Generalization of shadows and fixed point theorems for fuzzy sets

In this paper, we define a fuzzy set concerning shadows without assuming th e existence of the metric projection and extend a well-known result in a Hi lbert space to that in a normed linear space. We shall observe that the fuz zy set presented here is in essence a generalization of shadows. Further, w e study fuzzy mappings in the framework of fixed point theory. We first pro ve two theorems extending Nadler's fixed point theorem. Lots of fixed point theorems for fuzzy mappings have been proved, however, the method used her e is completely different from those in the previous results. Next, we prov e a theorem by applying Fan's fixed point theorem and conversely, we show t hat this is a generalization of this theorem. (C) 2000 Elsevier Science B.V . All rights reserved.