarrinfin
120588119899
= 0
The following is our main result
Theorem 12 Let 119870 be a nonempty closed convex subset of areal Banach space 119864 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623)Let 120573
119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) lim
119899rarrinfin120573119899
= 0
For arbitrary 1199090
isin 119870 let 119909119899 be a sequence iteratively
defined by
119909119899
= 119878119910119899
119910119899
= (1 minus 120573119899) 119909119899minus1
+ 120573119899119879119909119899 119899 ge 1
(15)
Then the sequence 119909119899 converges strongly to a common fixed
point 119901 of 119878 and 119879
Proof For strongly pseudocontractive mappings the exis-tence of a fixed point follows from Deimling [21] It isshown in [15] that the set of fixed points for stronglypseudocontractions is a singleton
By (ii) since lim119899rarrinfin
120573119899
= 0 there exists 1198990
isin N such thatforall119899 ge 119899
0
120573119899
le min1
4119896
1 minus 119896
2 (1 + 119871) (1 + 2119871) (16)
where 119896 lt 12 and 119871 is a Lipschitz constant of 119879 Consider1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
= ⟨119909119899
minus 119901 119895 (119909119899
minus 119901)⟩
= ⟨119878119910119899
minus 119901 119895 (119909119899
minus 119901)⟩
= ⟨119879119909119899
minus 119901 119895 (119909119899
minus 119901)⟩ + ⟨119878119910119899
minus 119879119909119899 119895 (119909119899
minus 119901)⟩
le 1198961003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
+1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
(17)
where1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119878119910119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119909119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119878119909119899
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
=1003817100381710038171003817119878119909119899
minus 119878119910119899
1003817100381710038171003817 + 21003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
le (1 + 2119871)1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(18)1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 le1003817100381710038171003817119909119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
=1003817100381710038171003817119878119910119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
le1003817100381710038171003817119878119909119899minus1
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
= 2120573119899
1003817100381710038171003817119909119899minus1
minus 119879119909119899
1003817100381710038171003817
le 2120573119899
(1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 +
1003817100381710038171003817119901 minus 119879119909119899
1003817100381710038171003817)
le 2120573119899
(1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 + 119871
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817)
(19)
and consequently from (18) and (19) we obtain
1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817 le 2 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
+ 2119871 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
(20)
Substituting (20) in (17) and using (16) we get
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 le
2 (1 + 2119871) 120573119899
1 minus 119896 minus 2119871 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 for 119899 ge 119899
0
(21)
So from the above discussion we can conclude that thesequence 119909
119899minus 119901 is bounded Since 119879 is Lipschitzian so
119879119909119899
minus 119901 is also bounded Let 1198721
= sup119899ge1
119909119899
minus 119901 +
sup119899ge1
119879119909119899
minus 119901 Also by (ii) we have
1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817 = 120573119899
1003817100381710038171003817119909119899minus1
minus 119879119909119899
1003817100381710038171003817
le 1198721120573119899
997888rarr 0
(22)
as 119899 rarr infin which implies that 119909119899minus1
minus 119910119899 is bounded so let
1198722
= sup119899ge1
119909119899minus1
minus 119910119899 + 119872
1 Further
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 le
1003817100381710038171003817119910119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
le 1198722
(23)
which implies that 119910119899
minus 119901 is bounded Therefore 119879119910119899
minus 119901
is also bounded
4 Abstract and Applied Analysis
Set
1198723
= sup119899ge1
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899
minus 1199011003817100381710038171003817 (24)
Denote 119872 = 1198721
+ 1198722
+ 1198723 Obviously 119872 lt infin
Now from (15) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817119878119910119899
minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
(25)
and by Lemma 10
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) 119909119899minus1
+ 120573119899119879119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus1
minus 119901) + 120573119899
(119879119909119899
minus 119901)1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119909119899
minus 119901 119895 (119910119899
minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119910119899
minus 119901 119895 (119910119899
minus 119901)⟩
+ 2120573119899
⟨119879119909119899
minus 119879119910119899 119895 (119910119899
minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 forall119895 (119910119899
minus 119901) isin 119869 (119910119899
minus 119901)
(26)
which implies that
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
le(1 minus 120573
119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+2119872119871120573
119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 for 119899 ge 1198990
(27)
Because of (16) we have (1 minus 120573119899)(1 minus 2119896120573
119899) le 1 and 1(1 minus
2119896120573119899) le 2 Also by (ii) and (19) 119909
119899minus119910119899 le 2119872(1+119871)120573
119899rarr 0
as 119899 rarr infinHence (25) and (27) give
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 (28)
For all 119899 ge 1 put
120588119899
=1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
120579119899
= 120573119899
119887119899
= 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(29)
then according to Lemma 11 we obtain from (28) that
lim119899rarrinfin
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 = 0 (30)
This completes the proof
Corollary 13 Let 119870 be a nonempty closed convex subset ofa real Hilbert space 119867 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623) Let 120573119899 be a sequence
in [0 1] satisfying conditions (i) and (ii) in Theorem 12For arbitrary 119909
0isin 119870 let 119909
119899 be a sequence iteratively
defined by (15) Then the sequence 119909119899 converges strongly to
a common fixed point 119901 of 119878 and 119879
Example 14 As a particular case wemay choose for instance120573119899
= 1119899
Remark 15 (1) Condition (1198622) is due to Kang et al [17] andcondition (1198621) with 119878 = 119879 becomes condition (1198622)
(2) Condition (1198623) is due to Kang et al [18] and condition(1198623) with 119878 = 119879 becomes condition (1198622)
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 all refereesfor their valuable comments and suggestions for improvingthe paper This study was supported by research funds fromDong-A University
References
[1] T Kato ldquoNonlinear semigroups and evolution equationsrdquoJournal of the Mathematical Society of Japan vol 19 pp 508ndash520 1967
[2] J Bogin ldquoOn strict pseudocontractions and a fixed pointtheoremrdquo Technion Preprint MT-29 Haifa Israel 1974
[3] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[4] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[5] D R Sahu ldquoApplications of the S-iteration process to con-strained minimization problems and split feasibility problemsrdquoFixed Point Theory vol 12 no 1 pp 187ndash204 2011
[6] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[7] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[8] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[9] C E Chidume and H Zegeye ldquoApproximate fixed pointsequences and convergence theorems for Lipschitz pseudo-contractive mapsrdquo Proceedings of the American MathematicalSociety vol 132 no 3 pp 831ndash840 2004
Abstract and Applied Analysis 5
[10] T Jitpeera andP Kumam ldquoThe shrinking projectionmethod forcommon solutions of generalized mixed equilibrium problemsand fixed point problems for strictly pseudocontractive map-pingsrdquo Journal of Inequalities and Applications vol 2011 ArticleID 840319 25 pages 2011
[11] N Onjai-Uea and P Kumam ldquoConvergence theorems for gen-eralized mixed equilibrium and variational inclusion problemsof strict-pseudocontractivemappingsrdquoBulletin of theMalaysianMathematical Sciences Society vol 36 no 4 pp 1049ndash10702013
[12] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[13] X Weng ldquoFixed point iteration for local strictly pseudo-contractive mappingrdquo Proceedings of the American Mathemat-ical Society vol 113 no 3 pp 727ndash731 1991
[14] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[15] H Zhou and Y Jia ldquoApproximation of fixed points of stronglypseudocontractive maps without Lipschitz assumptionrdquo Pro-ceedings of the American Mathematical Society vol 125 no 6pp 1705ndash1709 1997
[16] Z Liu C Feng J S Ume and S M Kang ldquoWeak and strongconvergence for common fixed points of a pair of nonexpansiveand asymptotically nonexpansivemappingsrdquoTaiwanese Journalof Mathematics vol 11 no 1 pp 27ndash42 2007
[17] S M Kang A Rafiq and S Lee ldquoStrong convergence of animplicit 119878-iterative process for Lipschitzian hemicontractivemappingsrdquo Abstract and Applied Analysis vol 2012 Article ID804745 7 pages 2012
[18] S M Kang A Rafiq and Y C Kwun ldquoStrong convergence forhybrid 119878-iteration schemerdquo Journal of AppliedMathematics vol2013 Article ID 705814 4 pages 2013
[19] S Chang ldquoSome problems and results in the study of nonlinearanalysisrdquo in Nonlinear Analysis vol 30 pp 4197ndash4208
[20] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis vol 16 no 12 pp 1127ndash1138 1991
[21] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 pp 365ndash374 1974
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal 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
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
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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
For a nonempty convex subset 119870 of a normed space 119864119879 119870 rarr 119870 is a mapping
(I) The sequence 119909119899 defined by for arbitrary 119909
1isin 119870
119909119899+1
= (1 minus 119886119899) 119909119899
+ 119886119899119879119910119899
119910119899
= (1 minus 119887119899) 119909119899
+ 119887119899119879119909119899 119899 ge 1
(8)
where 119886119899 and 119887
119899 are sequences in [0 1] is known as the
Ishikawa iteration process [3]If 119887119899
= 0 for 119899 ge 1 then the Ishikawa iteration schemebecomes the Mann iteration process [4]
(S) The sequence 119909119899 defined by for arbitrary 119909
1isin 119870
119909119899+1
= 119879119910119899
119910119899
= (1 minus 119887119899) 119909119899
+ 119887119899119879119909119899 119899 ge 1
(9)
where 119887119899 is a sequence in [0 1] is known as the 119878-iteration
process [5 6]In the last few years or so numerous papers have been
published on the iterative approximation of fixed pointsof Lipschitz strongly pseudocontractive mappings using theIshikawa iteration scheme (see eg [3]) Results which hadbeen known only in Hilbert spaces and only for Lipschitzmappings have been extended tomore general Banach spaces(see eg [7ndash13] and the references cited therein)
In 1974 Ishikawa [3] proved the following result
Theorem 6 Let 119870 be a compact convex subset of a Hilbertspace 119867 and let 119879 119870 rarr 119870 be a Lipschitzian pseudocon-tractive mapping For arbitrary 119909
1isin 119870 let 119909
119899 be a sequence
defined iteratively by
119909119899+1
= (1 minus 120572119899) 119909119899
+ 120572119899119879119910119899
119910119899
= (1 minus 120573119899) 119909119899
+ 120573119899119879119909119899 119899 ge 1
(10)
where 120572119899 and 120573
119899 are sequences satisfying
(i) 0 le 120572119899
le 120573119899
lt 1
(ii) lim119899rarrinfin
120573119899
= 0
(iii) suminfin
119899=1120572119899120573119899
= infin
Then the sequence 119909119899 converges strongly to a fixed point of 119879
In [7] Chidume extended the results of Schu [12] fromHilbert spaces to the much more general class of real Banachspaces and approximate the fixed points of pseudocontractivemappings Also in [14] he investigated the approximation ofthe fixed points of strongly pseudocontractive mappings
In [15] Zhou and Jia gave the answer of the questionraised by Chidume [14] and proved the following
If 119883 is a real Banach space with a uniformly convex dual119883lowast 119870 is a nonempty bounded closed convex subset of 119883 and
119879 119870 rarr 119870 is a continuous strongly pseudocontractivemapping then the Ishikawa iteration scheme convergesstrongly to the unique fixed point of 119879
In [16] Liu et al introduced the following condition
Remark 7 Let 119878 119879 119870 rarr 119870 be twomappingsThemappings119878 and 119879 are said to satisfy condition (1198621) if
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119878119909 minus 1198791199101003817100381710038171003817 (1198621)
for all 119909 119910 isin 119870
In 2012 Kang et al [17] established the strong conver-gence for the implicit 119878-iterative process associated withLipschitzian hemicontractive mappings in Hilbert spaces
Theorem8 Let 119870 be a compact convex subset of a real Hilbertspace 119867 and let 119879 119870 rarr 119870 be a Lipschitzian hemicontractivemapping satisfying
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 (1198622)
for all 119909 119910 isin 119870 Let 120573119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) sum
infin
119899=11205732
119899lt infin
For arbitrary 1199090
isin 119870 let 119909119899 be a sequence iteratively defined
by
119909119899
= 119879119910119899
119910119899
= (1 minus 120573119899) 119909119899minus1
+ 120573119899119879119909119899 119899 ge 1
(11)
Then the sequence 119909119899 converges strongly to the fixed point 119909
lowast
of 119879
In 2013 Kang et al [18] proved the following result
Theorem9 Let119870 be a nonempty closed convex subset of a realBanach space 119864 let 119878 119870 rarr 119870 be a nonexpansive mappingand let 119879 119870 rarr 119870 be a Lipschitz strongly pseudocontractivemapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin 119870 119878119909 = 119879119909 = 119909
and1003817100381710038171003817119909 minus 119878119910
1003817100381710038171003817 le1003817100381710038171003817119878119909 minus 119878119910
1003817100381710038171003817 1003817100381710038171003817119909 minus 119879119910
1003817100381710038171003817 le1003817100381710038171003817119879119909 minus 119879119910
1003817100381710038171003817 (1198623)
for all 119909 119910 isin 119870 Let 120573119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) lim
119899rarrinfin120573119899
= 0
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by
119909119899+1
= 119878119910119899
119910119899
= (1 minus 120573119899) 119909119899
+ 120573119899119879119909119899 119899 ge 1
(12)
Then the sequence 119909119899 converges strongly to a common fixed
point 119901 of 119878 and 119879
Keeping in view the importance of the implicit iterationschemes (see [17]) in this paper we establish the strongconvergence theorem for the hybrid implicit 119878-iterativescheme associated with nonexpansive and Lipschitz stronglypseudocontractive mappings in real Banach spaces
Abstract and Applied Analysis 3
2 Main Results
We will need the following results
Lemma 10 (see [19 20]) Let 119869 119864 rarr 2119864lowast
be the normalizedduality mapping Then for any 119909 119910 isin 119864 one has
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2
le 1199092
+ 2 ⟨119910 119895 (119909 + 119910)⟩
forall119895 (119909 + 119910) isin 119869 (119909 + 119910)
(13)
Lemma 11 (see [13]) Let 120588119899 and 120579
119899 be nonnegative
sequences satisfying
120588119899+1
le (1 minus 120579119899) 120588119899
+ 119887119899 (14)
where 120579119899
isin [0 1) suminfin
119899=1120579119899
= infin and 119887119899
= 119900(120579119899) Then
lim119899rarrinfin
120588119899
= 0
The following is our main result
Theorem 12 Let 119870 be a nonempty closed convex subset of areal Banach space 119864 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623)Let 120573
119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) lim
119899rarrinfin120573119899
= 0
For arbitrary 1199090
isin 119870 let 119909119899 be a sequence iteratively
defined by
119909119899
= 119878119910119899
119910119899
= (1 minus 120573119899) 119909119899minus1
+ 120573119899119879119909119899 119899 ge 1
(15)
Then the sequence 119909119899 converges strongly to a common fixed
point 119901 of 119878 and 119879
Proof For strongly pseudocontractive mappings the exis-tence of a fixed point follows from Deimling [21] It isshown in [15] that the set of fixed points for stronglypseudocontractions is a singleton
By (ii) since lim119899rarrinfin
120573119899
= 0 there exists 1198990
isin N such thatforall119899 ge 119899
0
120573119899
le min1
4119896
1 minus 119896
2 (1 + 119871) (1 + 2119871) (16)
where 119896 lt 12 and 119871 is a Lipschitz constant of 119879 Consider1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
= ⟨119909119899
minus 119901 119895 (119909119899
minus 119901)⟩
= ⟨119878119910119899
minus 119901 119895 (119909119899
minus 119901)⟩
= ⟨119879119909119899
minus 119901 119895 (119909119899
minus 119901)⟩ + ⟨119878119910119899
minus 119879119909119899 119895 (119909119899
minus 119901)⟩
le 1198961003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
+1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
(17)
where1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119878119910119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119909119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119878119909119899
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
=1003817100381710038171003817119878119909119899
minus 119878119910119899
1003817100381710038171003817 + 21003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
le (1 + 2119871)1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(18)1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 le1003817100381710038171003817119909119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
=1003817100381710038171003817119878119910119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
le1003817100381710038171003817119878119909119899minus1
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
= 2120573119899
1003817100381710038171003817119909119899minus1
minus 119879119909119899
1003817100381710038171003817
le 2120573119899
(1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 +
1003817100381710038171003817119901 minus 119879119909119899
1003817100381710038171003817)
le 2120573119899
(1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 + 119871
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817)
(19)
and consequently from (18) and (19) we obtain
1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817 le 2 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
+ 2119871 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
(20)
Substituting (20) in (17) and using (16) we get
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 le
2 (1 + 2119871) 120573119899
1 minus 119896 minus 2119871 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 for 119899 ge 119899
0
(21)
So from the above discussion we can conclude that thesequence 119909
119899minus 119901 is bounded Since 119879 is Lipschitzian so
119879119909119899
minus 119901 is also bounded Let 1198721
= sup119899ge1
119909119899
minus 119901 +
sup119899ge1
119879119909119899
minus 119901 Also by (ii) we have
1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817 = 120573119899
1003817100381710038171003817119909119899minus1
minus 119879119909119899
1003817100381710038171003817
le 1198721120573119899
997888rarr 0
(22)
as 119899 rarr infin which implies that 119909119899minus1
minus 119910119899 is bounded so let
1198722
= sup119899ge1
119909119899minus1
minus 119910119899 + 119872
1 Further
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 le
1003817100381710038171003817119910119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
le 1198722
(23)
which implies that 119910119899
minus 119901 is bounded Therefore 119879119910119899
minus 119901
is also bounded
4 Abstract and Applied Analysis
Set
1198723
= sup119899ge1
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899
minus 1199011003817100381710038171003817 (24)
Denote 119872 = 1198721
+ 1198722
+ 1198723 Obviously 119872 lt infin
Now from (15) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817119878119910119899
minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
(25)
and by Lemma 10
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) 119909119899minus1
+ 120573119899119879119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus1
minus 119901) + 120573119899
(119879119909119899
minus 119901)1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119909119899
minus 119901 119895 (119910119899
minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119910119899
minus 119901 119895 (119910119899
minus 119901)⟩
+ 2120573119899
⟨119879119909119899
minus 119879119910119899 119895 (119910119899
minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 forall119895 (119910119899
minus 119901) isin 119869 (119910119899
minus 119901)
(26)
which implies that
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
le(1 minus 120573
119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+2119872119871120573
119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 for 119899 ge 1198990
(27)
Because of (16) we have (1 minus 120573119899)(1 minus 2119896120573
119899) le 1 and 1(1 minus
2119896120573119899) le 2 Also by (ii) and (19) 119909
119899minus119910119899 le 2119872(1+119871)120573
119899rarr 0
as 119899 rarr infinHence (25) and (27) give
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 (28)
For all 119899 ge 1 put
120588119899
=1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
120579119899
= 120573119899
119887119899
= 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(29)
then according to Lemma 11 we obtain from (28) that
lim119899rarrinfin
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 = 0 (30)
This completes the proof
Corollary 13 Let 119870 be a nonempty closed convex subset ofa real Hilbert space 119867 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623) Let 120573119899 be a sequence
in [0 1] satisfying conditions (i) and (ii) in Theorem 12For arbitrary 119909
0isin 119870 let 119909
119899 be a sequence iteratively
defined by (15) Then the sequence 119909119899 converges strongly to
a common fixed point 119901 of 119878 and 119879
Example 14 As a particular case wemay choose for instance120573119899
= 1119899
Remark 15 (1) Condition (1198622) is due to Kang et al [17] andcondition (1198621) with 119878 = 119879 becomes condition (1198622)
(2) Condition (1198623) is due to Kang et al [18] and condition(1198623) with 119878 = 119879 becomes condition (1198622)
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 all refereesfor their valuable comments and suggestions for improvingthe paper This study was supported by research funds fromDong-A University
References
[1] T Kato ldquoNonlinear semigroups and evolution equationsrdquoJournal of the Mathematical Society of Japan vol 19 pp 508ndash520 1967
[2] J Bogin ldquoOn strict pseudocontractions and a fixed pointtheoremrdquo Technion Preprint MT-29 Haifa Israel 1974
[3] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[4] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[5] D R Sahu ldquoApplications of the S-iteration process to con-strained minimization problems and split feasibility problemsrdquoFixed Point Theory vol 12 no 1 pp 187ndash204 2011
[6] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[7] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[8] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[9] C E Chidume and H Zegeye ldquoApproximate fixed pointsequences and convergence theorems for Lipschitz pseudo-contractive mapsrdquo Proceedings of the American MathematicalSociety vol 132 no 3 pp 831ndash840 2004
Abstract and Applied Analysis 5
[10] T Jitpeera andP Kumam ldquoThe shrinking projectionmethod forcommon solutions of generalized mixed equilibrium problemsand fixed point problems for strictly pseudocontractive map-pingsrdquo Journal of Inequalities and Applications vol 2011 ArticleID 840319 25 pages 2011
[11] N Onjai-Uea and P Kumam ldquoConvergence theorems for gen-eralized mixed equilibrium and variational inclusion problemsof strict-pseudocontractivemappingsrdquoBulletin of theMalaysianMathematical Sciences Society vol 36 no 4 pp 1049ndash10702013
[12] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[13] X Weng ldquoFixed point iteration for local strictly pseudo-contractive mappingrdquo Proceedings of the American Mathemat-ical Society vol 113 no 3 pp 727ndash731 1991
[14] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[15] H Zhou and Y Jia ldquoApproximation of fixed points of stronglypseudocontractive maps without Lipschitz assumptionrdquo Pro-ceedings of the American Mathematical Society vol 125 no 6pp 1705ndash1709 1997
[16] Z Liu C Feng J S Ume and S M Kang ldquoWeak and strongconvergence for common fixed points of a pair of nonexpansiveand asymptotically nonexpansivemappingsrdquoTaiwanese Journalof Mathematics vol 11 no 1 pp 27ndash42 2007
[17] S M Kang A Rafiq and S Lee ldquoStrong convergence of animplicit 119878-iterative process for Lipschitzian hemicontractivemappingsrdquo Abstract and Applied Analysis vol 2012 Article ID804745 7 pages 2012
[18] S M Kang A Rafiq and Y C Kwun ldquoStrong convergence forhybrid 119878-iteration schemerdquo Journal of AppliedMathematics vol2013 Article ID 705814 4 pages 2013
[19] S Chang ldquoSome problems and results in the study of nonlinearanalysisrdquo in Nonlinear Analysis vol 30 pp 4197ndash4208
[20] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis vol 16 no 12 pp 1127ndash1138 1991
[21] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 pp 365ndash374 1974
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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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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 3
2 Main Results
We will need the following results
Lemma 10 (see [19 20]) Let 119869 119864 rarr 2119864lowast
be the normalizedduality mapping Then for any 119909 119910 isin 119864 one has
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2
le 1199092
+ 2 ⟨119910 119895 (119909 + 119910)⟩
forall119895 (119909 + 119910) isin 119869 (119909 + 119910)
(13)
Lemma 11 (see [13]) Let 120588119899 and 120579
119899 be nonnegative
sequences satisfying
120588119899+1
le (1 minus 120579119899) 120588119899
+ 119887119899 (14)
where 120579119899
isin [0 1) suminfin
119899=1120579119899
= infin and 119887119899
= 119900(120579119899) Then
lim119899rarrinfin
120588119899
= 0
The following is our main result
Theorem 12 Let 119870 be a nonempty closed convex subset of areal Banach space 119864 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623)Let 120573
119899 be a sequence in [0 1] satisfying
(i) suminfin
119899=1120573119899
= infin(ii) lim
119899rarrinfin120573119899
= 0
For arbitrary 1199090
isin 119870 let 119909119899 be a sequence iteratively
defined by
119909119899
= 119878119910119899
119910119899
= (1 minus 120573119899) 119909119899minus1
+ 120573119899119879119909119899 119899 ge 1
(15)
Then the sequence 119909119899 converges strongly to a common fixed
point 119901 of 119878 and 119879
Proof For strongly pseudocontractive mappings the exis-tence of a fixed point follows from Deimling [21] It isshown in [15] that the set of fixed points for stronglypseudocontractions is a singleton
By (ii) since lim119899rarrinfin
120573119899
= 0 there exists 1198990
isin N such thatforall119899 ge 119899
0
120573119899
le min1
4119896
1 minus 119896
2 (1 + 119871) (1 + 2119871) (16)
where 119896 lt 12 and 119871 is a Lipschitz constant of 119879 Consider1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
= ⟨119909119899
minus 119901 119895 (119909119899
minus 119901)⟩
= ⟨119878119910119899
minus 119901 119895 (119909119899
minus 119901)⟩
= ⟨119879119909119899
minus 119901 119895 (119909119899
minus 119901)⟩ + ⟨119878119910119899
minus 119879119909119899 119895 (119909119899
minus 119901)⟩
le 1198961003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
+1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
(17)
where1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119878119910119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119909119899
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119909119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
le1003817100381710038171003817119878119909119899
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899
minus 119879119909119899
1003817100381710038171003817
=1003817100381710038171003817119878119909119899
minus 119878119910119899
1003817100381710038171003817 + 21003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
le (1 + 2119871)1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(18)1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 le1003817100381710038171003817119909119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
=1003817100381710038171003817119878119910119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
le1003817100381710038171003817119878119909119899minus1
minus 119878119910119899
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
le 21003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817
= 2120573119899
1003817100381710038171003817119909119899minus1
minus 119879119909119899
1003817100381710038171003817
le 2120573119899
(1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 +
1003817100381710038171003817119901 minus 119879119909119899
1003817100381710038171003817)
le 2120573119899
(1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 + 119871
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817)
(19)
and consequently from (18) and (19) we obtain
1003817100381710038171003817119878119910119899
minus 119879119909119899
1003817100381710038171003817 le 2 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
+ 2119871 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
(20)
Substituting (20) in (17) and using (16) we get
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 le
2 (1 + 2119871) 120573119899
1 minus 119896 minus 2119871 (1 + 2119871) 120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817 for 119899 ge 119899
0
(21)
So from the above discussion we can conclude that thesequence 119909
119899minus 119901 is bounded Since 119879 is Lipschitzian so
119879119909119899
minus 119901 is also bounded Let 1198721
= sup119899ge1
119909119899
minus 119901 +
sup119899ge1
119879119909119899
minus 119901 Also by (ii) we have
1003817100381710038171003817119909119899minus1
minus 119910119899
1003817100381710038171003817 = 120573119899
1003817100381710038171003817119909119899minus1
minus 119879119909119899
1003817100381710038171003817
le 1198721120573119899
997888rarr 0
(22)
as 119899 rarr infin which implies that 119909119899minus1
minus 119910119899 is bounded so let
1198722
= sup119899ge1
119909119899minus1
minus 119910119899 + 119872
1 Further
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 le
1003817100381710038171003817119910119899
minus 119909119899minus1
1003817100381710038171003817 +1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
le 1198722
(23)
which implies that 119910119899
minus 119901 is bounded Therefore 119879119910119899
minus 119901
is also bounded
4 Abstract and Applied Analysis
Set
1198723
= sup119899ge1
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899
minus 1199011003817100381710038171003817 (24)
Denote 119872 = 1198721
+ 1198722
+ 1198723 Obviously 119872 lt infin
Now from (15) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817119878119910119899
minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
(25)
and by Lemma 10
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) 119909119899minus1
+ 120573119899119879119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus1
minus 119901) + 120573119899
(119879119909119899
minus 119901)1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119909119899
minus 119901 119895 (119910119899
minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119910119899
minus 119901 119895 (119910119899
minus 119901)⟩
+ 2120573119899
⟨119879119909119899
minus 119879119910119899 119895 (119910119899
minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 forall119895 (119910119899
minus 119901) isin 119869 (119910119899
minus 119901)
(26)
which implies that
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
le(1 minus 120573
119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+2119872119871120573
119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 for 119899 ge 1198990
(27)
Because of (16) we have (1 minus 120573119899)(1 minus 2119896120573
119899) le 1 and 1(1 minus
2119896120573119899) le 2 Also by (ii) and (19) 119909
119899minus119910119899 le 2119872(1+119871)120573
119899rarr 0
as 119899 rarr infinHence (25) and (27) give
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 (28)
For all 119899 ge 1 put
120588119899
=1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
120579119899
= 120573119899
119887119899
= 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(29)
then according to Lemma 11 we obtain from (28) that
lim119899rarrinfin
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 = 0 (30)
This completes the proof
Corollary 13 Let 119870 be a nonempty closed convex subset ofa real Hilbert space 119867 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623) Let 120573119899 be a sequence
in [0 1] satisfying conditions (i) and (ii) in Theorem 12For arbitrary 119909
0isin 119870 let 119909
119899 be a sequence iteratively
defined by (15) Then the sequence 119909119899 converges strongly to
a common fixed point 119901 of 119878 and 119879
Example 14 As a particular case wemay choose for instance120573119899
= 1119899
Remark 15 (1) Condition (1198622) is due to Kang et al [17] andcondition (1198621) with 119878 = 119879 becomes condition (1198622)
(2) Condition (1198623) is due to Kang et al [18] and condition(1198623) with 119878 = 119879 becomes condition (1198622)
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 all refereesfor their valuable comments and suggestions for improvingthe paper This study was supported by research funds fromDong-A University
References
[1] T Kato ldquoNonlinear semigroups and evolution equationsrdquoJournal of the Mathematical Society of Japan vol 19 pp 508ndash520 1967
[2] J Bogin ldquoOn strict pseudocontractions and a fixed pointtheoremrdquo Technion Preprint MT-29 Haifa Israel 1974
[3] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[4] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[5] D R Sahu ldquoApplications of the S-iteration process to con-strained minimization problems and split feasibility problemsrdquoFixed Point Theory vol 12 no 1 pp 187ndash204 2011
[6] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[7] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[8] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[9] C E Chidume and H Zegeye ldquoApproximate fixed pointsequences and convergence theorems for Lipschitz pseudo-contractive mapsrdquo Proceedings of the American MathematicalSociety vol 132 no 3 pp 831ndash840 2004
Abstract and Applied Analysis 5
[10] T Jitpeera andP Kumam ldquoThe shrinking projectionmethod forcommon solutions of generalized mixed equilibrium problemsand fixed point problems for strictly pseudocontractive map-pingsrdquo Journal of Inequalities and Applications vol 2011 ArticleID 840319 25 pages 2011
[11] N Onjai-Uea and P Kumam ldquoConvergence theorems for gen-eralized mixed equilibrium and variational inclusion problemsof strict-pseudocontractivemappingsrdquoBulletin of theMalaysianMathematical Sciences Society vol 36 no 4 pp 1049ndash10702013
[12] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[13] X Weng ldquoFixed point iteration for local strictly pseudo-contractive mappingrdquo Proceedings of the American Mathemat-ical Society vol 113 no 3 pp 727ndash731 1991
[14] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[15] H Zhou and Y Jia ldquoApproximation of fixed points of stronglypseudocontractive maps without Lipschitz assumptionrdquo Pro-ceedings of the American Mathematical Society vol 125 no 6pp 1705ndash1709 1997
[16] Z Liu C Feng J S Ume and S M Kang ldquoWeak and strongconvergence for common fixed points of a pair of nonexpansiveand asymptotically nonexpansivemappingsrdquoTaiwanese Journalof Mathematics vol 11 no 1 pp 27ndash42 2007
[17] S M Kang A Rafiq and S Lee ldquoStrong convergence of animplicit 119878-iterative process for Lipschitzian hemicontractivemappingsrdquo Abstract and Applied Analysis vol 2012 Article ID804745 7 pages 2012
[18] S M Kang A Rafiq and Y C Kwun ldquoStrong convergence forhybrid 119878-iteration schemerdquo Journal of AppliedMathematics vol2013 Article ID 705814 4 pages 2013
[19] S Chang ldquoSome problems and results in the study of nonlinearanalysisrdquo in Nonlinear Analysis vol 30 pp 4197ndash4208
[20] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis vol 16 no 12 pp 1127ndash1138 1991
[21] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 pp 365ndash374 1974
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
4 Abstract and Applied Analysis
Set
1198723
= sup119899ge1
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899
minus 1199011003817100381710038171003817 (24)
Denote 119872 = 1198721
+ 1198722
+ 1198723 Obviously 119872 lt infin
Now from (15) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817119878119910119899
minus 1199011003817100381710038171003817
2
le1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
(25)
and by Lemma 10
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) 119909119899minus1
+ 120573119899119879119909119899
minus 1199011003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus1
minus 119901) + 120573119899
(119879119909119899
minus 119901)1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119909119899
minus 119901 119895 (119910119899
minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2120573119899
⟨119879119910119899
minus 119901 119895 (119910119899
minus 119901)⟩
+ 2120573119899
⟨119879119909119899
minus 119879119910119899 119895 (119910119899
minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899
minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 2119896120573119899
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 forall119895 (119910119899
minus 119901) isin 119869 (119910119899
minus 119901)
(26)
which implies that
1003817100381710038171003817119910119899
minus 1199011003817100381710038171003817
2
le(1 minus 120573
119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+2119872119871120573
119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 for 119899 ge 1198990
(27)
Because of (16) we have (1 minus 120573119899)(1 minus 2119896120573
119899) le 1 and 1(1 minus
2119896120573119899) le 2 Also by (ii) and (19) 119909
119899minus119910119899 le 2119872(1+119871)120573
119899rarr 0
as 119899 rarr infinHence (25) and (27) give
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817
2
le (1 minus 120573119899)
1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
2
+ 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817 (28)
For all 119899 ge 1 put
120588119899
=1003817100381710038171003817119909119899minus1
minus 1199011003817100381710038171003817
120579119899
= 120573119899
119887119899
= 4119872119871120573119899
1003817100381710038171003817119909119899
minus 119910119899
1003817100381710038171003817
(29)
then according to Lemma 11 we obtain from (28) that
lim119899rarrinfin
1003817100381710038171003817119909119899
minus 1199011003817100381710038171003817 = 0 (30)
This completes the proof
Corollary 13 Let 119870 be a nonempty closed convex subset ofa real Hilbert space 119867 let 119878 119870 rarr 119870 be a nonexpansivemapping and let 119879 119870 rarr 119870 be a Lipschitz stronglypseudocontractive mapping such that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin
119870 119878119909 = 119879119909 = 119909 and condition (1198623) Let 120573119899 be a sequence
in [0 1] satisfying conditions (i) and (ii) in Theorem 12For arbitrary 119909
0isin 119870 let 119909
119899 be a sequence iteratively
defined by (15) Then the sequence 119909119899 converges strongly to
a common fixed point 119901 of 119878 and 119879
Example 14 As a particular case wemay choose for instance120573119899
= 1119899
Remark 15 (1) Condition (1198622) is due to Kang et al [17] andcondition (1198621) with 119878 = 119879 becomes condition (1198622)
(2) Condition (1198623) is due to Kang et al [18] and condition(1198623) with 119878 = 119879 becomes condition (1198622)
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 all refereesfor their valuable comments and suggestions for improvingthe paper This study was supported by research funds fromDong-A University
References
[1] T Kato ldquoNonlinear semigroups and evolution equationsrdquoJournal of the Mathematical Society of Japan vol 19 pp 508ndash520 1967
[2] J Bogin ldquoOn strict pseudocontractions and a fixed pointtheoremrdquo Technion Preprint MT-29 Haifa Israel 1974
[3] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[4] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[5] D R Sahu ldquoApplications of the S-iteration process to con-strained minimization problems and split feasibility problemsrdquoFixed Point Theory vol 12 no 1 pp 187ndash204 2011
[6] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[7] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[8] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[9] C E Chidume and H Zegeye ldquoApproximate fixed pointsequences and convergence theorems for Lipschitz pseudo-contractive mapsrdquo Proceedings of the American MathematicalSociety vol 132 no 3 pp 831ndash840 2004
Abstract and Applied Analysis 5
[10] T Jitpeera andP Kumam ldquoThe shrinking projectionmethod forcommon solutions of generalized mixed equilibrium problemsand fixed point problems for strictly pseudocontractive map-pingsrdquo Journal of Inequalities and Applications vol 2011 ArticleID 840319 25 pages 2011
[11] N Onjai-Uea and P Kumam ldquoConvergence theorems for gen-eralized mixed equilibrium and variational inclusion problemsof strict-pseudocontractivemappingsrdquoBulletin of theMalaysianMathematical Sciences Society vol 36 no 4 pp 1049ndash10702013
[12] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[13] X Weng ldquoFixed point iteration for local strictly pseudo-contractive mappingrdquo Proceedings of the American Mathemat-ical Society vol 113 no 3 pp 727ndash731 1991
[14] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[15] H Zhou and Y Jia ldquoApproximation of fixed points of stronglypseudocontractive maps without Lipschitz assumptionrdquo Pro-ceedings of the American Mathematical Society vol 125 no 6pp 1705ndash1709 1997
[16] Z Liu C Feng J S Ume and S M Kang ldquoWeak and strongconvergence for common fixed points of a pair of nonexpansiveand asymptotically nonexpansivemappingsrdquoTaiwanese Journalof Mathematics vol 11 no 1 pp 27ndash42 2007
[17] S M Kang A Rafiq and S Lee ldquoStrong convergence of animplicit 119878-iterative process for Lipschitzian hemicontractivemappingsrdquo Abstract and Applied Analysis vol 2012 Article ID804745 7 pages 2012
[18] S M Kang A Rafiq and Y C Kwun ldquoStrong convergence forhybrid 119878-iteration schemerdquo Journal of AppliedMathematics vol2013 Article ID 705814 4 pages 2013
[19] S Chang ldquoSome problems and results in the study of nonlinearanalysisrdquo in Nonlinear Analysis vol 30 pp 4197ndash4208
[20] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis vol 16 no 12 pp 1127ndash1138 1991
[21] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 pp 365ndash374 1974
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 5
[10] T Jitpeera andP Kumam ldquoThe shrinking projectionmethod forcommon solutions of generalized mixed equilibrium problemsand fixed point problems for strictly pseudocontractive map-pingsrdquo Journal of Inequalities and Applications vol 2011 ArticleID 840319 25 pages 2011
[11] N Onjai-Uea and P Kumam ldquoConvergence theorems for gen-eralized mixed equilibrium and variational inclusion problemsof strict-pseudocontractivemappingsrdquoBulletin of theMalaysianMathematical Sciences Society vol 36 no 4 pp 1049ndash10702013
[12] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[13] X Weng ldquoFixed point iteration for local strictly pseudo-contractive mappingrdquo Proceedings of the American Mathemat-ical Society vol 113 no 3 pp 727ndash731 1991
[14] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[15] H Zhou and Y Jia ldquoApproximation of fixed points of stronglypseudocontractive maps without Lipschitz assumptionrdquo Pro-ceedings of the American Mathematical Society vol 125 no 6pp 1705ndash1709 1997
[16] Z Liu C Feng J S Ume and S M Kang ldquoWeak and strongconvergence for common fixed points of a pair of nonexpansiveand asymptotically nonexpansivemappingsrdquoTaiwanese Journalof Mathematics vol 11 no 1 pp 27ndash42 2007
[17] S M Kang A Rafiq and S Lee ldquoStrong convergence of animplicit 119878-iterative process for Lipschitzian hemicontractivemappingsrdquo Abstract and Applied Analysis vol 2012 Article ID804745 7 pages 2012
[18] S M Kang A Rafiq and Y C Kwun ldquoStrong convergence forhybrid 119878-iteration schemerdquo Journal of AppliedMathematics vol2013 Article ID 705814 4 pages 2013
[19] S Chang ldquoSome problems and results in the study of nonlinearanalysisrdquo in Nonlinear Analysis vol 30 pp 4197ndash4208
[20] H K Xu ldquoInequalities in Banach spaces with applicationsrdquoNonlinear Analysis vol 16 no 12 pp 1127ndash1138 1991
[21] K Deimling ldquoZeros of accretive operatorsrdquoManuscriptaMath-ematica vol 13 pp 365ndash374 1974
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