Some geometric properties in modular spaces and application to fixed point theory

Abstract

We prove that modular spaces L ρ have the uniform Kadec-Klee property w.r.t. the convergence ρ-a.e. when they are endowed with the Luxemburg norm. We also prove that these spaces have the uniform Opial condition w.r.t. the convergence ρ-a.e. for both the Luxemburg norm and the Amemiya norm. Some assumptions over the modular ρ need to be assumed. The above geometric properties will enable us to obtain some fixed point results in modular spaces for different kind of mappings.  2004 Elsevier Inc. All rights reserved.In this paper we consider modular function spaces L ρ , where ρ is a convex additive modular satisfying the ∆ 2 -type condition. Our main goal is to prove that these spaces satisfy certain geometric properties which let us deduce some fixed point results for different classes of mappings. In order to do that we endow L ρ with either the Luxemburg norm or the Amemiya norm. The convergence ρ-a.e. will be essential in our purpose.The organization of this paper is the following: We begin by introducing the definitions and notations which will be used later. In Section 2 we prove some technical results which are our main tools throughout the paper. In Section 3 we prove that the modular space L ρ ✩

Convexity and boundedness relaxation for fixed point theorems in modular spaces

“…(H2) ∆ 2 -type condition: A modular ρ is said to satisfy the ∆ 2 -type condition [45,46] if there exists k > 0 such that ρ(2x) ≤ kρ(x) for all x ∈ X ρ . (H3)s-convex modulars: If condition (3) in the modular definition is replaced by ρ(αx + βy) ≤ αsρ(x) + βsρ(y) for all α, β ∈ [0, ∞) with αs + βs = 1 with ans ∈ (0, 1], the modular ρ is called ans-convex modular [22]. In particular, a 1-convex modular is simply called convex.…”

“…Some excellent overviews of (H1)-(H4) conditions are provided in [27,22,54]. It is shown that a modular ρ implies that…”

Nadler’s Theorem on Incomplete Modular Space

“….s-convex modular [39]: If condition (3) in the modular definition is replaced by ρ(αx + βy) ≤ αsρ(x) + βsρ(y) for all α, β ∈ [0, +∞) with αs + βs = 1 with ans ∈ (0, 1], the modular ρ is called ans-convex modular. In particular, a 1-convex modular is simply called convex.…”

On Fuzzy Modular Spaces

“…In 1990, Khamsi et al [7] initiated the study of fixed point theory for nonexpansive mappings defined on some subsets of modular function spaces. More researches on fixed point theory in modular function spaces can be found in [8][9][10][11][12][13].…”