Approximation of solution set of a variational inequality for (u, v)-cocoercive mappings in Banach spaces

TL;DR

This paper extends the concept of (u, v)-cocoercive mappings and variational inequalities from Hilbert spaces to Banach spaces, proving uniqueness of solutions under new assumptions.

Contribution

It introduces generalized definitions for (u, v)-cocoercive and v-strongly monotone mappings in Banach spaces and proves the singleton property of the variational inequality in this setting.

Findings

Generalized (u, v)-cocoercive mappings to Banach spaces.

Proved the variational inequality has a unique solution.

Extended and improved previous propositions in the literature.

Abstract

Let C be a nonempty closed convex subset of a real normed linear space $E$ and u, v are positive numbers. In this paper we introduce some new definitions that generalize the analogue definitions from real Hilbert spaces to real normed linear spaces. Indeed, we generalize (u, v)-cocoercive mappings and v-strongly monotone mappings and V I (C, B) for a mapping $B$, from real Hilbert spaces to real normed linear spaces. Then we prove that the generalized variational inequality V I (C, B) is singleton for (u, v)-cocoercive mappings under appropriate assumptions on Banach spaces that extends and improves Propositions 2, 3 in [S. Saeidi, Comments on relaxed (u, v)-cocoercive mappings. Int. J. Nonlinear Anal. Appl. 1 (2010) No. 1, 54-57].