A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory

It is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; and the set of fixed points of a continuous pseudocontractive mapping. Moreover, the common element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize, improve, and unify several recently announced results. 1. Introduction Let be a real normed space with dual . The normalized duality mapping from to is defined by where denotes the generalized duality pairing. It is well known (see, e.g., [1]) that if is strictly convex, then is single-valued and if is a Hilbert space, then is the identity mapping. In the sequel, we shall denote the single-valued normalized duality mapping by . A mapping with domain , and range , in is called a strict contraction or simply a contraction if and only if there exists such that for all , and is called nonexpansive if and only if for all , A point is called a fixed point of an operator if and only if . The set of fixed points of an operator is denoted by , that is, . The most important generalization of the class of nonexpansive mappings is, perhaps, the class of pseudocontractive mappings. These mappings are intimately connected with the important class of nonlinear accretive operators. This connection will be made precise in what follows. A mapping with domain , and range , in is called pseudocontractive if and only if for all , the following inequality holds: for all . As a consequence of a result of Kato [2], the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows: the mapping is called pseudocontractive if and only if for all , there exists such that It now follows trivially from (5) that every nonexpansive mapping is pseudocontractive. We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings. For examples of pseudocontractive mappings which are not nonexpansive, the reader may see [1]. To see the connection between the pseudocontractive mappings and the accretive mappings, we introduce the following definition: a mapping with domain, , and range, , in is called accretive if and only if for all , the following inequality is satisfied: for all . Again, as a consequence of Kato [2], it follows that is accretive if and only if for all