Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 618767, 9 pages doi:10.1155/2010/618767 ResearchArticleIshikawaIterativeProcessforaPairofSingle-valuedandMultivaluedNonexpansiveMappingsinBanach Spaces K. Sokhuma 1 and A. Kaewkhao 2 1 Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand 2 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Correspondence should be addressed to A. Kaewkhao, akaewkhao@yahoo.com Received 8 August 2010; Accepted 24 September 2010 Academic Editor: T. D. Benavides Copyright q 2010 K. Sokhuma and A. Kaewkhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X,andlet t : E → E and T : E → KCE be asingle-valuednonexpansive mapping andamultivaluednonexpansive mapping, respectively. Assume in addition that Fixt ∩ FixT / ∅ and Tw {w} for all w ∈ Fixt ∩ FixT. We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary x 0 ∈ E by y n 1 − β n x n β n z n , x n1 1 − α n x n α n ty n ,where z n ∈ Tx n and {α n }, {β n } are sequences of positive numbers satisfying 0 <a≤ α n , β n ≤ b<1, converges strongly to a common fixed point of t and T; that is, there exists x ∈ E such that x tx ∈ Tx. 1. Introduction Let X be aBanach space, and let E be a nonempty subset of X. We will denote by FBE the family of nonempty boundedclosedsubsetsofE and by KCE the family of nonempty compact convex subsets of E.LetH·, · be the Hausdorff distance on FBX,thatis, H A, B max sup a∈A dist a, B , sup b∈B dist b, A ,A,B∈ FB X , 1.1 where dista, Binf{a − b : b ∈ B} is the distance from the point a to the subset B. 2 Fixed Point Theory and Applications A mapping t : E → E is said to be nonexpansive if tx − ty ≤ x − y , ∀x, y ∈ E. 1.2 Apointx is called a fixed point of t if tx x. Amultivalued mapping T : E → FBX is said to be nonexpansive if H Tx,Ty ≤ x − y , ∀x, y ∈ E. 1.3 Apointx is called a fixed point foramultivalued mapping T if x ∈ Tx. We use the notation FixT standing for the set of fixed points ofa mapping T and Fixt ∩ FixT standing for the set of common fixed points of t and T. Precisely, a point x is called a common fixed point of t and T if x tx ∈ Tx. In 2006, S. Dhompongsa et al. 1 proved a common fixed point theorem for two nonexpansive commuting mappings. Theorem 1.1 see 1,Theorem4.2. Let E be a nonempty bounded closed convex subset ofa uniformly Banach space X,andlett : E → E,andT : E → KCE be anonexpansive mapping andamultivaluednonexpansive mapping, respectively. Assume that t and T are commuting; that is, if for every x, y ∈ E such that x ∈ Ty and ty ∈ E, there holds tx ∈ Tty.Then,t and T have a common fixed point. In this paper, we introduce an iterativeprocessina new sense, called the modified Ishikawa iteration method with respect to apairofsingle-valuedandmultivaluednonexpansive mappings. We also establish the strong convergence theorem ofa sequence from such processina nonempty compact convex subset ofa uniformly convex Banach space. 2. Preliminaries The important pr operty of the uniformly convex Banach space we use is the following lemma proved by Schu 2 in 1991. Lemma 2.1 see 2. Let X be a uniformly convex Banach space, let {u n } be a sequence of real numbers such that 0 <b≤ u n ≤ c<1 for all n ≥ 1,andlet{x n } and {y n } be sequences of X such that lim sup n →∞ x n ≤a, lim sup n →∞ y n ≤a,andlim n →∞ u n x n 1 − u n y n afor some a ≥ 0.Then,lim n →∞ x n − y n 0. The following observation will be used in proving our results, and the proof is straightforward. Lemma 2.2. Let X be aBanach space, and let E be a nonempty closed convex subset of X.Then, dist y, Ty ≤ y − x dist x, Tx H Tx,Ty , 2.1 where x, y ∈ E and T is amultivaluednonexpansive mapping from E into FBE. Fixed Point Theory and Applications 3 A fundamental principle which plays a key role in e rgodic theory is the demiclosed- ness principle. A mapping t defined on a subset E ofaBanach space X is said to be demiclosed if any sequence {x n } in E the following implication holds: x n xand tx n → y implies tx y. Theorem 2.3 see 3. Let E be a nonempty closed convex subset ofa uniformly convex Banach space X,andlett : E → E be anonexpansive mapping. If a sequence {x n } in E converges weakly to p and {x n − tx n } converges to 0 as n →∞,thenp ∈ Fixt. In 1974, Ishikawa introduced the following well-known iteration. Definition 2.4 see 4.LetX be aBanach space, let E be a closed convex subset of X,andlet t be a selfmap on E.Forx 0 ∈ E,thesequence{x n } ofIshikawa iterates of t is defined by y n 1 − β n x n β n tx n , x n1 1 − α n x n α n ty n ,n≥ 0, 2.2 where {α n } and {β n } are real sequences. AnonemptysubsetK of E is said to be proximinal if, for any x ∈ E,thereexistsan element y ∈ K such that x−y distx, K. We will denote PK by the family of nonempty proximinal bounded subsets of K. In 2005, Sastry and Babu 5 defined the Ishikawaiterative scheme formultivaluedmappings as follows. Let E be a compact convex subset ofa Hilbert space X,andletT : E → PE be amultivalued mapping, and fix p ∈ FixT. x 0 ∈ E, y n 1 − β n x n β n z n , x n1 1 − α n x n α n z n , ∀n ≥ 0, 2.3 where {α n }, {β n } are sequences in 0, 1 with z n ∈ Tx n such that z n − p distp, Tx n and z n − p distp, Ty n . They also proved the strong convergence of the above Ishikawaiterative scheme foramultivaluednonexpansive mapping T with a fixed point p under some certain conditions ina Hilbert space. Recently, Panyanak 6 extended the results of Sastry and Babu 5 to a uniformly convex Banach space and also modified the above Ishikawaiterative scheme as follows. Let E be a n onempty convex subset ofa uniformly convex Banach space X,andlet T : E → PE be amultivalued mapping x 0 ∈ E, y n 1 − β n x n β n z n , x n1 1 − α n x n α n z n , ∀n ≥ 0, 2.4 4 Fixed Point Theory and Applications where {α n }, {β n } are sequences in 0, 1 with z n ∈ Tx n and u n ∈ FixT such that z n − u n distu n ,Tx n and x n − u n distx n , FixT, respectively. Moreover, z n ∈ Tx n and v n ∈ FixT such that z n − v n distv n ,Tx n and y n − v n disty n , FixT, respectively. Very recently, Song and Wang 7, 8 improved the results of 5, 6 by means of the following Ishikawaiterative scheme. Let T : E → FBE be amultivalued mapping, where α n ,β n ∈ 0, 1. The Ishikawaiterative scheme {x n } is defined by x 0 ∈ E, y n 1 − β n x n β n z n , x n1 1 − α n x n α n z n , ∀n ≥ 0, 2.5 where z n ∈ Tx n and z n ∈ Ty n such that z n − z n ≤HTx n ,Ty n γ n and z n1 − z n ≤ HTx n1 ,Ty n γ n , respectively. Moreover, γ n ∈ 0, ∞ such that lim n →∞ γ n 0. At the same period, Shahzad and Zegeye 9 modified the Ishikawaiterative scheme {x n } andextendedtheresultof7,Theorem2 to amultivalued quasinonexpansive mapping as follows. Let K be a nonempty convex subset ofaBanach space X,andletT : E → FBE be amultivalued mapping, where α n ,β n ∈ 0, 1 . The Ishikawaiterative scheme {x n } is defined by x 0 ∈ E, y n 1 − β n x n β n z n , x n1 1 − α n x n α n z n , ∀n ≥ 0, 2.6 where z n ∈ Tx n and z n ∈ Ty n . In this paper, we introduce a new iteration method modifying the above ones and call it the modified Ishikawa iteration method. Definition 2.5. Let E be a nonempty closed bounded convex subset ofaBanach space X,lett : E → E be asingle-valuednonexpansive mapping, and let T : E → FBE be amultivaluednonexpansive mapping. The sequence {x n } of the modified Ishikawa iteration is defined by y n 1 − β n x n β n z n , x n1 1 − α n x n α n ty n , 2.7 where x 0 ∈ E, z n ∈ Tx n ,and0<a≤ α n , β n ≤ b<1. 3. Main Results We first prove the following lemmas, which play very important roles in this section. Lemma 3.1. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X,and let t : E → E and T : E → FBE be asingle-valuedandamultivaluednonexpansive mapping, Fixed Point Theory and Applications 5 respectively, and Fixt ∩ FixT / ∅ satisfying Tw {w} for all w ∈ Fixt ∩ FixT.Let{x n } be the sequence of the modified Ishikawa iteration defined by 2.7.Then,lim n →∞ x n − w exists for all w ∈ Fixt ∩ FixT. Proof. Letting x 0 ∈ E and w ∈ Fixt ∩ FixT,wehave x n1 − w 1 − α n x n α n t 1 − β n x n β n z n − w 1 − α n x n α n t 1 − β n x n β n z n − 1 − α n w − α n w ≤ 1 − α n x n − w α n t 1 − β n x n β n z n − w ≤ 1 − α n x n − w α n 1 − β n x n β n z n − w 1 − α n x n − w α n 1 − β n x n β n z n − 1 − β n w − β n w ≤ 1 − α n x n − w α n 1 − β n x n − w α n β n z n − w 1 − α n x n − w α n 1 − β n x n − w α n β n dist z n ,Tw ≤ 1 − α n x n − w α n 1 − β n x n − w α n β n H Tx n ,Tw ≤ 1 − α n x n − w α n 1 − β n x n − w α n β n x n − w x n − w . 3.1 Since {x n − w} is a decreasing and bounded s equence, we can conclude that the limit of {x n − w} exists. We can see how Lemma 2.1 is useful via the following lemma. Lemma 3.2. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X,and let t : E → E and T : E → FBE be asingle-valuedandamultivaluednonexpansive mapping, respectively, and Fixt∩FixT / ∅ satisfying Tw {w} for all w ∈ Fixt ∩FixT.Let{x n } be the sequence of the modified Ishikawa iteration defined by 2.7.If 0 <a≤ α n ≤ b<1 for some a, b ∈ , then, lim n →∞ ty n − x n 0. Proof. Let w ∈ Fixt ∩ FixT.ByLemma 3.1, we put lim n →∞ x n − w c and consider ty n − w ≤ y n − w 1 − β n x n β n z n − w ≤ 1 − β n x n − w β n z n − w 1 − β n x n − w β n dist z n ,Tw ≤ 1 − β n x n − w β n H Tx n ,Tw ≤ 1 − β n x n − w β n x n − w x n − w . 3.2 6 Fixed Point Theory and Applications Then, we have lim sup n →∞ ty n − w ≤ lim sup n →∞ y n − w ≤ lim sup n →∞ x n − w c. 3.3 Further, we have c lim n →∞ x n1 − w lim n →∞ 1 − α n x n α n ty n − w lim n →∞ α n ty n − α n w x n − α n x n α n w − w lim n →∞ α n ty n − w 1 − α n x n − w . 3.4 By Lemma 2.1, we can conclude that lim n →∞ ty n −w−x n −w lim n →∞ ty n −x n 0. Lemma 3.3. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X,and let t : E → E and T : E → FBE be asingle-valuedandamultivaluednonexpansive mapping, respectively, and Fixt ∩ FixT / ∅ satisfying Tw {w} for all w ∈ Fixt ∩ FixT.Let{x n } be the sequence of the modified Ishikawa iteration d efined by 2.7.If0 <a≤ α n , β n ≤ b<1 for some a, b ∈ ,thenlim n →∞ x n − z n 0. Proof. Let w ∈ Fixt ∩ FixT.Weput,asinLemma 3.2, lim n →∞ x n − w c.Forn ≥ 0, we have x n1 − w 1 − α n x n α n ty n − w 1 − α n x n α n ty n − 1 − α n w − α n w ≤ 1 − α n x n − w α n ty n − w ≤ 1 − α n x n − w α n y n − w , 3.5 and hence x n1 − w − x n − w ≤−α n x n − w α n y n − w , x n1 − w − x n − w ≤ α n y n − w − x n − w , x n1 − w − x n − w α n ≤ y n − w − x n − w . 3.6 Therefore, since 0 <a≤ α n ≤ b<1, x n1 − w − x n − w α n x n − w ≤ y n − w . 3.7 Fixed Point Theory and Applications 7 Thus, lim inf n →∞ x n1 − w − x n − w α n x n − w ≤ lim inf n →∞ y n − w . 3.8 It follows that c ≤ lim inf n →∞ y n − w . 3.9 Since, from 3.3, lim sup n →∞ y n − w≤c,wehave c lim n →∞ y n − w lim n →∞ 1 − β n x n β n z n − w lim n →∞ 1 − β n x n − w β n z n − w . 3.10 Recall that z n − w dist z n ,Tw ≤ H Tx n ,Tw ≤ x n − w . 3.11 Hence, we have lim sup n →∞ z n − w ≤ lim sup n →∞ x n − w c. 3.12 Using the fact that 0 <a≤ β n ≤ b<1andby3.10, we can conclude that lim n →∞ x n − z n 0. The following lemma allows us to go on. Lemma 3.4. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X,and let t : E → E and T : E → FBE be asingle-valuedandamultivaluednonexpansive mapping, respectively, and Fixt ∩ FixT / ∅ satisfying Tw {w} for all w ∈ Fixt ∩ FixT.Let{x n } be the sequence of the modified Ishikawa iteration defined by 2.7.If0 <a≤ α n , β n ≤ b<1,then lim n →∞ tx n − x n 0. 8 Fixed Point Theory and Applications Proof. Consider tx n − x n tx n − ty n ty n − x n ≤ tx n − ty n ty n − x n ≤ x n − y n ty n − x n x n − 1 − β n x n − β n z n ty n − x n x n − x n β n x n − β n z n ty n − x n β n x n − z n ty n − x n . 3.13 Then, we have lim n →∞ tx n − x n ≤ lim n →∞ β n x n − z n lim n →∞ ty n − x n . 3.14 Hence, by Lemmas 3.2 and 3.3, lim n →∞ tx n − x n 0. We give the sufficient conditions which imply the existence of common fixed points forsingle-valuedmappingsandmultivaluednonexpansive mappings, respectively, as follows Theorem 3.5. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X, and let t : E → E and T : E → FBE be asingle-valuedandamultivaluednonexpansive mapping, respectively, and Fixt∩FixT / ∅ satisfying Tw {w} for all w ∈ Fixt ∩FixT.Let{x n } be the sequence of the modified Ishikawa iteration defined by 2.7.If0 <a≤ α n , β n ≤ b<1,thenx n i → y for some subsequence {x n i } of {x n } implies y ∈ Fixt ∩ FixT. Proof. Assume that lim n →∞ x n i − y 0. From Lemma 3.4,wehave 0 lim n →∞ tx n i − x n i lim n →∞ I − t x n i . 3.15 Since I − t is demiclosed at 0, we have I − ty0, and hence y ty,thatis,y ∈ Fixt.By Lemma 2.2 and by Lemma 3.4,wehave dist y, Ty ≤ y − x n i dist x n i ,Tx n i H Tx n i ,Ty ≤ y − x n i x n i − z n i x n i − y −→ 0, as i →∞. 3.16 It follows that y ∈ FixT. Therefore y ∈ Fixt ∩ FixT as desired. Hereafter, we arrive at the convergence theorem of the sequence of the modified Ishikawa iteration. We conclude this paper with the following theorem. Theorem 3.6. Let E be a nonempty compact convex subset ofa uniformly convex Banach space X, and let t : E → E and T : E → FBE be asingle-valuedandamultivaluednonexpansive mapping, respectively, and Fixt ∩ FixT / ∅ satisfying Tw {w} for all w ∈ Fixt ∩ FixT.Let{x n } be Fixed Point Theory and Applications 9 the sequence of the modified Ishikawa iteration defined by 2.7 with 0 <a≤ α n , β n ≤ b<1.Then {x n } converges strongly to a common fixed point of t and T. Proof. Since {x n } is contained in E which is compact, there exists a subsequence {x n i } of {x n } such that {x n i } converges strongly to some point y ∈ E, that is, lim i →∞ x n i − y 0. By Theorem 3.5,wehavey ∈ Fixt ∩ FixT,andbyLemma 3.1,wehavethatlim n →∞ x n − y exists. It must be the case in which lim n →∞ x n − y lim i →∞ x n i − y 0. Therefore, {x n } converges strongly to a common fixed point y of t and T. Acknowledgments The first author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree for this research. The authors would like to express their deep gratitude to Prof. Dr. Sompong Dhompongsa whose guidance and support were valuable for the completion of the paper. This work was completed with the support of the Commission on Higher Education and The Thailand Research Fund under Grant no. MRG5180213. References 1 S.Dhompongsa,A.Kaewcharoen,andA.Kaewkhao,“TheDom ´ ınguez-Lorenzo condition andmultivaluednonexpansive mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 958–970, 2006. 2 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991. 3 F. E. 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