Abstract
The purpose of this paper is to study the existence of fixed points for contractive-type multivalued maps in the setting of modular metric spaces. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we investigate the existence of fixed points of multivalued modular contractive mappings in modular metric spaces. Consequently, our results either generalize or improve fixed point results of Nadler (Pac. J. Math. 30:475-488, 1969) and Edelstein (Proc. Am. Math. Soc. 12:7-10, 1961).
Highlights
1 Introduction The aim of this paper is to give an outline of a fixed point theory for multivalued Lipschitzian mappings defined on some subsets of modular metric spaces
We look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano [ ] on vector spaces and modular function spaces introduced by Musielak [ ] and Orlicz [ ]
In [ ] the authors have defined and investigated the fixed point property in the framework of modular metric space and introduced the analog of the Banach contraction principle theorem in modular metric space
Summary
The aim of this paper is to give an outline of a fixed point theory for multivalued Lipschitzian mappings defined on some subsets of modular metric spaces. [ , ] A function ω : ( , ∞) × X × X → [ , ∞] is said to be a modular metric on X if it satisfies the following axioms: (i) x = y if and only if ωλ(x, y) = , for all λ > ; (ii) ωλ(x, y) = ωλ(y, x), for all λ > , and x, y ∈ X; (iii) ωλ+μ(x, y) ≤ ωλ(x, z) + ωμ(z, y), for all λ, μ > and x, y, z ∈ X.
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