Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁p(·)

Abstract

Let C denote a convex subset within the vector space 𝓁p(·), and let T represent a mapping from C onto itself. Assume α=(α1,⋯,αn) is a multi-index in [0,1]n such that ∑i=1nαi=1, where α1>0 and αn>0. We define Tα:C→C as Tα=∑i=1nαiTi, known as the mean average of the mapping T. While every fixed point of T remains fixed for Tα, 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 Tα and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in R. Our focus then shifts to variable exponent modular vector spaces 𝓁p(·), 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.