Bregman f -Projection Operator with Applications to Variational Inequalities in Banach Spaces

Abstract

Using Bregman functions, we introduce the new concept of Bregman generalized $f$-projection operator ${\text{Proj}}_C^{f, g}:E^{*}\to C$, where $E$ is a reflexive Banach space with dual space $E^{*}; f: E\to ℝ\cup \left\{+\infty \right\}$ is a proper, convex, lower semicontinuous and bounded from below function; $g: E\to ℝ$ is a strictly convex and Gâteaux differentiable function; and $C$ is a nonempty, closed, and convex subset of $E$. The existence of a solution for a class of variational inequalities in Banach spaces is presented.