Generalization of shadows and fixed point theorems for fuzzy sets

摘要

In this paper, we define a fuzzy set concerning shadows without assuming the existence of the metric projection and extend a well-known result in a Hilbert space to that in a formed linear space. We shall observe that the fuzzy set presented here is in essence a generalization of shadows. Further, we study fuzzy mappings in the framework of fixed point theory. We first prove two theorems extending Nadler's fixed point theorem. Lots of fixed point theorems for fuzzy mappings have been proved, however, the method used here is completely different from those in the previous results. Next, we prove a theorem by applying Fan's fixed point theorem and conversely, we show that this is a generalization of this theorem.