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International Journal of Mathematics and Mathematical Sciences
Volume 3, Issue 3, Pages 455-460

Some fixed point theorems for set valued directional contraction mappings

Department of Mathematics, University of Wyoming, Laramie 82071, Wyoming, USA

Received 5 July 1979

Copyright © 1980 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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:SB(X) is a directional contraction iff there exists a real α[0,1) such that for each xS and yF(x), H(F(x),F(z))αd(x,z) for each z[x,y]S, where [x,y]={zX:d(x,z)+d(z,y)=d(x,y)}. In this paper, sufficient conditions are given under which such mappings have a fixed point.