Strong convergence inertial projection and contraction method with self adaptive stepsize for pseudomonotone variational inequalities and fixed point problems

Lateef Olakunle Jolaoso,Maggie Aphane

doi:10.1186/s13660-020-02536-0

Abstract

In this paper, we introduce a new inertial self-adaptive projection method for finding a common element in the set of solution of pseudomonotone variational inequality problem and set of fixed point of a pseudocontractive mapping in real Hilbert spaces. The self-adaptive technique ensures the convergence of the algorithm without any prior estimate of the Lipschitz constant. With the aid of Moudafi’s viscosity approximation method, we prove a strong convergence result for the sequence generated by our algorithm under some mild conditions. We also provide some numerical examples to illustrate the accuracy and efficiency of the algorithm by comparing with other recent methods in the literature.

Highlights

Let H be a real Hilbert space with inner product ·, · and norm ·
It is well known that the VIP is a very fundamental problem in nonlinear analysis. It serves as a useful mathematical model which unifies in several ways, many important concepts in applied mathematics such as optimization, equilibrium problem, Nash equilibrium problem, complementarity problem, fixed point problems and system of nonlinear equations; see for instance [19,20,21, 31, 33]
We introduce a new self-adaptive inertial projection and contraction method for finding common element in the set of solution of pseudomonotone variational inequalities and the set of fixed points of strictly pseudocontractive mappings in real Hilbert spaces

Summary

Introduction

Let H be a real Hilbert space with inner product ·, · and norm ·. It is well known that the gradient projection method converges weakly to a solution of the VIP if and only if the operator When A is monotone, the gradient projection method fails to converge to solution of the VIP.

Results

Conclusion