Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

and Applied Analysis 3 by 1.5 to a common fixed point of a countable infinite family of nonexpansive mappings in convex metric spaces and CAT 0 spaces under certain suitable conditions. 2. Preliminaries We recall some definitions and useful lemmas used in the main results. Lemma 2.1 see 9, 10 . Let X, d,W be a convex metric space. For each x, y ∈ X and λ, λ1, λ2 ∈ 0, 1 , we have the following. i W x, x, λ x,W x, y, 0 y and W x, y, 1 x. ii d x,W x, y, λ 1 − λ d x, y and d y,W x, y, λ λd x, y . iii d x, y d x,W x, y, λ d W x, y, λ , y . iv |λ1 − λ2|d x, y ≤ d W x, y, λ1 ,W x, y, λ2 . We say that a convex metric space X, d,W has the property: C ifW x, y, λ W y, x, 1 − λ for all x, y ∈ X and λ ∈ 0, 1 , I if d W x, y, λ1 ,W x, y, λ2 ≤ |λ1 − λ2|d x, y for all x, y ∈ X and λ1, λ2 ∈ 0, 1 , H if d W x, y, λ ,W x, z, λ ≤ 1 − λ d y, z for all x, y, z ∈ X and λ ∈ 0, 1 , S if d W x, y, λ ,W z,w, λ ≤ λd x, z 1 − λ d y,w for all x, y, z,w ∈ X and λ ∈ 0, 1 . From the above properties, it is obvious that the property C and H imply continuity of a convex structure W : X × X × 0, 1 → X. Clearly, the property S implies the property H . In 10 , Aoyama et al. showed that a convex metric space with the property C and H has the property S . In 1996, Shimizu and Takahashi 11 introduced the concept of uniform convexity in convex metric spaces and studied some properties of these spaces. A convex metric space X, d,W is said to be uniformly convex if for any ε > 0, there exists δ δ ε > 0 such that for all r > 0 and x, y, z ∈ X with d z, x ≤ r, d z, y ≤ r and d x, y ≥ rε imply that d z,W x, y, 1/2 ≤ 1 − δ r. Obviously, uniformly convex Banach spaces are uniformly convex metric spaces. In fact, the property I holds in uniformly convex metric spaces, see 12 . Lemma 2.2. Property (C) holds in uniformly convex metric spaces. Proof. Suppose that X, d,W is a uniformly convexmetric space. Let x, y ∈ X and λ ∈ 0, 1 . It is obvious that the conclusion holds if λ 0 or λ 1. So, suppose λ ∈ 0, 1 . By Lemma 2.1 ii , we have d ( x,W ( x, y, λ )) 1 − λ dx, y, dy,Wx, y, λ λdx, y, d ( x,W ( y, x, 1 − λ 1 − λ dx, y, dy,Wy, x, 1 − λ λdx, y. 2.1 We will show that W x, y, λ W y, x, 1 − λ . To show this, suppose not. Put z1 W x, y, λ and z2 W y, x, 1 − λ . Let r1 1 − λ d x, y > 0, r2 λd x, y > 0, 4 Abstract and Applied Analysis ε1 d z1, z2 /r1, and ε2 d z1, z2 /r2. It is easy to see that ε1, ε2 > 0. Since X, d,W is uniformly convex, we have d ( x,W ( z1, z2, 1 2 )) ≤ r1 1 − δ ε1 , d ( y,W ( z1, z2, 1 2 )) ≤ r2 1 − δ ε2 . 2.2 By λ ∈ 0, 1 , we get x / y. Since δ ε1 > 0 and δ ε2 > 0, then d ( x, y ) ≤ d ( x,W ( z1, z2, 1 2 )) d ( y,W ( z1, z2, 1 2 )) ≤ r1 1 − δ ε1 r2 1 − δ ε2 < r1 r2 d ( x, y ) . 2.3 This is a contradiction. Hence, W x, y, λ W y, x, 1 − λ . By Lemma 2.2, it is clear that a uniformly convex metric space X, d,W with the property H has the property S , and the convex structure W is also continuous. Next, we recall the special space of convex metric spaces, namely, CAT 0 spaces. 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 and d c t1 , c t2 |t1 − t2| for all t1, t2 ∈ 0, l . In particular, 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 unique, this geodesic is denoted x, y . The space X, d is said to be a geodesic metric space if every two points ofX are joined by a geodesic, andX is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y of X is said to be convex if Y includes every geodesic segment joining any two of its points. A geodesic triangle x1, x2, x3 in a geodesic metric space X, d consists of three points x1, x2, x3 in X the vertices of and a geodesic segment between each pair of vertices the edges of . A comparison triangle for geodesic triangle x1, x2, x3 in X, d is a triangle x1, x2, x3 : x1, x2, x3 in the Euclidean plane E2 such that dE2 xi, xj d xi, xj for i, j ∈ {1, 2, 3}. A geodesic metric space is said to be a CAT 0 space if all geodesic triangles satisfy the following comparison axiom. Let be a geodesic triangle in X, and let be a comparison triangle for . Then is said to satisfy the CAT 0 inequality if for all x, y ∈ and all comparison points x, y ∈ , d x, y ≤ dE2 x, y . If z, x, y are points in a CAT 0 space and if m is the midpoint of the segment x, y , then the CAT 0 inequality implies d z,m 2 ≤ 1 2 d z, x 2 1 2 d ( z, y )2 − 1 4 d ( x, y )2 . CN This is the CN inequality of Bruhat and Tits 13 , which is equivalent to d ( z, λx ⊕ 1 − λ y2 ≤ λd z, x 2 1 − λ dz, y2 − λ 1 − λ dx, y2, CN∗ Abstract and Applied Analysis 5 for any λ ∈ 0, 1 , where λx⊕ 1−λ y denotes the unique point in x, y . The CN∗ inequality has appeared in 14 . By using the CN inequality, it is easy to see that the CAT 0 spaces are uniformly convex. In fact 15 , a geodesic metric space is a CAT 0 space if and only if it satisfies the CN inequality. Moreover, if X is CAT 0 space and x, y ∈ X, then for any λ ∈ 0, 1 , there exists a unique point λx ⊕ 1 − λ y ∈ x, y such thatand Applied Analysis 5 for any λ ∈ 0, 1 , where λx⊕ 1−λ y denotes the unique point in x, y . The CN∗ inequality has appeared in 14 . By using the CN inequality, it is easy to see that the CAT 0 spaces are uniformly convex. In fact 15 , a geodesic metric space is a CAT 0 space if and only if it satisfies the CN inequality. Moreover, if X is CAT 0 space and x, y ∈ X, then for any λ ∈ 0, 1 , there exists a unique point λx ⊕ 1 − λ y ∈ x, y such that d ( z, λx ⊕ 1 − λ y ≤ λd z, x 1 − λ dz, y, 2.4 for any z ∈ X. It follows that CAT 0 spaces have convex structureW x, y, λ λx ⊕ 1− λ y. It is clear that the properties C , I , and S are satisfied for CAT 0 spaces, see 15, 16 . This is also true for Banach spaces. Let μ be a continuous linear functional on l∞, the Banach space of bounded real sequences, and let a1, a2, . . . ∈ l∞. We write μn an instead of μ a1, a2, . . . . We call μ a Banach limit if μ satisfies ‖μ‖ μ 1, 1, . . . 1 and μn an μn an 1 for each a1, a2, . . . ∈ l∞. For a Banach limit μ, we know that lim infn→∞an ≤ μn an ≤ lim supn→∞an for all a1, a2, . . . ∈ l∞. So if a1, a2, . . . ∈ l∞ with limn→∞an c, then μn an c, see also 17 . Lemma 2.3 4 , Proposition 2 . Let a1, a2, . . . ∈ l∞ be such that μn an ≤ 0 for all Banach limit μ. If lim supn→∞ an 1 − an ≤ 0, then lim supn→∞an ≤ 0. Lemma 2.4 6 , Lemma 2.3 . Let {sn} be a sequence of nonnegative real numbers, let {αn} be a sequence of real numbers in 0, 1 with ∑∞ n 1 αn ∞, let {δn} be a sequence of nonnegative real numbers with ∑∞ n 1 δn < ∞, and let {γn}be a sequence of real numbers with lim supn→∞γn ≤ 0. Suppose that sn 1 ≤ 1 − αn sn αnγn δn ∀n ∈ N. 2.5 Then limn→∞sn 0. Lemma 2.5 18 , Lemma 1 . Let X, d,W be a uniformly convex metric space with a continuous convex structure W : X × X × 0, 1 → X. Then for arbitrary positive number ε and r, there exists η η ε > 0 such that d ( z,W ( x, y, λ )) ≤ r1 − 2min{λ, 1 − λ}η, 2.6 for all x, y, z ∈ X, d z, x ≤ r, d z, y ≤ r, d x, y ≥ rε, and λ ∈ 0, 1 . Remark 2.6. The above lemma also holds for a uniformly convex metric space with the property H . 3. Main Results The following condition was introduced by Aoyama et al. 6 . Let C be a subset of a complete convex metric space X, d,W , and let {Tn} be a countable infinite family of mappings from 6 Abstract and Applied Analysis C into itself. We say that {Tn} satisfies AKTT-condition if