A new definition for variational inequalities on real normed linear spaces and the case that it is singelton for (u, v)-cocoercive mappings

TL;DR

This paper introduces a new generalized definition of variational inequalities on Banach spaces, extending concepts from Hilbert spaces, and proves singleton solutions for (u, v)-cocoercive mappings under certain conditions.

Contribution

It generalizes variational inequalities and (u, v)-cocoercive mappings from Hilbert to Banach spaces and establishes conditions for unique solutions.

Findings

New definition of variational inequality on Banach spaces

Extension of (u, v)-cocoercive mappings to Banach spaces

Proved singleton solutions for generalized variational inequalities

Abstract

Let C be a nonempty closed convex subset of a Banach space $E$. In this paper we introduce a new definition for variational inequality V I (C, B) on E that generalizes the analogue definition on Hilbert spaces. We generalize (u, v)-cocoercive mappings and v-strongly monotone mappings from Hilbert spaces to Banach spaces. Then we prove 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 [S. Saeidi, Comments on relaxed (u, v)-cocoercive mappings. Int. J. Nonlinear Anal. Appl. 1 (2010) No. 1, 54-57].