The generalised f-projection operator with an application

Abstract

In this paper, we introduce a new concept of generalised /-projection operator which extends the generalised projection operator -KK • B* -» K, where B is a reflexive Banach space with dual space B* and K is a nonempty, closed and convex subset of B. Some properties of the generalised /-projection operator are given. As an application, we study the existence of solution for a class of variational inequalities in Banach spaces.

Strong Convergence Theorems for Generalized Variational Inequalities and Relatively Weak Nonexpansive Mappings in Banach Spaces

“…Let f : K → R {+∞} be proper, convex and lower semi-continuous, Wu and Huang [21] introduced the functional G : B * × K → R {+∞} defined as follows:…”

“…But, due to the presence of nonlinear terms, the projection methods presented in [6,9,13,15,17] cannot be applied to suggest any iterative scheme for generalized variational inequality (1.1) in Banach spaces. Fortunately, Wu and Huang in [20,21] introduced a new generalized f −projection operator. They in [20] proposed an iterative method of approximate solutions for the generalized variational inequality (1.1) when T is single-valued and f is convex lower semi-continuous and positively homogeneous in compact subsets of Banach spaces.…”

Existence and convergence theorems for the new system of generalized mixed variational inequalities in Banach spaces

“…[16]. Let B be a reflexive Banach space with its dual B* and let C be a nonempty closed convex subset of B.…”

A modified Mann iterative scheme by generalized f-projection for a countable family of relatively quasi-nonexpansive mappings and a system of generalized mixed equilibrium problems

“…In 2005, Li [22] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [24] introduced a new generalized f-projection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by Abler [23] and proved some properties of the generalized fprojection operator.…”

“…For the generalized f-projection operator, Wu and Hung [24] proved the following basic properties: Lemma 1.2.…”