Strong convergence of an inertial algorithm for maximal monotone inclusions with applications

C E Chidume,A Adamu,M O Nnakwe

doi:10.1186/s13663-020-00680-2

Abstract

An inertial iterative algorithm is proposed for approximating a solution of a maximal monotone inclusion in a uniformly convex and uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a solution of the inclusion. Moreover, the theorem proved is applied to approximate a solution of a convex optimization problem and a solution of a Hammerstein equation. Furthermore, numerical experiments are given to compare, in terms of CPU time and number of iterations, the performance of the sequence generated by our algorithm with the performance of the sequences generated by three recent inertial type algorithms for approximating zeros of maximal monotone operators. In addition, the performance of the sequence generated by our algorithm is compared with the performance of a sequence generated by another recent algorithm for approximating a solution of a Hammerstein equation. Finally, a numerical example is given to illustrate the implementability of our algorithm for approximating a solution of a convex optimization problem.

Highlights

The theorem proved is applied to approximate a solution of a convex optimization problem, and a solution of a Hammerstein equation
6 Conclusions An inertial iterative algorithm which does not involve the resolvent operator is proposed for approximating a solution of a maximal monotone inclusion in uniformly convex and uniformly smooth real Banach spaces
The theorem proved is applied to approximate a solution of a convex optimization problem, and a solution of a Hammerstein integral equation

Summary

Introduction

An inertial iterative algorithm is proposed for approximating a solution of a maximal monotone inclusion in a uniformly convex and uniformly smooth real Banach space. Numerical experiments are given to compare, in terms of CPU time and number of iterations, the performance of the sequence generated by our algorithm with the sequences generated by IPPA, RPPA, and RIPPA, respectively, for approximating a solution of a maximal monotone inclusion in Hilbert spaces. Lemma 2.7 (Reich [38]) Let E∗ be a strictly convex dual Banach space with a Fréchet differentiable norm, and let A : E ⇒ E∗ be a maximal monotone map with a zero and z ∈ E∗.

Main results

Conclusions