A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others. 1. Introduction Throughout this paper we denoted by and the set of all positive integers and all positive real numbers, respectively. Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . We know that is nonempty if is bounded; for more detail see [1]. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i) ;(ii) for all ;(iii)for each the mapping is continuous;(iv) for all and . We denote by the set of all common fixed points of , that is, . We know that is nonempty if is bounded; see [2]. Recall that a self-mapping is a contraction if there exists a constant such that for each . As in [3], we use the notation to denote the collection of all contractions on , that is, . Note that each has a unique fixed point in . In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [3–5]. Let be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let be a strongly positive bounded linear operator on : that is, there is a constant with property A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space : where is the fixed point set of a nonexpansive mapping on and is a given point in . In 2003, Xu [3] proved that the sequence generated by converges strongly to the unique solution of the minimization problem (1.2) provided that the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [6] introduced the iterative process for nonexpansive mappings (see [3, 7] for further developments in both Hilbert and Banach spaces) and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with constant and satisfies certain conditions, converges strongly to a fixed point of in which is unique solution of the variational inequality: In 2006, Marino and Xu [8] combined the iterative method (1.3)