A fixed point theorem for nonself set-valued mappings, Internat

ABSTRACT. Let X be a complete, metrically convex metric space, K a closed convex subset of X, CB(X) the set of closed and bounded subsets of X. Let F " K CB(X) satisfying definition (1) below, with the added condition that Fx C _ K for each x OK. Then F has a fixed point in K. This result is an extension to multivalued mappings of a result of (irid [1]. KEY WORDS AND PHRASES. Fixed point, multivalued map, non-self map 1991 AMS SUBJECT CLASSIFICATION CODE. 47H10 Let X be a complete metrically convex metric space. This means that, for each x,y in X, x y, there exists a z in OX such that d(x,y) d(x,z) + d(z,y). Let CB(X) denote the set of closed and bounded subsets of X, H denote the Hausdorff metric on CB(X). Let K be a nonempty closed, convex subset of X. Let F K CB(X) satisfying: for each x, y in K, H(Fx,Fy) < _ h max { d(x,Y)a D(x,Fx), D(y,Fy), [D(x,Fy) + D(y,Fx)]>a+h j (1) where 0 < _ h < (-1 + v/2,a> _ 1 + (2h2/(1 + h)), and F(x) C K for each x OK. (irid [1] proved a fixed point theorem for the single-valued version of (1). He also established a multivalued version. However, he used the S-distance, instead of the.Hausdorff distance, so tha