A splitting algorithm for fixed points of nonexpansive mappings and equilibrium problems

TL;DR

This paper introduces a splitting algorithm that combines gradient and Mann iteration methods to find fixed points of nonexpansive mappings satisfying equilibrium conditions, without requiring Lipschitz or H"older continuity assumptions.

Contribution

It presents a novel splitting algorithm for fixed points and equilibrium problems that converges under paramonotonicity without Lipschitz or H"older conditions.

Findings

Algorithm converges under paramonotonicity.

No Lipschitz or H"older continuity needed.

Requires solving two convex subprograms each iteration.

Abstract

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz type condition as well as H\"older continuity of the bifunctions involved.