Fixed point theorems for fuzzy mappings in complete metric spaces

Abstract

In this paper, we are concerned with mathematical properties of the previous contractive types of set-valued maps or fuzzy mappings on a metric space (see, for instance, Nadler, Pacific J. Math. 30 (1969) 475–488; Heilpern, J. Math. Anal. Appl. 83 (1981) 566–569; Papageorgiou, Nonlinear Anal. Theory Methods Appl. 7 (1983) 763–770; Bose and Sahani, Fuzzy Sets and Systems 21 (1987) 53–58; Som and Mukherjee, Fuzzy Sets and Systems 33 (1989) 213–219; Park and Jeong, Fuzzy Sets and Systems 59 (1993) 231–235; 87 (1997) 111–116; Lee et al., Fuzzy Sets and Systems 101 (1999) 143–152) and we prove some crucial theorems relating fixed points of fuzzy mappings on complete metric spaces. First, we prove a theorem by using the concept of w-distance (Kada et al., Math. Japonica 44 (1996) 381–391), which generalizes the results of Amemiya and Takahashi (Fuzzy Sets and Systems 114 (2000) 469–476), and Kada et al. (1996). Next, we prove two theorems for fuzzy mappings, which are connected with fixed point theorems for set-valued maps. Then we shall observe that our theorems can be used to obtain almost all results proved so far concerning fixed points of fuzzy mappings or set-valued maps on complete metric spaces and therefore, that our results imply the existence of fixed points of various contractive types of fuzzy mappings or set-valued maps on complete metric spaces.