Research Article A Generalization of Suzuki s Lemma

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1 Abstract and Applied Analysis Volume 011, Article ID 84718, 14 pages doi: /011/84718 Research Article A Generalization of Suzuki s Lemma B. Panyanak and A. Cuntavepanit Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 5000, Thailand Correspondence should be addressed to B. Panyanak, banchap@chiangmai.ac.th Received 4 February 011; Accepted 7 April 011 Academic Editor: D. Anderson Copyright q 011 B. Panyanak and A. Cuntavepanit. 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 {z n }, {w n },and{v n } be bounded sequences in a metric space of hyperbolic type X, d,andlet {α n } be a sequence in 0, 1 with 0 < lim inf n α n lim sup n α n < 1. If z n 1 α n w n 1 α n v n for all n N, lim n d z n,v n 0, and lim sup n d w n 1,w n d z n 1,z n 0, then lim n d w n,z n 0. This is a generalization of Lemma. in T. Suzuki, 005. As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT 0 spaces. 1. Introduction Suppose that X, d is a metric space which contains a family L of metric segments isometric images of real line segments such that distinct points x, y X lie on exactly one member S x, y of L. Letα 0, 1, weusethenotationαx 1 α y to denote the point of the segment S x, y with distance αd x, y from y, thatis, d αx 1 α y, y αd x, y. 1.1 We will say that X, d, L is of hyperbolic type if for each p, x, y X and α 0, 1, d αp 1 α x, αp 1 α y 1 α d x, y. 1. It is proved in 1 that 1. implies d p, 1 α x αy 1 α d p, x αd p, y. 1.3

2 Abstract and Applied Analysis It is well-known that Banach spaces are of hyperbolic type. Notice also that CAT 0 spaces and hyperconvex metric spaces are of hyperbolic type see, 3. In 1983, Goebel and Kirk 4 proved that if {z n } and {w n } are sequences in a metric space of hyperbolic type X, d and {α n } 0, 1 which satisfy for all i, n N, i z n 1 α n w n 1 α n z n, ii d w n 1,w n d z n 1,z n, iii d w i n,x i a<, iv α n b<1, and v n 1 α n, then lim n d w n,z n 0. It was proved by Suzuki 5 that one obtains the same conclusion if the conditions i v are replaced by the conditions S1 S4 as follows: S1 z n 1 α n w n 1 α n z n, S lim sup n d w n 1,w n d z n 1,z n 0, S3 {z n } and {w n } are bounded sequences, S4 0 < lim inf n α n lim sup n α n < 1. Both Goebel-Kirk s and Suzuki s results have been used to prove weak and strong convergence theorems for approximating fixed points of various types of mappings. The purpose of this paper is to generalize Suzuki s result by relaxing the condition S1, namely, we can define z n 1 in terms of w n and v n such that lim n d z n,v n 0. Precisely, we are going to prove the following lemma. Lemma 1.1. Let {z n }, {w n },and{v n } be bounded sequences in a metric space of hyperbolic type X, d, andlet{α n } be a sequence in 0, 1 with 0 < lim inf n α n lim sup n α n < 1. Suppose that z n 1 α n w n 1 α n v n for all n N, lim n d z n,v n 0, andlim sup n d w n 1,w n d z n 1,z n 0,thenlim n d w n,z n 0.. Proof of Lemma 1.1 We begin by proving a crucial lemma. Lemma.1. Let {z n }, {w n },and{v n } be sequences in a metric space of hyperbolic type X, d, and let {α n } be a sequence in 0, 1 with lim sup n α n < 1.Put r lim sup d w n,z n or r lim inf d w n,z n..1 Suppose that r<, z n 1 α n w n 1 α n v n for all n N, lim n d z n,v n 0,and lim sup d w n 1,w n d z n 1,z n 0,. then lim inf d w n k,z n 1 α n α n 1 α n k 1 r 0.3 holds for all k N.

3 Abstract and Applied Analysis 3 Proof. This proof is patterned after the proof of 5, Lemma 1.1. Foreachn N, letu n α n w n 1 α n z n,thenby 1.,wehave d u n,z n 1 1 α n d z n,v n d z n,v n..4 This implies d w n 1,z n 1 d w n 1,w n d w n,u n d u n,z n 1 d w n 1,w n d w n,u n d z n,v n..5 Since d w n,u n d u n,z n d w n,z n,then d w n 1,z n 1 d w n,z n d w n 1,w n d w n,u n d z n,v n d w n,u n d u n,z n d w n 1,w n d z n,v n d u n,z n..6 This fact and.4 yield d w n 1,z n 1 d w n,z n d z n,v n d w n 1,z n 1 d w n,z n d u n,z n 1 d w n 1,w n d z n,v n d u n,z n d u n,z n 1 d w n 1,w n d z n,v n d z n 1,z n d w n 1,w n d z n 1,z n d z n,v n..7 Since lim n d z n,v n 0, we have lim sup d w n 1,z n 1 d w n,z n lim sup d w n 1,w n d z n 1,z n..8 By using this fact, we have, for j N, lim sup d wn j,z n j d wn,z n lim sup j 1 d w n i 1,z n i 1 d w n i,z n i j 1 lim sup d w n i 1,z n i 1 d w n i,z n i j 1 lim sup d w n i 1,w n i d z n i 1,z n i 0..9 Put a 1 lim sup n α n /. We note that 0 <a 1/. Fix k, l N and ε>0, then there exists m l such that a 1 α n, d z n,v n ε/, d w n 1,w n d z n 1,z n ε/, and

4 4 Abstract and Applied Analysis d w n j,z n j d w n,z n ε/4, for all n m and j lim sup n d w n,z n, we choose m m satisfying 1,,...,k.Inthecaseofr d w m k,z m k r ε 4,.10 and d w n,z n r ε/ for all n m. Wenotethat d w m j,z m j d wm k,z m k ε 4 r ε,.11 for j 0, 1,...,k 1. In the case of r lim inf n d w n,z n, we choose m m satisfying d w m,z m r ε 4,.1 and d w n,z n r ε/ for all n m. Wenotethat d w m j,z m j d wm,z m ε 4 r ε,.13 for j 1,,...,k.Inbothcases,suchm satisfies that m l, a 1 α n ε/,d w n 1,w n d z n 1,z n ε/ foralln m, and 1, d z n,v n for j 0, 1,...,k. We next show that r ε d w m j,z m j r ε,.14 d w m k,z m j 1 αm j α m j 1 α m k 1 r k j k a k j ε,.15 for j 0, 1,...,k 1. Since r ε d w m k,z m k d w m k,α m k 1 w m k 1 1 α m k 1 v m k 1 α m k 1 d w m k,w m k 1 1 α m k 1 d w m k,v m k 1 α m k 1 d z m k,z m k 1 ε 1 α m k 1 d w m k,z m k 1 1 α m k 1 d z m k 1,v m k 1 α m k 1 d z m k,u m k 1 d u m k 1,z m k 1 ε 1 α m k 1 d w m k,z m k 1 1 α m k 1 d z m k 1,v m k 1

5 Abstract and Applied Analysis 5 α m k 1 1 α m k 1 d z m k 1,v m k 1 α m k 1 d w m k 1,z m k 1 ε 1 α m k 1 d w m k,z m k 1 1 α m k 1 d z m k 1,v m k 1 α m k 1 r ε ε 1 α m k 1 d w m k,z m k 1 1 α m k 1 d z m k 1,v m k 1 ε α m k 1 r ε 1 α m k 1 d w m k,z m k 1 1 α m k 1.16 and a 1 α m k 1,wehave 1 α m k 1 r 3/ ε 1 α d w m k,z m k 1 m k 1 ε/ 1 α m k 1 k 1 1 α m k 1 r ε ε a a k 1 α m k 1 r ε. a.17 Hence,.15 holds for j k 1. We assume that.15 holds for some j {1,,...,k 1}. Then, since k j k 1 α m i r ε i j a k j d w m k,z m j d w m k,α m j 1 w m j 1 1 α m j 1 vm j 1 α m j 1 d w m k,w m j 1 1 αm j 1 d wm k,v m j 1 α m j 1 d w m i 1,w m i 1 α m j 1 d wm k,v m j 1 α m j 1 d z m i 1,z m i ε 1 α m j 1 d wm k,v m j 1 α m j 1 d z m i 1,z m i kε 1 α m j 1 d wm k,v m j 1 α m j 1 α m i d w m i,z m i 1 α m i d z m i,v m i kε 1 α m j 1 d wm k,v m j 1 α m j 1 α m i d w m i,z m i α m j 1 1 α m i d z m i,v m i kε 1 α m j 1 d wm k,z m j 1 1 αm j 1 d zm j 1,v m j 1

6 6 Abstract and Applied Analysis α m j 1 α m i r ε k 1 ε α m j 1 α m i r 3k 1 ε kε 1 α m j 1 d wm k,z m j 1 1 α m j 1 d wm k,z m j 1,.18 we obtain d w m k,z m j 1 1 k 1 i j α m i α k 1 m j 1 α m i r 1 α m j 1 1 α m i r k j 1 k a k j 1 ε. k j k /a k j / 3k 1 / 1 α m j 1 ε.19 Hence,.15 holds for j : j 1. Therefore,.15 holds for all j 0, 1,...,k 1. Specially, we have On the other hand, we have d w m k,z m 1 α m α m 1 α m k 1 r k k a k ε..0 d w m k,z m d w m k,z m k d z m i 1,z m i d w m k,z m k d z m i 1,u m i d u m i,z m i d w m k,z m k d v m i,z m i α m i d w m i,z m i r ε k 1 kε α m i r ε k 1 1 α m i r ε..1 This fact and.0 imply d w m k,z m 1 α m α m 1 α m k 1 r k k a k ε.. Since l N and ε > 0 are arbitrary, we obtain the desired result.

7 Abstract and Applied Analysis 7 By using Lemma.1 together with the argument in the proof of 5, Lemma.,simply replacing by d,, we can obtain Lemma 1.1 as desired. 3. Applications In this section, we apply Lemma 1.1 to prove two strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT 0 spaces. The results we obtain are analogs of the Banach space results of Song and Li 6. AmetricspaceX is a CAT 0 space if it is geodesically connected, and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT 0 space. Other examples include pre-hilbert spaces see 7, R trees see 8, Euclidean buildings see 9, the complex Hilbert ball with a hyperbolic metric see 10, and many others. For a thorough discussion of these spaces and of the fundamental role they play in geometry, we refer the reader to Bridson and Haefliger 7. Fixed-point theory in CAT 0 spaces was first studied by Kirk see, 11.Heshowed that every nonexpansive single-valued mapping defined on a bounded closed convex subset of a complete CAT 0 space always has a fixed point. Since then, the fixed-point theory for single-valued and multivalued mappings in CAT 0 spaces has been rapidly developed, and many papers have appeared see, e.g., 1 4 and the references therein. Itisworth mentioning that fixed-point theorems in CAT 0 spaces specially in R trees can be applied to graph theory, biology, and computer science see, e.g., 8, 5 8. Let X, d be a metric space. A geodesic path joining x X to y X or, more briefly, a geodesic from x to y is a map c from a closed interval 0,l R to X such that c 0 x, c l y, andd c t,c t t t for all t, t 0,l. Inparticular,c is an isometry and d x, y l. The image α of c is called a geodesic or metric segment joining x and y. When it is unique, this geodesic is denoted by x, y. Thespace X, d is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y X. AsubsetY X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle M x 1,x,x 3 in a geodesic space X, d consists of three points x 1,x,x 3 in X the vertices of M and a geodesic segment between each pair of vertices the edges of M. Acomparison triangle for geodesic triangle M x 1,x,x 3 in X, d is a triangle M x 1,x,x 3 : M x 1, x, x 3 in the Euclidean plane E such that d E x i, x j d x i,x j for i, j {1,, 3}. A geodesic space is said to be a CAT 0 space if all geodesic triangles satisfy the following comparison axiom. CAT 0 : LetM be a geodesic triangle in X, andletm be a comparison triangle for M, then M is said to satisfy the CAT 0 inequality if for all x, y M and all comparison points x, y M, d x, y d E x, y. 3.1 We now collect some elementary facts about CAT 0 spaces.

8 8 Abstract and Applied Analysis Lemma 3.1. Let X, d be a CAT0 space. i see [7, Proposition.4] Let C be a closed-convex subset of X, then, for every x X,there exists a unique point Px C such that d x, Px inf {d x, y : y C}. The mapping P : X C is called the nearest point or metric projection from X onto C. ii see [15, Lemma.5] For x, y, z X and t 0, 1,onehas d 1 t x ty, z 1 t d x, z td y, z t 1 t d x, y. 3. Recall that a mapping T on a CAT 0 space X, d is called nonexpansive if d Tx,Ty d x, y x, y X. 3.3 Apointx X is called a fixed point of T if x Tx. We will denote by F T the set of fixed points of T. The following result can be found in 13 see also, Theorem1. Theorem 3.. Let C be a convex subset of a CAT0 space, and let T : C C be a nonexpansive mapping whose fixed-point set is nonempty, then F T is closed and convex. A continuous linear functional μ on l, the Banach space of bounded real sequences, is called a Banach limit if μ μ 1, 1,... 1andμ n a n μ n a n 1 for all {a n } l. Lemma 3.3 see 9,Proposition. Let {a 1,a, } l be such that μ n a n 0 for all Banach limits μ and lim sup n a n 1 a n 0,thenlim sup n a n 0. Lemma 3.4 see 1, Lemma.1. Let C be a closed-convex subset of a complete CAT0 space X, and let T : C C be a nonexpansive mapping. Let u C be fixed. For each t 0, 1, the mapping S t : C C defined by S t z tu 1 t Tz for z C 3.4 has a unique fixed-point z t C, thatis, z t S t z t tu 1 t T z t. 3.5 Lemma 3.5 see 1, Lemma.. Let C and T be as the preceding lemma, then F T / if and only if {z t } given by the formula 3.5 remains bounded as t 0. In this case, the following statements hold: 1 {z t } converges to the unique fixed-point z of T which is the nearest u, d u, z μ n d u, x n for all Banach limits μ and all bounded sequences {x n } with lim n d x n,tx n 0. Lemma 3.6 see 30, Lemma.1. Let {α n } n 1 satisfying the condition be a sequence of nonnegative real numbers α n 1 1 γ n αn γ n σ n, n 1, 3.6

9 Abstract and Applied Analysis 9 where {γ n } and {σ n } are sequences of real numbers, such that i {γ n } 0, 1 and n 1 γ n, ii either lim sup σ n 0 or n 1 γ nσ n <, then lim α n 0. The following result is an analog of 6,Theorem3.1. Theorem 3.7. Let C be a nonempty closed-convex subset of a complete CAT0 space X, andlet T : C C be a nonexpansive mapping such that F T /. Givenapointu C and sequences {α n } and {λ n } in 0, 1, the following conditions are satisfied: C1 lim α n 0, C n 1 α n, C3 0 < lim inf λ n lim sup λ n < 1. Define a sequence {x n } in C by x 1 x C arbitrarily, and x n 1 λ n x n 1 λ n T α n u 1 α n x n, n 1, 3.7 then {x n } converges to a fixed-point Pu of T, wherep is the nearest point projection from C onto F T. Proof. For each n 1, we let y n T α n u 1 α n x n. We divide the proof into 3 steps. i We show that {x n } and {y n } are bounded sequences. ii We show that lim n d x n,tx n 0. iii We show that {x n } converges to a fixed-point z F T which is the nearest u. i Let p F T, thenwehave d x n 1,p d λ n x n 1 λ n y n,p λ n d x n,p 1 λ n d T α n u 1 α n x n,p λ n d x n,p 1 λ n α n d u, p 1 λ n 1 α n d x n,p λ n 1 λ n 1 α n d x n,p 1 λ n α n d u, p 1 1 λ n α n d x n,p 1 λ n α n d u, p 3.8 max { d x n,p,d u, p }. Now, an induction yields d x n,p max { d x 1,p,d u, p }, n Hence, {x n } is bounded and so is {y n }.

10 10 Abstract and Applied Analysis ii First, we show that lim n d x n,y n 0. Consider d y n 1,y n d T αn 1 u 1 α n 1 x n 1,T α n u 1 α n x n d α n 1 u 1 α n 1 x n 1,α n u 1 α n x n α n 1 d u, α n u 1 α n x n 1 α n 1 d x n 1,α n u 1 α n x n α n 1 1 α n d u, x n 1 α n 1 α n d u, x n 1 1 α n 1 1 α n d x n 1,x n This implies d y n 1,y n d xn 1,x n α n 1 1 α n d u, x n 1 α n 1 α n d u, x n By the condition C1,wehave lim sup d yn 1,y n d xn 1,x n It follows from Lemma 1.1 that lim d x n,y n 0. Now, d x n,tx n d x n,y n d yn,tx n d x n,y n d T αn u 1 α n x n,tx n d x n,y n d αn u 1 α n x n,x n 3.13 d x n,y n αn d u, x n 0 as n. iii From Lemma 3.4, letz lim t 0 z t where z t is given by the formula 3.5. Thenz is the point of F T which is the nearest u. By applying Lemma 3.1, wehave d x n 1,z d λ n x n 1 λ n y n,z λ n d x n,z 1 λ n d y n,z λ n 1 λ n d x n,y n λ n d x n,z 1 λ n d α n u 1 α n x n,z λ n 1 λ n d x n,y n [ ] λ n d x n,z 1 λ n α n d u, z 1 α n d x n,z α n 1 α n d u, x n [ ] λ n 1 λ n 1 α n d x n,z α n 1 λ n d u, z 1 α n d u, x n ] 1 1 λ n α n d x n,z 1 λ n α n [d u, z 1 α n d u, x n. 3.14

11 Abstract and Applied Analysis 11 By Lemma 3.5, wehaveμ n d u, z d u, x n 0 for all Banach limit μ. Moreover,since d x n 1,x n d λ n x n 1 λ n y n,x n 1 λ n d y n,x n 0 as n, [ ] lim sup d u, z d u, x n 1 d u, z d u, x n It follows from condition C1 and Lemma 3.3 that lim sup d u, z 1 α n d u, x n lim sup d u, z d u, x n Hence, the conclusion follows from Lemma 3.6. Remark 3.8. In the proof of Theorem 3.7, one may observe that it is not necessary to use Lemma 1.1 because Suzuki s original lemma is sufficient. However, in 6, thereisastrong convergence theorem for another type of modified Halpern iteration see 6, Theorem3.. We show that the proof is quite easy when we use Lemma 1.1. Theorem 3.9. Let C be a nonempty closed-convex subset of a complete CAT0 space X, andlet T : C C be a nonexpansive mapping such that F T /. Givenapointu C and sequences {α n } and {λ n } in 0, 1, the following conditions are satisfied: C1 lim α n 0, C n 1 α n, C3 0 < lim inf λ n lim sup λ n < 1. Define a sequence {x n } in C by x 1 x C arbitrarily, and x n 1 λ n α n u 1 α n x n 1 λ n Tx n, n 1, 3.17 Then {x n } converges to a fixed-point Pu of T, wherep is the nearest point projection from C onto F T. Proof. Using the same technique as in the proof of Theorem 3.7, we easily obtain that both {x n } and {Tx n } are bounded. Let y n α n u 1 α n x n,thenx n 1 λ n y n 1 λ n Tx n.bythe condition C1,wehave d x n,y n d xn,α n u 1 α n x n α n d x n,u 0 as n It follows from the nonexpansiveness of T that lim sup d Tx n 1,Tx n d x n 1,x n

12 1 Abstract and Applied Analysis By Lemma 1.1, wehave lim d Tx n,x n From 3.18 and 3.0,wegetthat d x n 1,x n d λ n y n 1 λ n Tx n,x n λ n d y n,x n 1 λn d Tx n,x n 0 as n. 3.1 Let z lim t 0 z t where z t is given by 3.5, thenz is the point of F T which is the nearest u. Consider d x n 1,z d λ n y n 1 λ n Tx n,z λ n d y n,z 1 λ n d Tx n,z λ n 1 λ n d y n,tx n λ n d α n u 1 α n x n,z 1 λ n d Tx n,z λ n α n d u, z 1 α n d x n,z α n 1 α n d u, x n 1 λ n d x n,z λ n 1 α n 1 λ n d x n,z λ n α n d u, z λ n α n 1 α n d u, x n 1 λ n α n d x n,z λ n α n d u, z 1 α n d u, x n. 3. By Lemma 3.5, wehaveμ n d u, z d u, x n 0 for all Banach limit μ. Moreover,since d x n 1,x n 0, then [ ] lim sup d u, z d u, x n 1 d u, z d u, x n It follows from condition C1 and Lemma 3.3 that lim sup d u, z 1 α n d u, x n lim sup d u, z d u, x n Hence, the conclusion follows from Lemma 3.6. Acknowledgment This research was supported by the Faculty of Science, Chiang Mai University, Chiang Mai, Thailand.

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