A fixed point theorem for uniformly Lipschitzian mappings in modular vector spaces

Abstract

We give a fixed point theorem for uniformly Lipschitzian mappings defined in modular vector spaces which have the uniform normal structure property in the modular sense. We also discuss this result in the variable exponent space

On Suzuki Mappings in Modular Spaces

“…Let us prove it by induction on . If = 1, one has A 0 1 = ω(1) = 1 and relation (11) becomes in this particular case…”

“…More extensively, if ρ is assumed to be convex and to satisfy condition ∆ 2 , we can define the growth function (see [11]):…”

“…Then, 1 < ω (2). The arguments for this statement can be extracted from Lemma 2.6 in [11]. In addition, let us notice that ρ(αx) ≤ ω(α)ρ(x), ∀α ≥ 0, ∀x ∈ X ρ and also that, for each positive integer l and arbitrary elements x 1 , x 2 , .…”

“…In time, the notion was reactivated in the expanded framework of vector spaces and one important direction was settled by Khamsi [6] in connection with the fixed point theory. Nowadays, this approach is fructified in several works: Okeke et al [7], Khan [8], Abbas et al [9], Abdou and Khamsi [10], Alfuraidan et al [11] and the papers referenced there.…”

Feng-Liu Type Fixed Point Results for Multivalued Mappings in GMMS

“…In our related paper [22], τ X D meets properties of usual topology. For example, if we take modular vector spaces as in [23], the ρ-ball…”