An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

Wanna Sriprad,Somnuk Srisawat,Narendra K Govil

doi:10.1155/2021/4273851

Abstract

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.

Highlights

Let H1 and H2 be real Hilbert spaces whose inner product and norm are denoted by 〈·, ·〉 and ‖ · ‖, respectively, and letC, Q be nonempty closed convex subsets of H1 and H2, respectively
It is known that the variational inequality, as a strong and great tool, has already been investigated for an extensive class of optimization problems in economics and equilibrium problems arising in physics and many other branches of pure and applied sciences
Question: can we prove the strong convergence theorem of two sequences of split monotone variational inclusion problems and fixed-point problems of nonlinear mappings in real Hilbert spaces?

Summary

Introduction

Let H1 and H2 be real Hilbert spaces whose inner product and norm are denoted by 〈·, ·〉 and ‖ · ‖, respectively, and let. Many authors have developed and used this algorithm to solve the fixed-point problems of many nonlinear operators in real Hilbert spaces (see for example [21,22,23,24,25,26,27]). Question: can we prove the strong convergence theorem of two sequences of split monotone variational inclusion problems and fixed-point problems of nonlinear mappings in real Hilbert spaces?. E purpose of this paper is to modify an intermixed algorithm to answer the question above and prove a strong convergence theorem of two sequences for finding a common element of the set of solutions of (SMVI) (4) and (5) and the set of solutions of a finite family of variational inequality problems in real Hilbert spaces. We give some numerical examples for supporting our main theorem

Preliminaries

Main Results

Applications

Numerical Examples

Conclusion