Strong convergence analysis for solving quasi-monotone variational inequalities and fixed point problems in reflexive Banach spaces

Abstract

In this paper, we propose a modified inertial hybrid Tseng's extragradient algorithm with self-adaptive step sizes for finding a common solution of variational inequalities with quasimonotone operators and the fixed point problems of a finite family of Bregman quasi-nonexpansive mappings. By using the Bregman-distance approach, we prove a strong convergence result under some appropriate conditions on the control parameters in real reflexive Banach spaces. Our algorithm is based on a self-adaptive step size which generates a non-monotonic sequence. Unlike the existing results in the literature, our algorithm does not require any linesearch technique which uses inner loops and might consume additional computational time for determining the step size. Finally, we present some numerical examples to illustrate the efficiency of our algorithm in comparison with related methods in the literature.