%0 Journal Article
%T Path Convergence and Approximation of Common Zeroes of a Finite Family of -Accretive Mappings in Banach Spaces
%A Yekini Shehu
%A Jerry N. Ezeora
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/285376
%X Let be a real Banach space which is uniformly smooth and uniformly convex. Let be a nonempty, closed, and convex sunny nonexpansive retract of , where is the sunny nonexpansive retraction. If admits weakly sequentially continuous duality mapping , path convergence is proved for a nonexpansive mapping . As an application, we prove strong convergence theorem for common zeroes of a finite family of -accretive mappings of to . As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from to under certain mild conditions. 1. Introduction Let be a real Banach space with dual and a nonempty, closed and convex subset of . A mapping is said to be nonexpansive if for all , we have A point is called a fixed point of if . The fixed points set of is the set . Construction of fixed points of nonexpansive mappings is an important subject in nonlinear mapping theory and its applications; in particular, in image recovery and signal processing (see, e.g., [1¨C3]). Many authors have worked extensively on the approximation of fixed points of nonexpansive mappings. For example, the reader can consult the recent monographs of Berinde [4] and Chidume [5]. We denote by the normalized duality mapping from to defined by where denotes the generalized duality pairing between members of and . It is well known that if is strictly convex then is single valued (see, e.g., [5, 6]). In the sequel, we will denote the single-valued normalized duality mapping by . A mapping is called accretive if, for all , there exists such that By the results of Kato [7], (1.3) is equivalent to If is a Hilbert space, accretive mappings are also called monotone. A mapping is called m-accretive if it is accretive and , range of , is for all ; and is said to satisfy the range condition if , where denotes the closure of the domain of . is said to be maximal accretive if it is accretive and the inclusion , where is a graph of , with accretive, implies . It is known (see e.g., [8]) that every maximal accretive mapping is -accretive and the converse holds if is a Hilbert space. Interest in accretive mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., [9]) that many physically significant problems can be modelled by initial-value problems of the following form: where is an accretive mapping in an appropriate Banach space. Typical examples where such evolution equations occur can be found in the heat, wave, or Schr£żdinger equations. One of the fundamental results
%U http://www.hindawi.com/journals/aaa/2010/285376/