On the S-Iteration Processes for Multivalued Mappings in Some CAT(κ) Spaces

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1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 18, HIKARI Ltd, On the S-Iteration Processes for Multivalued Mappings in Some CAT(κ) Spaces Kritsana Sokhuma Department of Mathematics, Faculty of Science and Technology Muban Chom Bueng Rajabhat University, Ratchaburi 70150, Thailand Copyright c 2014 Kritsana Sokhuma. 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. Abstract The purpose of this paper is to prove the strong convergence of the S-iteration processes for some generalized multivalued nonexpansive mappings in the framework of CAT(1) spaces. Our results extend the corresponding results given by Shahzad and Zegeye [5], Şahin and Başarir [10], Puttasontiphot [11], Song and Cho [12]. Mathematics Subject Classification: 47H09, 47H10 Keywords: Multivalued mappings, S-iteration, Fixed points, CAT(0) spaces, CAT(κ) spaces 1 Introduction Fixed point theory in CAT(κ) spaces was first studied by Kirk ([2], [3]). His works were followed by a series of new works by many authors (see, e.g., [4]-[6]) mainly focusing on CAT(0) spaces. Since any CAT(κ) space is a CAT(κ ) space for κ >κ, all results for CAT(0) spaces immediately apply to any CAT(κ) space with κ 0. Notice also that all CAT(κ) spaces are uniformly convex metric spaces in the sense of [7]. Thus, the results in [7] concerning uniformly convex metric spaces also hold in CAT(κ) spaces as well. Agarwal, O Regan and Sahu [8] introduced the S-iteration process in a

2 858 Kritsana Sokhuma Banach space, (A) x 1 K, y n =(1 β n )x n + β n Tx n, x n+1 =(1 α n )Tx n + α n Ty n,n N, where and throughout the paper {α n }, {β n } are the sequences such that 0 α n,β n 1 for all n N. Recently, Panyanak [9] introduced the modified Ishikawa iteration process in a CAT(1) space, (B) x 1 K, y n = β n z n (1 β n )x n, x n+1 = α n z n (1 α n)x n,n N, where z n Tx n and z n Ty n. We now modify (A) in a CAT(κ) spaces with κ 0 as follows. Let (X, d) be a CAT(1) space with convex metric, E be a nonempty closed convex subset of X, and T : E 2 E be a multivalued mapping with Fix(T ). Suppose that {x n } is a sequence generated iteratively by, (C) where u n Tx n and v n Ty n. x 1 K, y n = β n u n (1 β n )x n, x n+1 = α n v n (1 α n )u n,n N, 2 Preliminaries Let (X, d) be a metric space, and let x X, E X. The distance from x to E is defined by dist(x, E) = inf{d(x, y) :y E}. The diameter of E is defined by diam(e) = sup{d(u, v) :u, v E}. Let H(, ) be the Hausdorff (generalized) distance on 2 E, i.e., H(A, B) = max{sup a A dist(a, B), sup dist(b, A)}, A,B 2 E. b B Definition 2.1. Let E be a nonempty subset of a metric space (X, d) and T : E 2 E. Then T is said to (i) be nonexpansive if H(Tx,Ty) d(x, y) for all x, y E; (ii) be quasi-nonexpansive if Fix(T ) and H(Tx,Tp) d(x, p) for all x E and p Fix(T ); (iii) satisfy condition (I) if there is a nondecreasing function f :[0, ] [0, ] with f(0) = 0,f(r) > 0 for r (0, ) such that

3 On S-iteration processes 859 dist(x, T x) f(dist(x, F ix(t ))) forallx E; (iv) be hemicompact if for any sequence {x n } in E such that lim dist(x n,tx n )=0, n there exists a subsequence {x nk } of {x n } and q E such that lim x nk = q. k A point x E is called a fixed point of T if x T (x). We denote by Fix(T ) the set of all fixed points of T. Let (X, d) be a metric space. A geodesic path joining x X to y X is a map c from a closed interval [0,l] R to X such that c(0) = x, c(l) =y, and d(c(t),c(t )) = t t for all t, t [0,l]. In particular, c is an isometry and d(x, y) =l. The image α of c is called a geodesic segment joining x and y. When it is unique this geodesic segment is denoted by [x, y]. The space (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. When it is unique, this geodesic segment is denoted by [x, y]. This mean that z [x, y] if and only if there exists α [0, 1] such that d(x, z) =(1 α)d(x, y) and d(y, z) =αd(x, y). In this case we write z = αx (1 α)y. A subset E of X is said to be convex if E includes every geodesic segment joining any two of its points. In a geodesic space (X, d), the metric d : X X R is convex if for any x, y, z X and α [0, 1], one has d(x, αy (1 α)z) αd(x, y)+(1 α)d(x, z). Let D (0, ], then (X, d) is call D-geodesic space if any two points of X with their distance smaller than D are joined by a geodesic segment. Notice that (X, d) is a geodesic space if and only if it is a D-geodesic space. Let n N, we denote by the Euclidean scalar product in R n, that is, x y = x 1 y 1 + x 2 y x n y n where x =(x 1,x 2,x 3,..., x n ) and y =(y 1,y 2,y 3,..., y n ). Let S n denote the n-dimentional sphere defined by S n = {x =(x 1,x 2,x 3,..., x n+1 ) R n+1 : x x =1}, with metric d(x, y) = arccos x y,x,y S n. From now on, we assume that κ 0 and define D κ := π κ if κ>0 and D κ := if κ =0. We denote by Mκ n the following metric spaces: (i) if κ = 0 then M0 n is Euclidean space Rn ; (ii) if κ>0then Mκ n is obtained from S n by multiplying the distance function by the constant 1/ κ. A geodesic triangle (x, y, z) in a geodesic metric space (X, d) consists of three points x, y, z in X and a geodesic segment between each pair of vertices. We write p (x, y, z) where p [x, y] [y, z] [z, x]. For (x, y, z) ina geodesic space X satisfying d(x, y)+d(y, z)+d(z, x) < 2D κ, there exist point

4 860 Kritsana Sokhuma x, ȳ, z Mκ 2 such that d(x, y) =d Mκ 2( x, ȳ), d(y, z) =d Mκ 2 (ȳ, z), and d(z, x) = d M 2 κ ( z, x). We call the triangle having vertices x, ȳ, z in Mκ 2 a comparison triangle of (x, y, z). Notice that it is unique up to an isometry of Mκ 2, and we denote it by ( x, ȳ, z). A point p [ x, ȳ] is called a comparison point for p [x, y] if d(x, p) =d M 2 κ ( x, p). A geodesic triangle (x, y, z) inx with d(x, y)+d(y, z)+ d(z, x) < 2D κ is said to satisfy the CAT(κ) inequality if for any p, q (x, y, z) and for their comparison points p, q ( x, ȳ, z), one has d(p, q) d M 2 κ ( p, q). Definition 2.2. A metric space (X, d) is called a CAT(κ) space if it is D κ -geodesic and any geodesic triangle (x, y, z) in X with d(x, y)+d(y, z)+ d(z, x) < 2D κ satisfies the CAT (κ) inequality. It follows from Proposition 1.4 of [1] that CAT(κ) spaces are uniquely geodesic spaces. In this paper, we consider CAT(κ) spaces with κ 0. Since most of the result for such spaces are easily deduced from those for CAT(1) spaces, in what follows, we mainly focus on CAT(1) spaces. Lemma 2.3. If (X, d) is a CAT(1) spaces with diam(x) <π/2, then there is a constant K>0 such that d 2 ((1 α)x αy, z) (1 α)d 2 (x, z)+αd 2 (y, z) K 2 α(1 α)d2 (x, y) for any α [0, 1] and any points x, y, z X. Lemma 2.4. Let {α n }, {β n } be two real sequences such that (i) 0 α n,β n < 1; (ii) β n 0 as n ; (iii) α n β n =. Let {γ n } be a nonnegative real sequence such that n=1 α nβ n (1 β n )γ n is bounded. Then {γ n } has a subsequence which converges to zero. 3 Main Results In this section we prove the strong convergence theorems of the S-iterative in a CAT(1) space defined by (C). We begin this section by proving a crucial lemma. Lemma 3.1. Let (X, d) be a CAT(1) space with convex metric, E be a nonempty closed convex subset of X, and T : E 2 E be a quasi-nonexpansive mapping with Fix(T) and T (p) =p for each p Fix(T ). Let {x n } be the sequence of S-iterative defined by (C). Then lim n d(x n,p) exists for each p Fix(T ).

5 On S-iteration processes 861 Proof. Let p Fix(T ) and for each n 1, we get d(y n,p) = d(β n u n (1 β n )x n,p) β n d(u n,p)+(1 β n )d(x n,p) β n H(Tx n,tp)+(1 β n )d(x n,p) β n d(x n,p)+(1 β n )d(x n,p) = d(x n,p). Also, d(x n+1,p) = d(α n v n (1 α n )u n,p) α n d(v n,p)+(1 α n )d(u n,p) α n H(Ty n,tp)+(1 α n )H(Tx n,tp) α n d(y n,p)+(1 α n )d(x n,p) = α n d(β n u n (1 β n )x n,p)+(1 α n )d(x n,p) α n [β n d(u n,p)+(1 β n )d(x n,p)] + (1 α n )d(x n,p) = α n β n d(u n,p)+α n (1 β n )d(x n,p)+(1 α n )d(x n,p) α n β n H(Tx n,tp)+α n d(x n,p) α n β n d(x n,p)+(1 α n )d(x n,p) α n β n d(x n,p)+α n d(x n,p) α n β n d(x n,p)+d(x n,p) α n d(x n,p) = d(x n,p). This show that the sequence {d(x n,p)} is decreasing and bounded below. Thus lim n d(x n,p) exists for each p Fix(T ). Theorem 3.2. Let (X, d) be a complete CAT(1) space with convex metric and diam(x) <π/2, E be a nonempty closed convex subset of X, and T : E C(E) be a quasi-nonexpansive mapping with Fix(T ) and Tp = {p} for each p Fix(T ). Let {α n }, {β n } [a, b] (0, 1) and {x n } be the sequence of S-iterative defined by (C). If T satisfies condition (I), then {x n } converges strongly to a fixed point of T. Proof. Let p Fix(T ) and by using Lemma 3.1, we get d 2 (x n+1,p) = d 2 (α n v n (1 α n )u n,p) (1 α n )d 2 (u n,p)+α n d 2 (v n,p) K 2 α n(1 α n )d 2 (u n,v n ) (1 α n )H 2 (Tx n,tp)+α n H 2 (Ty n,tp) (1 α n )d 2 (x n,p)+α n d 2 (y n,p).

6 862 Kritsana Sokhuma Then since T is quasi-nonexpansive, we have d 2 (y n,p) = d 2 (β n u n (1 β n )x n,p) So that, (1 β n )d 2 (x n,p)+β n d 2 (u n,p) K 2 β n(1 β n )d 2 (u n,x n ) (1 β n )d 2 (x n,p)+β n H 2 (Tx n,tp) K 2 β n(1 β n )d 2 (x n,u n ) (1 β n )d 2 (x n,p)+β n d 2 (x n,p) K 2 β n(1 β n )d 2 (x n,u n ) = d 2 (x n,p) K 2 β n(1 β n )d 2 (x n,u n ). d 2 (x n+1,p) (1 α n )d 2 (x n,p)+α n d 2 (x n,p) K 2 α nβ n (1 β n )d 2 (x n,u n ). This implies that K 2 a2 (1 b)d 2 (x n,u n ) K 2 α nβ n (1 β n )d 2 (x n,u n ) d 2 (x n,p) d 2 (x n+1,p) (1) and so K n=1 2 a2 (1 b)d 2 (x n,u n ) <. Thus, lim n d(x n,u n )=0. Also dist(x n,tx n ) d(x n,u n ) 0asn. Since T satisfies condition(i), we have lim n dist(x n, Fix(T ))=0. The proof of the remaining part follow the proof of Theorem 3.2 in [11], therefore we omit it. Theorem 3.3. Let (X, d) be a complete CAT(1) space with convex metric and diam(x) <π/2, E be a nonempty closed convex subset of X, and T : E C(E) be a quasi-nonexpansive mapping with Fix(T ) and Tp = {p} for each p Fix(T ). Assume that (i) 0 α n,β n < 1; (ii) β n 0 as n ; (iii) αn β n =, and let {x n } be the sequence of S-iterative defined by (C). If T is hemicompact and continuous, then {x n } converges strongly to a fixed point of T. Proof. Let p Fix(T ) and by using (1) we get, K 2 n=1 α nβ n (1 β n )d 2 (x n,u n ) <. By Lemma 2.4, there exist subsequence {x nk } and {u nk } of {x n } and {u n } respectively such that lim k d(x nk,u nk )=0. Hence lim k dist(x n k,tx nk ) lim k d(x nk,u nk )=0.

7 On S-iteration processes 863 Since T is hemicompact, by passing through a subsequence we may assume that x nk q for some q E. Since T is continuous, dist(q,tq) d(q, x nk ) + dist(x nk,tx nk )+H(Tx nk,tq) 0 as k. This implies that q Fix(T ). Since Tq is closed. Thus lim n d(x n,q) exist by Lemma 3.1 and hence q is the limit of {x n } itself. References [1] M.R. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer-Verlag, Berlin-Heidelberg, (1999). [2] W.A. Kirk, Geodesic geometry and fixed point theory. In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), Colecc. Abierta, Univ. Sevilla Secr. Publ., Seville, 64(2003), [3] W.A. Kirk, Geodesic geometry and fixed point theory II. In: International Conference on Fixed Point Theory and Applications, Yokohama Publ., (2004), [4] S. Dhompongsa, A. Kaewkhao, B. Panyanak, Lim s theorems for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl., 312(2005), [5] N. Shahzad, H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal., 71(2009), [6] A. Abkar, M. Eslamian, Common fixed point results in CAT(0) spaces, Nonlinear Anal., 74 (2011), [7] R. Espínola, P. Lorenzo, A. Nicolae, Fixed points selections and common fixed points for nonexpansive-type mappings, J. Math. Anal. Appl., 382(2011), [8] R.P. Agarwal, D. O Regan, D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex. Anal., 8(2007), [9] B. Panyanak, On the Ishikawa iteration processes for multivalued mappings in some CAT(κ) spaces, Fixed Point Theory and Applications, (2014), 2014:1. [10] A. Şahin, M. Başarir, On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a CAT(0) space, Fixed Point Theory and Applications, (2013), 2013:12.

8 864 Kritsana Sokhuma [11] T. Puttasontiphot, Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces, Appl. Math. Sci., 39(2010), [12] Y. Song, Y.J. Cho, Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc., 40(2011), Received: March 15, 2014