Comments on relaxed $(\gamma, r)$-cocoercive mappings

TL;DR

This paper proves the unique solvability of variational inequalities involving relaxed $(, )$-cocoercive, Lipschitzian mappings under certain conditions, highlighting limitations of recent algorithms in this context.

Contribution

It establishes a new uniqueness result for variational inequalities with relaxed cocoercive mappings, clarifying the scope of existing algorithms.

Findings

Unique solution exists for the variational inequality under specified conditions.

Recent algorithms are not as general as they should be despite increased complexity.

Provides theoretical foundation for the solvability of certain variational inequalities.

Abstract

We show that the variational inequality $VI(C,A)$ has a unique solution for a relaxed $(\gamma, r)$-cocoercive, $\mu$-Lipschitzian mapping $A: C\to H$ with $r>\gamma \mu^2$, where $C$ is a nonempty closed convex subset of a Hilbert space $H$. From this result, it can be derived that, for example, the recent algorithms given in the references of this paper, despite their becoming more complicated, are not general as they should be.