ELSEVIER Fuzzy Sets and Systems 87 (1997) 325-334 FU2ZY sets and systems Fixed degree and fixed point theorems for fuzzy mappings in probabilistic metric spaces Shih-Sen Chang a, Yeol Je Cho b, Byung Soo Lee c'*, Gue Myung Lee d a Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, People's Republic of China b Department of Mathematics, Gyeon9san 9 National University, College of Education, Chinju 660-701, South Korea c Department of Mathematics, Kyunosun9 University, 110-1 Daeyeon-don 9, Nam-9 u, Pusan 608-736, South Korea d Department of Natural Sciences, Pusan National University of Technolo9y, Pusan 608-739, South Korea Received May 1995; revised October 1995 Abstract This paper introduces the concept and properties of fixed degrees for fuzzy mappings in probabilistic metric spaces. By virtue of this concept, some theorems about common fixed degree of a sequence of fuzzy mappings in probabilistic metric spaces are obtained. These new results are a unified approach to generalize several fixed point theorems for fuzzy mappings. @ 1997 Elsevier Science B.V. Keywords: Analysis; Fuzzy sets; Membership function; Fixed degree for a fuzzy mapping; Fixed point for a fuzzy mapping; Probabilistic metric space In recent years, the fixed point theorems for fuzzy mappings have been studied by many authors [1,2,4-8, 11-13, 15-19,23]. The purpose of this paper is to introduce the concept of fixed degree for fuzzy mappings in probabilistic metric spaces. By virtue of this concept, some fixed degree theorems and fixed point theorems for fuzzy mappings in probabilistic metric spaces are obtained. The results given in this paper generalize and unify some recent results of Butnariu [2], Chang [4-8], Fan [11], Grabiec [13], Hadzic [15], Heilpern [16] and Lee et al. [18, 19]. 1. Preliminaries Throughout this paper, let ~ = (-cx~, +cxz), ~+= [0, +cx~) and 7/+ be the set of all positive integers. Definition 1. A function f:~---, [0, 1] is called a distribution function if it is nondecreasing and left- continuous with suptca f(t) = 1, inftER f(t) = O. * Corresponding author. 0165-0114/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved SSDI 01 65-01 1 4(95 )003 73 -8