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

Let C denote a convex subset within the vector space p (·) , and let T represent a mapping from C onto itself. Assume α = (α 1 , ⋯ , α n) is a multi-index in [ 0 , 1 ] n such that ∑ i = 1 n α i = 1 , where α 1 > 0 and α n > 0 . We define T α : C → C as T α = ∑ i = 1 n α i T i , known as the mean average of the mapping T. While every fixed point of T remains fixed for T α , 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 T α 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 p (·) , 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. [ABSTRACT FROM AUTHOR]