Path Convergence and Approximation of Common Zeroes of a Finite Family of -Accretive Mappings in Banach Spaces

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., [1–3]). 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？dinger equations. One of the fundamental results