Abstract
Abstract In this paper, we prove a coupled fixed point theorem for a multivalued fuzzy contraction mapping in complete Hausdorff fuzzy metric spaces. As an application of the first theorem, a coupled coincidence and coupled common fixed point theorem has been proved for a hybrid pair of multivalued and single-valued mappings. It is worth mentioning that to find coupled coincidence points, we do not employ the condition of continuity of any mapping involved therein. Also, coupled coincidence points are obtained without exploiting any type of commutativity condition. Our results extend, improve, and unify some well-known results in the literature. MSC:47H10, 47H04, 47H07.
Highlights
Introduction and preliminaries Bhaskar andLakshmikantham [ ] introduced the concept of a coupled fixed point of a mapping F from X ×X to X and established some coupled fixed point theorems in partially ordered sets
The concept of fuzzy sets was initiated by Zadeh [ ] in
A number of authors started the study of fixed point theory in fuzzy metric spaces; for a detailed survey, we refer to [ – ] and the references therein
Summary
Introduction and preliminaries Bhaskar andLakshmikantham [ ] introduced the concept of a coupled fixed point of a mapping F from X ×X to X and established some coupled fixed point theorems in partially ordered sets. If a mapping M : X × [ , ∞) → [ , ] satisfies the following conditions: (F ) M(x, y, t) > ; (F ) M(x, y, t) = if and only if x = y; (F ) M(x, y, t) = M(y, x, t); (F ) M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s); (F ) M(x, y, t) : ( , ∞) → [ , ] is continuous; for each x, y, z ∈ X and s, t > , -tuple (X, M, ∗) is called a fuzzy metric space.
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