Let be a metric space and the set of all nonempty bounded subsets of . If then : and and . The authors extend the concepts of compatible (weak) maps and generalized Meir-Keeler contractions.
Definition 1. Let be a metric space and let . Then and are contractions relative to maps if , and there exist functions , such that for all and for :
Definition 2. Let be a metric space and let and . The pair is a weakly compatible pair if implies . Every compatible pair [G. Jungck and B. E. Rhoades, Int. J. Math. Math. Sci. 16, No. 3, 417-428 (1993; Zbl 0783.54038)] is weakly compatible. Examples of weakly compatible pairs which are not compatible are given in the paper. In this paper the authors prove some fixed point theorems for set valued functions without appeal to continuity. These theorems extend results of T.-H. Chang [Math. Jap. 38, No. 4, 675-690 (1993; Zbl 0805.47049)], generalize results by J. Jachymski [ibid. 42, No. 1, 131-136 (1995; Zbl 0845.47044)] and by S. M. Kang and B. E. Rhoades [ibid. 37, No. 6, 1053-1059 (1992; Zbl 0767.54037)] and produce as byproducts generalizations of theorems for point valued functions.
Theorem 4.1: Let and be self maps of a metric space and let . Suppose , , and one of , is complete. Let and : be maps, and suppose that for . If for , then there exists a unique point such that provided that both and are weakly compatible pairs, and one of (a), (b) below is true:
(a) , , whenever , and is u.s.c. from the right; and
(b) , , , and is u.s.c.