New strong convergence theorems for split variational inclusion problems in Hilbert spaces

Chih-Sheng Chuang,Ing-Jer Lin

doi:10.1186/s13660-015-0697-1

Abstract

The split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present a descent-conjugate gradient algorithm for the split variational inclusion problems in Hilbert spaces. Next, a strong convergence theorem of the proposed algorithm is proved under suitable conditions. As an application, we give a new strong convergence theorem for the split feasibility problem in Hilbert spaces. Finally, we give numerical results for split variational inclusion problems to demonstrate the efficiency of the proposed algorithm.

Highlights

1 Introduction Let H be a real Hilbert space, and B : H H be a set-valued mapping with domain D(B) := {x ∈ H : B(x) = ∅}
Recall that B is called monotone if u – v, x – y ≥ for any u ∈ Bx and v ∈ By; B is maximal monotone if its graph {(x, y) : x ∈ D(B), y ∈ Bx} is not properly contained in the graph of any other monotone mapping
Let H and H be two real Hilbert spaces, B : H H and B : H H be two setvalued maximal monotone mappings, A : H → H be a linear and bounded operator, and A∗ be the adjoint of A

Summary

Introduction

Let H be a real Hilbert space, and B : H H be a set-valued mapping with domain D(B) := {x ∈ H : B(x) = ∅}. In this paper, motivated by the above works and related results, we present the following algorithm with conjugate gradient method for the split variational inclusion problems in Hilbert spaces. Let H and H be two real Hilbert spaces, A : H → H be a linear operator, and A∗ be the adjoint of A, and let β > be fixed, and let ρ ∈ Let C and Q be nonempty, closed, and convex subsets of infinite dimensional real Hilbert spaces H and H , respectively. Time Iteration (0.9977524, –0.002246182) (1.000870, 0.0008703675) (1.020805, 0.02122562) (1.029428, 0.03002226) (1.000037, 4.018706e–05)

Approximate solution

Time Iteration Approximate solution