A strong convergence theorem for generalized-\u03a6-strongly monotone maps, with applications

C E Chidume,A Adamu,M O Nnakwe

doi:10.1186/s13663-019-0660-9

Abstract

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X^{*}. In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.

Highlights

Let X be a real Banach space with dual space X∗
Using the fact that A is a generalized-Φ-strongly monotone map and Lemma 2.5, it follows from inequality (3.4) that ψ v∗, vn+1 ≤ ψ v∗, vn – 2βnΦ vn – v∗ + 2βnδX–1 4RLβn Avn Avn ≤ ψ v∗, vn – 2βnΦ vn – v∗ + 2βnδX–1(βnM)M
Using the fact that A is a generalized-Φ-strongly monotone map and Lemma 2.5, it follows from inequality (6.4) that ψ v∗, vn+1 ≤ ψ v∗, vn – 2βnΦ ≤ ψ v∗, vn – 2βnΦ

Summary

Introduction

Let X be a real Banach space with dual space X∗. Let A : D(A) ⊂ X → X be a map, where D(A) denotes the domain of A. In 2015, Diop et al [33] studied an iterative scheme of Mann type to approximate the zero of a strongly monotone bounded map in a 2-uniformly convex real Banach space with a uniformly Gâteaux differentiable norm.

Results

Conclusion