research article strong convergence for hybrid s-iteration...
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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 705814 4 pageshttpdxdoiorg1011552013705814
Research ArticleStrong Convergence for Hybrid S-Iteration Scheme
Shin Min Kang1 Arif Rafiq2 and Young Chel Kwun3
1 Department of Mathematics and RINS Gyeongsang National University Jinju 660-701 Republic of Korea2 School of CS and Mathematics Hajvery University 43-52 Industrial Area Gulberg-III Lahore 54660 Pakistan3 Department of Mathematics Dong-A University Pusan 614-714 Republic of Korea
Correspondence should be addressed to Young Chel Kwun yckwundauackr
Received 19 November 2012 Accepted 4 February 2013
Academic Editor D R Sahu
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
We establish a strong convergence for the hybrid S-iterative scheme associated with nonexpansive and Lipschitz strongly pseudo-contractive mappings in real Banach spaces
1 Introduction and Preliminaries
Let 119864 be a real Banach space and let119870 be a nonempty convexsubset of 119864 Let 119869 denote the normalized duality mappingfrom 119864 to 2
119864lowast
defined by
119869 (119909) = 119891lowastisin 119864lowast ⟨119909 119891
lowast⟩ = 119909
2
1003817100381710038171003817119891lowast1003817100381710038171003817 = 119909 forall119909 119910 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 a constant 119871 gt 1 such that
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le 119871
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119870 (2)
Definition 2 Themapping 119879 is said to be nonexpansive if
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119909 minus 1199101003817100381710038171003817 forall119909 119910 isin 119870 (3)
Definition 3 Themapping 119879 is said to be pseudocontractive iffor all 119909 119910 isin 119870 there exists 119895(119909 minus 119910) isin 119869(119909 minus 119910) such that
⟨119879119909 minus 119879119910 119895 (119909 minus 119910)⟩ le1003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 (4)
Definition 4 Themapping 119879 is said to be strongly pseudocon-tractive if for all 119909 119910 isin 119870 there exists 119896 isin (0 1) such that
⟨119879119909 minus 119879119910 119895 (119909 minus 119910)⟩ le 1198961003817100381710038171003817119909 minus 119910
1003817100381710038171003817
2 (5)
Let 119870 be a nonempty convex subset 119862 of a normed space119864
(a) The sequence 119909119899 defined by for arbitrary 119909
1isin 119870
119909119899+1
= (1 minus 120572119899) 119909119899+ 120572119899119879119910119899
119910119899= (1 minus 120573
119899) 119909119899+ 120573119899119879119909119899 119899 ge 1
(6)
where 120572119899 and 120573
119899 are sequences in [0 1] is known
as the Ishikawa iteration process [1]If 120573119899= 0 for 119899 ge 1 then the Ishikawa iteration process
becomes the Mann iteration process [2](b) The sequence 119909
119899 defined by for arbitrary 119909
1isin 119870
119909119899+1
= 119879119910119899
119910119899= (1 minus 120573
119899) 119909119899+ 120573119899119879119909119899 119899 ge 1
(7)
where 120573119899 is a sequence in [0 1] is known as the 119878-
iteration process [3 4]
In the last few years or so numerous papers have beenpublished on the iterative approximation of fixed pointsof Lipschitz strongly pseudocontractive mappings using theIshikawa iteration scheme (see eg [1]) Results which had
2 Journal of Applied Mathematics
been known only in Hilbert spaces and only for Lipschitzmappings have been extended to more general Banach spaces(see eg [5ndash10] and the references cited therein)
In 1974 Ishikawa [1] proved the following result
Theorem 5 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 120573
119899) 119909119899+ 120573119899119879119909119899 119899 ge 1
(8)
where 120572119899 and 120573
119899 are sequences satisfying
(i) 0 le 120572119899le 120573119899le 1
(ii) lim119899rarrinfin
120573119899= 0
(iii) sum119899ge1
120572119899120573119899= infin
Then the sequence 119909119899 converges strongly at a fixed point of 119879
In [6] Chidume extended the results of Schu [9] fromHilbert spaces to the much more general class of real Banachspaces and approximated the fixed points of (strongly) pseu-docontractive mappings
In [11] Zhou and Jia gave the more general answer of thequestion raised by Chidume [5] and proved the following
If 119883 is a real Banach space with a uniformly convex dual119883lowast119870 is a nonempty bounded closed convex subset of119883 and
119879 119870 rarr 119870 is a continuous strongly pseudocontractivemapping then the Ishikawa iteration scheme convergesstrongly at the unique fixed point of 119879
In this paper we establish the strong convergence forthe hybrid 119878-iterative scheme associated with nonexpansiveand Lipschitz strongly pseudocontractive mappings in realBanach spaces We also improve the result of Zhou and Jia[11]
2 Main Results
We will need the following lemmas
Lemma 6 (see [12]) Let 119869 119864 rarr 2119864 be the normalized
duality mapping Then for any 119909 119910 isin 119864 one has
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2le 119909
2+ 2 ⟨119910 119895 (119909 + 119910)⟩
forall119895 (119909 + 119910) isin 119869 (119909 + 119910)
(9)
Lemma 7 (see [10]) Let 120588119899 be nonnegative sequence satisfy-
ing
120588119899+1
le (1 minus 120579119899) 120588119899+ 120596119899 (10)
where 120579119899isin [0 1]sum
119899ge1120579119899= infin and 120596
119899= o(120579119899) Then
lim119899rarrinfin
120588119899= 0 (11)
The following is our main result
Theorem 8 Let 119870 be a nonempty closed convex subset of areal Banach space 119864 let 119878 119870 rarr 119870 be nonexpansive and let119879 119870 rarr 119870 be Lipschitz strongly pseudocontractive mappingssuch that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin 119870 119878119909 = 119879119909 = 119909 and
1003817100381710038171003817119909 minus 1198781199101003817100381710038171003817 le
1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 forall119909 119910 isin 119870
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 forall119909 119910 isin 119870
(C)
Let 120573119899 be a sequence in [0 1] satisfying
(iv) sum119899ge1
120573119899= infin
(v) lim119899rarrinfin
120573119899= 0
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by119909119899+1
= 119878119910119899
119910119899= (1 minus 120573
119899) 119909119899+ 120573119899119879119909119899 119899 ge 1
(12)
Then the sequence 119909119899 converges strongly at the common fixed
point 119901 of 119878 and 119879
Proof For strongly pseudocontractive mappings the exis-tence of a fixed point follows from Delmling [13] It is shownin [11] that the set of fixed points for strongly pseudocontrac-tions is a singleton
By (v) since lim119899rarrinfin
120573119899= 0 there exists 119899
0isin N such that
for all 119899 ge 1198990
120573119899le min 1
4119896
1 minus 119896
(1 + 119871) (1 + 3119871) (13)
where 119896 lt 12 Consider1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2= ⟨119909119899+1
minus 119901 119895 (119909119899+1
minus 119901)⟩
= ⟨119878119910119899minus 119901 119895 (119909
119899+1minus 119901)⟩
= ⟨119879119909119899+1
minus 119901 119895 (119909119899+1
minus 119901)⟩
+ ⟨119878119910119899minus 119879119909119899+1
119895 (119909119899+1
minus 119901)⟩
le 1198961003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2+1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817
(14)which implies that
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 le
1
1 minus 119896
1003817100381710038171003817119878119910119899 minus 119879119909119899+1
1003817100381710038171003817 (15)
where1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817 le1003817100381710038171003817119878119910119899 minus 119879119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899 minus 119879119909
119899+1
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119878119910
119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119879119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899 minus 119879119909
119899+1
1003817100381710038171003817
le1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119909119899 minus 119879119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899 minus 119879119909
119899+1
1003817100381710038171003817
le1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 + 119871 (1003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119909
119899+1
1003817100381710038171003817)
(16)1003817100381710038171003817119910119899 minus 119909
119899+1
1003817100381710038171003817 le1003817100381710038171003817119910119899 minus 119909
119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119909
119899+1
1003817100381710038171003817
=1003817100381710038171003817119910119899 minus 119909
119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119878119910
119899
1003817100381710038171003817
le1003817100381710038171003817119910119899 minus 119909
119899
1003817100381710038171003817 +1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817
(17)
Journal of Applied Mathematics 3
and consequently from (16) we obtain1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817 le (1 + 119871)1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 + 21198711003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817
le (1 + 3119871)1003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817
= (1 + 3119871) 1205731198991003817100381710038171003817119909119899 minus 119879119909
119899
1003817100381710038171003817
le (1 + 119871) (1 + 3119871) 1205731198991003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
(18)
Substituting (18) in (15) and using (13) we get
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 le
(1 + 119871) (1 + 3119871)
1 minus 119896120573119899
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
(19)
So from the above discussion we can conclude that thesequence 119909
119899minus 119901 is bounded Since 119879 is Lipschitzian so
119879119909119899minus 119901 is also bounded Let 119872
1= sup
119899ge1119909119899minus 119901 +
sup119899ge1
119879119909119899minus 119901 Also by (ii) we have
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817 = 120573119899
1003817100381710038171003817119909119899 minus 119879119909119899
1003817100381710038171003817
le 1198721120573119899
997888rarr 0
(20)
as 119899 rarr infin implying that 119909119899minus 119910119899 is bounded so let 119872
2=
sup119899ge1
119909119899minus 119910119899 + 119872
1 Further
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 le
1003817100381710038171003817119910119899 minus 119909119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
le 1198722
(21)
which implies that 119910119899minus 119901 is bounded Therefore 119879119910
119899minus 119901
is also boundedSet
1198723= sup119899ge1
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899 minus 1199011003817100381710038171003817 (22)
Denote119872 = 1198721+1198722+1198723 Obviously119872 lt infin
Now from (12) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2=1003817100381710038171003817119878119910119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2 (23)
and by Lemma 6 we get1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2=1003817100381710038171003817(1 minus 120573
119899) 119909119899+ 120573119899119879119909119899minus 119901
1003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus 119901) + 120573
119899(119879119909119899minus 119901)
1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2120573119899⟨119879119909119899minus 119901 119895 (119910
119899minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2120573119899⟨119879119910119899minus 119901 119895 (119910
119899minus 119901)⟩
+ 2120573119899⟨119879119909119899minus 119879119910119899 119895 (119910119899minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2119896120573
119899
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899 minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2119896120573
119899
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(24)
which implies that
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2le
(1 minus 120573119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817
2+
2119872119871120573119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(25)
because by (13) we have ((1minus120573119899)(1minus2119896120573
119899)) le 1 and (1(1minus
2119896120573119899)) le 2 Hence (23) gives us
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817 (26)
For all 119899 ge 1 put
120588119899=1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
120579119899= 120573119899
120596119899= 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(27)
then according to Lemma 7 we obtain from (26) that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (28)
This completes the proof
Corollary 9 Let 119870 be a nonempty closed convex subset of areal Hilbert space119867 let 119878 119870 rarr 119870 be nonexpansive and let119879 119870 rarr 119870 be Lipschitz strongly pseudocontractive mappingssuch that 119901 isin 119865(119878) cap 119865(119879) and the condition (C) Let 120573119899 be asequence in [0 1] satisfying the conditions (iv) and (v)
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by (12) Then the sequence 119909119899 converges strongly at
the common fixed point 119901 of 119878 and 119879
Example 10 As a particular case wemay choose for instance120573119899= 1119899
Remark 11 (1) The condition (C) is not new and it is due toLiu et al [14]
(2) We prove our results for a hybrid iteration schemewhich is simple in comparison to the previously knowniteration schemes
Acknowledgment
This study was supported by research funds from Dong-AUniversity
References
[1] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[2] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[3] 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
4 Journal of Applied Mathematics
[4] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[5] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[6] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[7] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[8] 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
[9] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[10] 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
[11] 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
[12] S S Chang ldquoSome problems and results in the study ofnonlinear analysisrdquo Nonlinear Analysis vol 30 no 7 pp 4197ndash4208 1997
[13] K Delmling ldquoZeros of accretive operatorsrdquoManuscripta Math-ematica vol 13 pp 283ndash288 1974
[14] 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
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2 Journal of Applied Mathematics
been known only in Hilbert spaces and only for Lipschitzmappings have been extended to more general Banach spaces(see eg [5ndash10] and the references cited therein)
In 1974 Ishikawa [1] proved the following result
Theorem 5 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 120573
119899) 119909119899+ 120573119899119879119909119899 119899 ge 1
(8)
where 120572119899 and 120573
119899 are sequences satisfying
(i) 0 le 120572119899le 120573119899le 1
(ii) lim119899rarrinfin
120573119899= 0
(iii) sum119899ge1
120572119899120573119899= infin
Then the sequence 119909119899 converges strongly at a fixed point of 119879
In [6] Chidume extended the results of Schu [9] fromHilbert spaces to the much more general class of real Banachspaces and approximated the fixed points of (strongly) pseu-docontractive mappings
In [11] Zhou and Jia gave the more general answer of thequestion raised by Chidume [5] and proved the following
If 119883 is a real Banach space with a uniformly convex dual119883lowast119870 is a nonempty bounded closed convex subset of119883 and
119879 119870 rarr 119870 is a continuous strongly pseudocontractivemapping then the Ishikawa iteration scheme convergesstrongly at the unique fixed point of 119879
In this paper we establish the strong convergence forthe hybrid 119878-iterative scheme associated with nonexpansiveand Lipschitz strongly pseudocontractive mappings in realBanach spaces We also improve the result of Zhou and Jia[11]
2 Main Results
We will need the following lemmas
Lemma 6 (see [12]) Let 119869 119864 rarr 2119864 be the normalized
duality mapping Then for any 119909 119910 isin 119864 one has
1003817100381710038171003817119909 + 1199101003817100381710038171003817
2le 119909
2+ 2 ⟨119910 119895 (119909 + 119910)⟩
forall119895 (119909 + 119910) isin 119869 (119909 + 119910)
(9)
Lemma 7 (see [10]) Let 120588119899 be nonnegative sequence satisfy-
ing
120588119899+1
le (1 minus 120579119899) 120588119899+ 120596119899 (10)
where 120579119899isin [0 1]sum
119899ge1120579119899= infin and 120596
119899= o(120579119899) Then
lim119899rarrinfin
120588119899= 0 (11)
The following is our main result
Theorem 8 Let 119870 be a nonempty closed convex subset of areal Banach space 119864 let 119878 119870 rarr 119870 be nonexpansive and let119879 119870 rarr 119870 be Lipschitz strongly pseudocontractive mappingssuch that 119901 isin 119865(119878) cap 119865(119879) = 119909 isin 119870 119878119909 = 119879119909 = 119909 and
1003817100381710038171003817119909 minus 1198781199101003817100381710038171003817 le
1003817100381710038171003817119878119909 minus 1198781199101003817100381710038171003817 forall119909 119910 isin 119870
1003817100381710038171003817119909 minus 1198791199101003817100381710038171003817 le
1003817100381710038171003817119879119909 minus 1198791199101003817100381710038171003817 forall119909 119910 isin 119870
(C)
Let 120573119899 be a sequence in [0 1] satisfying
(iv) sum119899ge1
120573119899= infin
(v) lim119899rarrinfin
120573119899= 0
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by119909119899+1
= 119878119910119899
119910119899= (1 minus 120573
119899) 119909119899+ 120573119899119879119909119899 119899 ge 1
(12)
Then the sequence 119909119899 converges strongly at the common fixed
point 119901 of 119878 and 119879
Proof For strongly pseudocontractive mappings the exis-tence of a fixed point follows from Delmling [13] It is shownin [11] that the set of fixed points for strongly pseudocontrac-tions is a singleton
By (v) since lim119899rarrinfin
120573119899= 0 there exists 119899
0isin N such that
for all 119899 ge 1198990
120573119899le min 1
4119896
1 minus 119896
(1 + 119871) (1 + 3119871) (13)
where 119896 lt 12 Consider1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2= ⟨119909119899+1
minus 119901 119895 (119909119899+1
minus 119901)⟩
= ⟨119878119910119899minus 119901 119895 (119909
119899+1minus 119901)⟩
= ⟨119879119909119899+1
minus 119901 119895 (119909119899+1
minus 119901)⟩
+ ⟨119878119910119899minus 119879119909119899+1
119895 (119909119899+1
minus 119901)⟩
le 1198961003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2+1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817
(14)which implies that
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 le
1
1 minus 119896
1003817100381710038171003817119878119910119899 minus 119879119909119899+1
1003817100381710038171003817 (15)
where1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817 le1003817100381710038171003817119878119910119899 minus 119879119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899 minus 119879119909
119899+1
1003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119878119910
119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119879119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899 minus 119879119909
119899+1
1003817100381710038171003817
le1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119909119899 minus 119879119910
119899
1003817100381710038171003817 +1003817100381710038171003817119879119910119899 minus 119879119909
119899+1
1003817100381710038171003817
le1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 + 119871 (1003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817 +1003817100381710038171003817119910119899 minus 119909
119899+1
1003817100381710038171003817)
(16)1003817100381710038171003817119910119899 minus 119909
119899+1
1003817100381710038171003817 le1003817100381710038171003817119910119899 minus 119909
119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119909
119899+1
1003817100381710038171003817
=1003817100381710038171003817119910119899 minus 119909
119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119878119910
119899
1003817100381710038171003817
le1003817100381710038171003817119910119899 minus 119909
119899
1003817100381710038171003817 +1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817
(17)
Journal of Applied Mathematics 3
and consequently from (16) we obtain1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817 le (1 + 119871)1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 + 21198711003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817
le (1 + 3119871)1003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817
= (1 + 3119871) 1205731198991003817100381710038171003817119909119899 minus 119879119909
119899
1003817100381710038171003817
le (1 + 119871) (1 + 3119871) 1205731198991003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
(18)
Substituting (18) in (15) and using (13) we get
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 le
(1 + 119871) (1 + 3119871)
1 minus 119896120573119899
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
(19)
So from the above discussion we can conclude that thesequence 119909
119899minus 119901 is bounded Since 119879 is Lipschitzian so
119879119909119899minus 119901 is also bounded Let 119872
1= sup
119899ge1119909119899minus 119901 +
sup119899ge1
119879119909119899minus 119901 Also by (ii) we have
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817 = 120573119899
1003817100381710038171003817119909119899 minus 119879119909119899
1003817100381710038171003817
le 1198721120573119899
997888rarr 0
(20)
as 119899 rarr infin implying that 119909119899minus 119910119899 is bounded so let 119872
2=
sup119899ge1
119909119899minus 119910119899 + 119872
1 Further
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 le
1003817100381710038171003817119910119899 minus 119909119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
le 1198722
(21)
which implies that 119910119899minus 119901 is bounded Therefore 119879119910
119899minus 119901
is also boundedSet
1198723= sup119899ge1
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899 minus 1199011003817100381710038171003817 (22)
Denote119872 = 1198721+1198722+1198723 Obviously119872 lt infin
Now from (12) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2=1003817100381710038171003817119878119910119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2 (23)
and by Lemma 6 we get1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2=1003817100381710038171003817(1 minus 120573
119899) 119909119899+ 120573119899119879119909119899minus 119901
1003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus 119901) + 120573
119899(119879119909119899minus 119901)
1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2120573119899⟨119879119909119899minus 119901 119895 (119910
119899minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2120573119899⟨119879119910119899minus 119901 119895 (119910
119899minus 119901)⟩
+ 2120573119899⟨119879119909119899minus 119879119910119899 119895 (119910119899minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2119896120573
119899
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899 minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2119896120573
119899
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(24)
which implies that
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2le
(1 minus 120573119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817
2+
2119872119871120573119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(25)
because by (13) we have ((1minus120573119899)(1minus2119896120573
119899)) le 1 and (1(1minus
2119896120573119899)) le 2 Hence (23) gives us
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817 (26)
For all 119899 ge 1 put
120588119899=1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
120579119899= 120573119899
120596119899= 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(27)
then according to Lemma 7 we obtain from (26) that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (28)
This completes the proof
Corollary 9 Let 119870 be a nonempty closed convex subset of areal Hilbert space119867 let 119878 119870 rarr 119870 be nonexpansive and let119879 119870 rarr 119870 be Lipschitz strongly pseudocontractive mappingssuch that 119901 isin 119865(119878) cap 119865(119879) and the condition (C) Let 120573119899 be asequence in [0 1] satisfying the conditions (iv) and (v)
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by (12) Then the sequence 119909119899 converges strongly at
the common fixed point 119901 of 119878 and 119879
Example 10 As a particular case wemay choose for instance120573119899= 1119899
Remark 11 (1) The condition (C) is not new and it is due toLiu et al [14]
(2) We prove our results for a hybrid iteration schemewhich is simple in comparison to the previously knowniteration schemes
Acknowledgment
This study was supported by research funds from Dong-AUniversity
References
[1] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[2] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[3] 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
4 Journal of Applied Mathematics
[4] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[5] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[6] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[7] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[8] 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
[9] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[10] 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
[11] 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
[12] S S Chang ldquoSome problems and results in the study ofnonlinear analysisrdquo Nonlinear Analysis vol 30 no 7 pp 4197ndash4208 1997
[13] K Delmling ldquoZeros of accretive operatorsrdquoManuscripta Math-ematica vol 13 pp 283ndash288 1974
[14] 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
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
and consequently from (16) we obtain1003817100381710038171003817119878119910119899 minus 119879119909
119899+1
1003817100381710038171003817 le (1 + 119871)1003817100381710038171003817119878119909119899 minus 119878119910
119899
1003817100381710038171003817 + 21198711003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817
le (1 + 3119871)1003817100381710038171003817119909119899 minus 119910
119899
1003817100381710038171003817
= (1 + 3119871) 1205731198991003817100381710038171003817119909119899 minus 119879119909
119899
1003817100381710038171003817
le (1 + 119871) (1 + 3119871) 1205731198991003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
(18)
Substituting (18) in (15) and using (13) we get
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817 le
(1 + 119871) (1 + 3119871)
1 minus 119896120573119899
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817
le1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
(19)
So from the above discussion we can conclude that thesequence 119909
119899minus 119901 is bounded Since 119879 is Lipschitzian so
119879119909119899minus 119901 is also bounded Let 119872
1= sup
119899ge1119909119899minus 119901 +
sup119899ge1
119879119909119899minus 119901 Also by (ii) we have
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817 = 120573119899
1003817100381710038171003817119909119899 minus 119879119909119899
1003817100381710038171003817
le 1198721120573119899
997888rarr 0
(20)
as 119899 rarr infin implying that 119909119899minus 119910119899 is bounded so let 119872
2=
sup119899ge1
119909119899minus 119910119899 + 119872
1 Further
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 le
1003817100381710038171003817119910119899 minus 119909119899
1003817100381710038171003817 +1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
le 1198722
(21)
which implies that 119910119899minus 119901 is bounded Therefore 119879119910
119899minus 119901
is also boundedSet
1198723= sup119899ge1
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817 + sup119899ge1
1003817100381710038171003817119879119910119899 minus 1199011003817100381710038171003817 (22)
Denote119872 = 1198721+1198722+1198723 Obviously119872 lt infin
Now from (12) for all 119899 ge 1 we obtain1003817100381710038171003817119909119899+1 minus 119901
1003817100381710038171003817
2=1003817100381710038171003817119878119910119899 minus 119901
1003817100381710038171003817
2le1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2 (23)
and by Lemma 6 we get1003817100381710038171003817119910119899 minus 119901
1003817100381710038171003817
2=1003817100381710038171003817(1 minus 120573
119899) 119909119899+ 120573119899119879119909119899minus 119901
1003817100381710038171003817
2
=1003817100381710038171003817(1 minus 120573
119899) (119909119899minus 119901) + 120573
119899(119879119909119899minus 119901)
1003817100381710038171003817
2
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2120573119899⟨119879119909119899minus 119901 119895 (119910
119899minus 119901)⟩
= (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2120573119899⟨119879119910119899minus 119901 119895 (119910
119899minus 119901)⟩
+ 2120573119899⟨119879119909119899minus 119879119910119899 119895 (119910119899minus 119901)⟩
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2119896120573
119899
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2
+ 2120573119899
1003817100381710038171003817119879119909119899 minus 119879119910119899
1003817100381710038171003817
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
le (1 minus 120573119899)21003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 2119896120573
119899
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2
+ 2119872119871120573119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(24)
which implies that
1003817100381710038171003817119910119899 minus 1199011003817100381710038171003817
2le
(1 minus 120573119899)2
1 minus 2119896120573119899
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817
2+
2119872119871120573119899
1 minus 2119896120573119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
le (1 minus 120573119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(25)
because by (13) we have ((1minus120573119899)(1minus2119896120573
119899)) le 1 and (1(1minus
2119896120573119899)) le 2 Hence (23) gives us
1003817100381710038171003817119909119899+1 minus 1199011003817100381710038171003817
2le (1 minus 120573
119899)1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
2+ 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817 (26)
For all 119899 ge 1 put
120588119899=1003817100381710038171003817119909119899 minus 119901
1003817100381710038171003817
120579119899= 120573119899
120596119899= 4119872119871120573
119899
1003817100381710038171003817119909119899 minus 119910119899
1003817100381710038171003817
(27)
then according to Lemma 7 we obtain from (26) that
lim119899rarrinfin
1003817100381710038171003817119909119899 minus 1199011003817100381710038171003817 = 0 (28)
This completes the proof
Corollary 9 Let 119870 be a nonempty closed convex subset of areal Hilbert space119867 let 119878 119870 rarr 119870 be nonexpansive and let119879 119870 rarr 119870 be Lipschitz strongly pseudocontractive mappingssuch that 119901 isin 119865(119878) cap 119865(119879) and the condition (C) Let 120573119899 be asequence in [0 1] satisfying the conditions (iv) and (v)
For arbitrary 1199091
isin 119870 let 119909119899 be a sequence iteratively
defined by (12) Then the sequence 119909119899 converges strongly at
the common fixed point 119901 of 119878 and 119879
Example 10 As a particular case wemay choose for instance120573119899= 1119899
Remark 11 (1) The condition (C) is not new and it is due toLiu et al [14]
(2) We prove our results for a hybrid iteration schemewhich is simple in comparison to the previously knowniteration schemes
Acknowledgment
This study was supported by research funds from Dong-AUniversity
References
[1] S Ishikawa ldquoFixed points by a new iteration methodrdquo Proceed-ings of the American Mathematical Society vol 44 pp 147ndash1501974
[2] W R Mann ldquoMean value methods in iterationrdquo Proceedings ofthe American Mathematical Society vol 4 pp 506ndash510 1953
[3] 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
4 Journal of Applied Mathematics
[4] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[5] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[6] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[7] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[8] 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
[9] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[10] 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
[11] 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
[12] S S Chang ldquoSome problems and results in the study ofnonlinear analysisrdquo Nonlinear Analysis vol 30 no 7 pp 4197ndash4208 1997
[13] K Delmling ldquoZeros of accretive operatorsrdquoManuscripta Math-ematica vol 13 pp 283ndash288 1974
[14] 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
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 Journal of Applied Mathematics
[4] D R Sahu and A Petrusel ldquoStrong convergence of iterativemethods by strictly pseudocontractive mappings in BanachspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol74 no 17 pp 6012ndash6023 2011
[5] C E Chidume ldquoApproximation of fixed points of stronglypseudocontractive mappingsrdquo Proceedings of the AmericanMathematical Society vol 120 no 2 pp 545ndash551 1994
[6] C E Chidume ldquoIterative approximation of fixed points ofLipschitz pseudocontractivemapsrdquo Proceedings of the AmericanMathematical Society vol 129 no 8 pp 2245ndash2251 2001
[7] C E Chidume and C Moore ldquoFixed point iteration for pseu-docontractive mapsrdquo Proceedings of the AmericanMathematicalSociety vol 127 no 4 pp 1163ndash1170 1999
[8] 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
[9] J Schu ldquoApproximating fixed points of Lipschitzian pseudocon-tractive mappingsrdquoHouston Journal of Mathematics vol 19 no1 pp 107ndash115 1993
[10] 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
[11] 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
[12] S S Chang ldquoSome problems and results in the study ofnonlinear analysisrdquo Nonlinear Analysis vol 30 no 7 pp 4197ndash4208 1997
[13] K Delmling ldquoZeros of accretive operatorsrdquoManuscripta Math-ematica vol 13 pp 283ndash288 1974
[14] 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
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