On monotone mappings in modular function spaces

Abstract

We prove the existence of fixed points of monotone ρ-nonexpansive mappings in ρ-uniformly convex modular function spaces. This is the modular version of Browder and Göhde fixed point theorems for monotone mappings. We also discuss the validity of this result in modular function spaces where the modular is uniformly convex in every direction. This property has never been considered in the context of modular spaces.

On Modulated Topological Vector Spaces and Applications

“…As it turns out, the same can be said of modulated topological vector spaces. To see this, let us first introduce a relevant notion of UCED, in which we will follow the relevant notion introduced for modular function spaces in [11] (see also a more recent application of UCED for function modulars in [2]). Definition 3.6.…”

Fixed Point Theorems for Noncyclic Monotone Relativelyρ-Nonexpansive Mappings in Modular Spaces

“…e following corollary is an immediate consequence of eorem 3, and it suffices to take A � B; therefore, A � A ⪯ 0 and B � B ⪯ 0 . It will be considered as a generalization of Browder and Göhde fixed point theorem for monotone ρ-nonexpansive mappings in modular spaces (see [7,11,12]). is corollary result has already been mentioned by Bin Dehaish and Khamsi (see eorem 1) [7] in the framework of modular function spaces.…”