Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

Mihai Postolache,Ashish Nandal,Renu Chugh

doi:10.3390/math7090773

Abstract

In this paper, based on the very recent work by Nandal et al. (Nandal, A.; Chugh, R.; Postolache, M. Iteration process for fixed point problems and zeros of maximal monotone operators. Symmetry 2019, 11, 655.), we propose a new generalized viscosity implicit rule for finding a common element of the fixed point sets of a finite family of nonexpansive mappings and the sets of zeros of maximal monotone operators. Utilizing the main result, we first propose and investigate a new general system of generalized equilibrium problems, which includes several equilibrium and variational inequality problems as special cases, and then we derive an implicit iterative method to solve constrained multiple-set split convex feasibility problem. We further combine forward–backward splitting method and generalized viscosity implicit rule for solving monotone inclusion problem. Moreover, we apply the main result to solve convex minimization problem.

Highlights

A problem which appears very often in different areas of mathematics and physical sciences consists of finding an element in the intersection of closed and convex subsets of a Hilbert space.This problem is generally named as convex feasibility problem (CFP)
The multiple-sets split feasibility problem (MSSFP) is stated as finding a point belonging to a family of closed convex subsets in one space whose image under a bounded linear transformation belongs to another family of closed convex subsets in the image space
Inspired by the above work, we introduce and study a new generalized viscosity implicit iterative rule based on Nandal, Chugh and Postolache’s [28] iterative method for approximating a common element of the fixed point sets of nonexpansive mappings and the sets of zeros of maximal monotone mappings

Summary

Introduction

A problem which appears very often in different areas of mathematics and physical sciences consists of finding an element in the intersection of closed and convex subsets of a Hilbert space. Monotone inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mapping is among the very important and extensively studied problems in recent years This problem includes various important problems such as convex minimization problem, variational inequality problem, linear inverse problem and split feasibility problem. In [32], Khuri and Sayfy established relation between variational iteration method and the fixed point theory Very recently, He and Ji [33] suggested a simple approach using Taylor series technology to solve approximately the Lane-Emden equation. Inspired by the above work, we introduce and study a new generalized viscosity implicit iterative rule based on Nandal, Chugh and Postolache’s [28] iterative method for approximating a common element of the fixed point sets of nonexpansive mappings and the sets of zeros of maximal monotone mappings. We utilize our main result to solve constrained multiple set split convex feasibility problem, monotone inclusion problem and convex minimization problem

Preliminaries

Main Results

New General System of Generalized Equilibrium Problems

Monotone Inclusion and Fixed Point Problem

Convex Minimization Problem

Concluding Remarks