Convergence Theorems for the Variational Inequality Problems and Split Feasibility Problems in Hilbert Spaces

Panisa Lohawech,Anchalee Kaewcharoen,Ali Farajzadeh,Sergejs Solovjovs

doi:10.1155/2021/9980309

Abstract

In this paper, we establish an iterative algorithm by combining Yamada’s hybrid steepest descent method and Wang’s algorithm for finding the common solutions of variational inequality problems and split feasibility problems. The strong convergence of the sequence generated by our suggested iterative algorithm to such a common solution is proved in the setting of Hilbert spaces under some suitable assumptions imposed on the parameters. Moreover, we propose iterative algorithms for finding the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.

Highlights

In 2005, Censor et al [1] introduced the multiple-sets split feasibility problem (MSSFP), which is formulated as follows: N M find x ∈∩ Ci such that Ax i 1 ∩ Qj, j 1 (1)where Ci (i 1, 2, . . . , N) and Qj(j 1, 2, . . . , M) are nonempty closed convex subsets of Hilbert spaces H1 and H2, respectively, and A: H1 ⟶ H2 is a bounded linear mapping
When N M 1, the MSSFP is known as the split feasibility problem (SFP); it was first introduced by Censor and Elfving [5], which is formulated as follows: find x ∈ C such that Ax ∈ Q
Buong [2] considered the sequence 􏼈xn􏼉 that is generated by the following algorithm, which is weakly convergent to a solution of MSSFP (1): xn+1 P1 I − cA∗ I − P2􏼁A􏼁xn, (9)

Summary

Introduction

In 2005, Censor et al [1] introduced the multiple-sets split feasibility problem (MSSFP), which is formulated as follows: N. where Ci When N M 1, the MSSFP is known as the split feasibility problem (SFP); it was first introduced by Censor and Elfving [5], which is formulated as follows: find x ∈ C such that Ax ∈ Q. Buong [2] considered the sequence 􏼈xn􏼉 that is generated by the following algorithm, which is weakly convergent to a solution of MSSFP (1): xn+1 P1 I − cA∗ I − P2􏼁A􏼁xn,. We propose iterative algorithms for solving the common solutions of variational inequality problems and multiple-sets split feasibility problems. We give numerical examples for illustrating our algorithms

Preliminaries

Main Results

Numerical Example