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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]

The purpose of this note is to discuss the recent paper of Espínola and Wiśnicki about the fixed point theory of monotone nonexpansive mappings. In their work, it is claimed that most of the fixed point results of this class of mappings boil down to the classical Knaster-Tarski fixed point theorem. We will show that their approach is very restrictive and fail to have any meaningful usefulness in applications.

We investigate the existence of solutions for the Fredholm integral equation Φ (ϑ) = G (ϑ , Φ (ϑ)) + ∫ 0 1 F (ϑ , ζ , Φ (ζ)) d ζ , for ϑ ∈ [ 0 , 1 ] , in the setting of the modular function spaces L ρ . We also derive an application of this research within the framework of variable exponent Lebesgue spaces L p (·) subject to specific conditions imposed on the exponent function p (·) and the functions F and G. [ABSTRACT FROM AUTHOR]

We establish a fixed point property for the Lebesgue spaces with variable exponents L p (·) , focusing on the scenario where the exponent closely approaches 1. The proof does not impose any additional conditions. In particular, our investigation centers on ρ -non-expansive mappings defined on convex subsets of L p (·) , satisfying the "condition of uniform decrease" that we define subsequently. [ABSTRACT FROM AUTHOR]