Viscosity Approximation Methods for Generalized Modification of the System of Equilibrium Problem and Fixed Point Problems of an Infinite Family of Nonexpansive Mappings

Abstract

Fixed points (FP) of infinite families of nonexpansive mappings find diverse applications across various disciplines. In economics, they help to find stable prices and quantities in markets. In game theory, fixed points help to find Nash equilibria. In computer science, fixed points are used to understand program meanings and help in making better algorithms for tasks like data analysis, checking models, and improving compilers. Solutions to equilibrium problems have practical uses in various areas. For instance, in physics, these solutions assist in analyzing systems at rest or in motion. In engineering, they aid in designing structures that can withstand forces without collapsing, ensuring safety and stability in construction projects. The main aim of the article is to present the concept of generalized modification of the system of equilibrium problems (GMSEP) for an infinite family of nonexpansive mappings. In this paper, we study viscosity approximation methods and present a new algorithm to find a common element of the fixed point of an infinite family of nonexpansive mappings and the set of solutions of generalized modification of the system of equilibrium problem in the setting of Hilbert spaces. Under some conditions, we prove that the sequence generated by the algorithm converges strongly to this common solution.