Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Let be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive semigroup on such that , and is a contraction on with coefficient . Let be -strongly accretive and -strictly pseudocontractive with and a positive real number such that . When the sequences of real numbers and satisfy some appropriate conditions, the three iterative processes given as follows: , , , , and , converge strongly to , where is the unique solution in of the variational inequality , . Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others. 1. Introduction Let be a real Banach space. A mapping of into itself is said to be nonexpansive if for each . We denote by the set of fixed points of . A mapping is called -contraction if there exists a constant such that for all . A family of mappings of into itself is called a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ; (ii) for all ; (iii) for all and ; (iv) for all , the mapping is continuous. We denote by the set of all common fixed points of , that is, In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space where is a sequence in and is a sequence of positive real numbers which diverges to . Under certain restrictions on the sequence , Shioji and Takahashi [1] proved strong convergence of the sequence to a member of . In [2], Shimizu and Takahashi studied the strong convergence of the sequence defined by in a real Hilbert space where is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset of a Banach space and . Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup with in a Banach space: where is a contraction. Note however that their iterate at step is constructed through the average of the semigroup over the interval . Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process: for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if satisfies an asymptotic regularity condition and fulfills the control conditions , , and , then both the implicit iteration process (1.5) and the explicit iteration process (1.6), converge to a same point of . In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping.