A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings

TL;DR

This paper provides simplified proofs for the convergence of algorithms involving relaxed $(u, v)$-cocoercive and $\alpha$-inverse strongly monotone mappings, refining and improving upon previous results.

Contribution

It offers new, simplified convergence proofs for existing algorithms related to these classes of mappings, enhancing understanding and applicability.

Findings

Simplified proof for convergence of algorithms with relaxed $(u, v)$-cocoercive mappings.

Simplified proof for convergence of algorithms with $\alpha$-inverse strongly monotone mappings.

Refinement and improvement of previously known convergence results.

Abstract

In this paper, a simple proof is presented for the convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings. Based on $\alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.