Strong convergence theorem for split monotone variational inclusion with constraints of variational inequalities and fixed point problems

Jin-Lin Guan,Lu-Chuan Ceng,Bing Hu

doi:10.1186/s13660-018-1905-6

Abstract

In this paper, inspired by Jitsupa et al. (J. Comput. Appl. Math. 318:293–306, 2017), we propose a general iterative scheme for finding a solution of a split monotone variational inclusion with the constraints of a variational inequality and a fixed point problem of a finite family of strict pseudo-contractions in real Hilbert spaces. Under very mild conditions, we prove a strong convergence theorem for this iterative scheme. Our result improves and extends the corresponding ones announced by some others in the earlier and recent literature.

Highlights

1 Introduction It is known that variational inequality, as a greatly important tool, has already been studied for a wide class of unilateral, obstacle, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework
Under some suitable assumptions on the sequences {αn}, {βn}, and {ηi(n)}Ni=1, we prove that the sequence {xn} defined by (1.7) converges strongly to a common solution of SMVIP (1.3) with the constraints of a variational inequality and a fixed point problem of a finite family of strict pseudo-contractions, which solves the following variational inequality: μVq – τ Fq, q – p ≤ 0, ∀p ∈ F, where F denotes the set of common solutions of SMVIP (1.3), a variational inequality, and a fixed point problem of a finite family of strict pseudo-contractions
5 Results and discussion In this paper, we propose a new iterative scheme for finding a solution of SMVIP (1.3) with the constraints of a variational inequality and a fixed point problem of a finite family of strict pseudo-contractions in real Hilbert spaces

Summary

Introduction

It is known that variational inequality, as a greatly important tool, has already been studied for a wide class of unilateral, obstacle, and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. Where A is a bounded linear operator, A∗ is the adjoint of A, {Ti}Ni=1 is a finite family of kistrictly pseudo-contractions, f is a contraction, D is a strong positive linear bounded operator They proved, under certain appropriate assumptions on the sequences {αn}, {βn}, and {ηi(n)}Ni=1, that {xn} defined by (1.6) converges strongly to a common solution of SVIP (1.2) and a fixed point of a finite family of ki-strictly pseudo-contractions, which solves some variational inequality problem. Under some suitable assumptions on the sequences {αn}, {βn}, and {ηi(n)}Ni=1, we prove that the sequence {xn} defined by (1.7) converges strongly to a common solution of SMVIP (1.3) with the constraints of a variational inequality and a fixed point problem of a finite family of strict pseudo-contractions, which solves the following variational inequality: μVq – τ Fq, q – p ≤ 0, ∀p ∈ F , where F denotes the set of common solutions of SMVIP (1.3), a variational inequality, and a fixed point problem of a finite family of strict pseudo-contractions. We provide a numerical example to support our strong convergence result

Preliminaries

Main results

Results and discussion

Conclusions