Some fixed point theorems for set valued directional contraction mappings

Abstract

Let S be a subset of a metric space X and let B(X) be the class of all nonempty bounded subsets of X with the Hausdorff pseudometric H. A mapping F:S→B(X) is a directional contraction iff there exists a real α∈[0,1) such that for each x∈S and y∈F(x), H(F(x),F(z))≤αd(x,z) for each z∈[x,y]∩S, where [x,y]={z∈X:d(x,z)+d(z,y)=d(x,y)}. In this paper, sufficient conditions are given under which such mappings have a fixed point.

Keywords

• directional contraction
• Hausdorff pseudometric.