Abstract
AbstractThe purpose of this paper is to study the existence of fixed points for contractive-type and nonexpansive-type multivalued maps in the setting of modular function spaces. We also discuss the concept of "Equation missing"-modular function and prove fixed point results for weakly-modular contractive maps in modular function spaces. These results extend several similar results proved in metric and Banach spaces settings.
Highlights
Introduction and PreliminariesThe well-known Banach fixed point theorem on complete metric spaces each contraction self-map of a complete metric space has a unique fixed point has been extended and generalized in different directions
Fixed point theorems for contractive and nonexpansive multivalued maps have been established by several authors
A multivalued map J : X → 2X where 2X denotes the collection of all nonempty subsets of X with bounded subsets as values is called contractive 10 if
Summary
The well-known Banach fixed point theorem on complete metric spaces each contraction self-map of a complete metric space has a unique fixed point has been extended and generalized in different directions. Without using the concept of the Hausdorff metric, Husain and Tarafdar 13 introduced the notion of a nonexpansive-type multivalued map and proved a fixed point theorem on compact intervals of the real line Using such type of notions Husain and Latif 14 extended their result to general Banach space setting. We define nonexpansive-type and contractive-type multivalued maps in modular function spaces, investigate the existence of fixed points of such mappings, and prove similar results found in 17. A function modular is said to satisfy the Δ2-type condition if there exists K > 0 such that for any f ∈ Lρ we have ρ 2f ≤ Kρ f. Let ρ be a convex function modular satisfying the Δ2-type condition.
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