Abstract
We introduce a new concept for multivalued maps, also called multivalued nonlinear F-contraction, and give a fixed point result. Our result is a proper generalization of some recent fixed point theorems including the famous theorem of Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl., 334(1):132–139, 2007].
Highlights
Introduction and preliminariesLet (X, d) be a metric space
It is well known that H is a metric on CB (X), which is called Pompeiu–Hausdorff metric induced by d
In 1969, Nadler [14] proved that every multivalued contraction on complete metric space has a fixed point
Summary
Introduction and preliminariesLet (X, d) be a metric space. P (X) denotes the family of all nonempty subsets of X, C(X) denotes the family of all nonempty, closed subsets of X, CB (X) denotes the family of all nonempty, closed, and bounded subsets of X, and K(X) denotes the family of all nonempty compact subsets of X. Let T : X → CB (X) be a map, T is called multivalued contraction (see [14]) if for all x, y ∈ X, there exists L ∈ [0, 1) such that In 1969, Nadler [14] proved that every multivalued contraction on complete metric space has a fixed point.
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