A self-adaptive inertial subgradient extragradient method for pseudomonotone equilibrium and common fixed point problems

Lateef Olakunle Jolaoso,Maggie Aphane

doi:10.1186/s13663-020-00676-y

Lateef Olakunle Jolaoso, Maggie Aphane

Open Access

https://doi.org/10.1186/s13663-020-00676-y

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Abstract

In this paper, we introduce a self-adaptive inertial subgradient extragradient method for solving pseudomonotone equilibrium problem and common fixed point problem in real Hilbert spaces. The algorithm consists of an inertial extrapolation process for speeding the rate of its convergence, a monotone nonincreasing stepsize rule, and a viscosity approximation method which guaranteed its strong convergence. More so, a strong convergence theorem is proved for the sequence generated by the algorithm under some mild conditions and without prior knowledge of the Lipschitz-like constants of the equilibrium bifunction. We further provide some numerical examples to illustrate the performance and accuracy of our method.

Highlights

Muu and Oettli [40] introduced the equilibrium problem as a generalization of many problems in nonlinear analysis, which include variational inequalities, convex minimization, saddle point problems, fixed point problems, and Nash-equilibrium problems; see, e.g., [13, 40]
It is important to study the approximation of common solutions of the EP and fixed point problem (i.e., find x ∈ C such that x ∈ EP ∩ F(T), where F(T) = {x ∈ H : x = Tx}) because of some mathematical models whose constraints are expressed as fixed point and equilibrium problems
In an attempt to provide an alternative method which does not require prior knowledge of the Lipschitz constants c1 and c2, Yang and Liu [48] proposed the following modified HSEM with a nonincreasing stepsize and proved a strong convergence theorem for finding a common solution of pseudomonotone EP and a fixed point of a quasi-nonexpansive mapping S in real Hilbert spaces (Algorithm 1.3)

Summary

Introduction

Muu and Oettli [40] introduced the equilibrium problem (shortly, EP) as a generalization of many problems in nonlinear analysis, which include variational inequalities, convex minimization, saddle point problems, fixed point problems, and Nash-equilibrium problems; see, e.g., [13, 40]. Tada and Takahashi [46] first proposed the following hybrid method for approximating a common solution of EP with monotone bifunction and fixed points of a nonexpansive mapping T in Hilbert spaces:

Results

Conclusion