Fixed Point Theorems for a Weaker Meir-Keeler Type -Set Contraction in Metric Spaces

In 1929, Knaster et al. 1 had proved the well-known KKM theorem on n-simplex. Besides, in 1961, Fan 2 had generalized the KKM theorem to an infinite dimensional topological vector space. Later, Amini et al. 3 had introduced the class of KKM-typemappings onmetric spaces and established some fixed point theorems for this class. In this paper, we define a weaker Meir-Keeler type function and establish the fixed point theorems for a weaker MeirKeeler type ψ-set contraction in metric spaces. Throughout this paper, byR we denote the set of all real nonnegative numbers, while N is the set of all natural numbers. We digress briefly to list some notations and review some definitions. Let X and Y be two Hausdorff topological spaces, and let T : X → 2 be a setvalued mapping. Then T is said to be closed if its graph GT { x, y ∈ X × Y : y ∈ T x } is closed. T is said to be compact if the image T X of X under T is contained in a compact subset of Y . IfD is a nonempty subset of X, then 〈D〉 denotes the class of all nonempty finite subsets of D. And, the following notations are used: