Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces l_p(.)

Let C denote a convex subset within the vector space (Formula presented.), and let T represent a mapping from C onto itself. Assume (Formula presented.) is a multi-index in (Formula presented.) such that (Formula presented.), where (Formula presented.) and (Formula presented.). We define (Formula presented.) as (Formula presented.), known as the mean average of the mapping T. While every fixed point of T remains fixed for (Formula presented.), the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for (Formula presented.) and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in (Formula presented.). Our focus then shifts to variable exponent modular vector spaces (Formula presented.), where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework.