A modified proximal contraction principle with applications to variational inequality problems

TL;DR

This paper introduces new concepts related to proximally completeness and continuity, improving the proximal contraction principle, and applies these to establish existence and uniqueness results for solutions to variational inequality problems in metric spaces.

Contribution

The paper develops modified notions of proximally completeness, closedness, and continuity, enhancing the proximal contraction principle and applying it to variational inequality problems.

Findings

Established conditions for unique solutions of variational inequalities.

Extended the proximal contraction principle to non-complete metric spaces.

Provided illustrative examples demonstrating the applicability of the results.

Abstract

In this paper, we introduce the notions of proximally completeness, proximally closedness and proximally continuity and utilize the same to prove a result on existence and uniqueness of best proximity points in the setting of metric space (not necessarily complete). Our newly proved result enriches, sharpens, improves and modifies the proximal contraction principle of Basha [J. Optim. Theory Appl. 2011:151 (2011), 210-216]. In order to illustrate the effectiveness of our finding, we discuss the sufficient conditions ensuring the existence of a unique solution of certain variational inequality problem.