Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces

Keywords: Implicit iterative algorithm Asymptotically quasi-nonexpansive mappings Common fixed point Convex metric space a b s t r a c t In this paper, we consider an implicit iteration process to approximate the common fixed points of two finite families of asymptotically quasi-nonexpansive mappings in convex metric spaces. As a consequence of our result, we obtain some related convergence theorems. Our results generalize some recent results of Khan and Ahmed [4], Khan et al. Throughout this paper, N denotes the set of natural numbers and J = {1, 2,. .. , N}, the set of first N natural numbers. Denote by F(T) the set of fixed points of T and by F :¼ \ N j¼1 FðT j Þ \ \ N j¼1 FðS j Þ the set of common fixed points of two finite families of mappings {T j : j 2 J} and {S j : j 2 J}. Takahashi [13] introduced the notion of convex metric spaces and studied the fixed point theory for nonexpansive map-pings in this setting. For convex metric spaces, Kirk [8] used the term ''hyperbolic type spaces'' and studied iteration processes for nonexpansive mappings in this abstract framework. Later on, many authors discussed the existence of the fixed point and the convergence of the iterative process for various mappings in convex metric spaces (see, for example, [1,4,7,9,13] and the references therein). We recall some definitions in a metric space (X, d). A metric space together with a convex structure is called a convex metric space. A nonempty subset C of X is said to be convex if W(x, y; k) 2 C for all (x, y; k) 2 C Â C Â [0, 1]. Definition 1.2 [4]. Let (X, d) be a metric space and q be a fixed element of X. A q-starshaped structure in X is a mapping W : A metric space together with a q-starshaped structure is called a q-starshaped metric space.