Convergence Results of Forward-Backward Algorithms for Sum of Monotone Operators in Banach Spaces

Yekini Shehu

doi:10.1007/s00025-019-1061-4

Abstract

It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper.

Highlights

Let E be a real Banach space with norm . , we denote by E∗ the dual of E and f, x the value of f ∈ E∗ at x ∈ E
A popular method for solving problem (1) in real Hilbert spaces, is the well-known forward-backward splitting method introduced by Passty [35] and Lions and Mercier [28]
We study the Tseng-type algorithm for finding a solution to monotone inclusion problem involving a sum of maximal monotone and a Lipschitz continuous monotone mapping in 2-uniformly convex Banach space which is uniformly smooth

Summary

Introduction

Let E be a real Banach space with norm . , we denote by E∗ the dual of E and f, x the value of f ∈ E∗ at x ∈ E. Let E be a real Banach space with norm . We denote by E∗ the dual of E and f, x the value of f ∈ E∗ at x ∈ E.

Results

Preliminaries

Approximation Method

Convergence Analysis

Application

Conclusion