Analysis of some convergence results for inertial variational inequalities problem and its application

Abstract

Some core aspects of nonlinear analysis, which is a major branch of mathematics, are the optimization problems, fixed point theory and its applications. These concepts, that is, optimization theory, fixed point theory and its applications are widely applied in several fields of science such as networking, inventory control, engineering, economics, policy modelling, transportation and mathematical sciences to mention but a few. Due to its relevance to different fields, the theory of optimization and fixed point has been a popular field of research for a long time. Given its expansive nature, researchers continue to make new discoveries and advancements, contributing to its enduring significance across various disciplines. The goal of this dissertation is to explore some convergence iterative methods for approximating optimization problems. We propose a new modified projection and contraction algorithm for approximating solutions of a variational inequality problem involving a quasi-monotone and Lipschitz continuous mapping in real Hilbert spaces. We incorporate the technique of two-step inertial into a single projection and contraction method and prove a weak convergence theorem for the proposed algorithm. The weak convergence theorem proved requires neither the prior knowledge of the Lipschitz constant nor the weak sequential continuity of the associated mapping. Under additional strong pseudomonotonicity, the R-linear convergence rate of the two-step inertial algorithm is presented. Finally, some numerical examples are given to illustrate the effectiveness and competitiveness of the proposed algorithm in comparison with some existing algorithms in the literature