Common Fixed Point Theorems for Compatible Mappings and Compatible Mappings of Type (a)
نویسندگان
چکیده
The purpose of this paper is to prove some theorems which generalize a theorem of M. Tel ci, K. Tas and B. Fisher on compatible mappings and compatible mappings of type (A). Let S, T be two self-mappings of a metric space (X; d). Sessa 7] deenes S and T to be weakly commuting if d(STx; TSx) 6 d(Tx; Sx) for all x in X. Jungck 3] deenes T and S to be compatible ii lim n d(STx n ; TSx n) = 0 whenever fx n g is a sequence in X such that lim n Sx n = lim n Tx n = x for some x 2 X. Clearly, commuting mappings are weakly commuting and weakly commuting maps are compatible, neither implication is reversible, Ex. 1 9] and Ex. 2.2 3]. Recently, Jungck, Muthy and Cho 4] have deened S and T to be compatible of type (A) if lim n d(TSx n ; S 2 x n) = 0 and lim n d(STx n ; T 2 x n) = 0 whenever fx n g is a sequence in X such that lim n Sx n = lim n Tx n = t for some t 2 X. Clearly, weakly commuting mappings are compatible of type (A). By 4], Ex. 2.2, it follows that this implication is not reversible. By 4], Ex. 2.1 and Ex. 2.2, it follows that the notions of compatible mappings and compatible mappings of type (A) are independent. Lemma 4] Let S; T : (X; d) ! (X; d) be two mappings. If A and T are compatible of type (A) and S(t) = T(t) for some t 2 X, then ST(t) = TT(t) = TS(t) = SS(t). Let F be the set of all functions f : R + ! R + such that: (i) f is isotone, i.e. if t 1 6 t 2 , then f(t 1) 6 f(t 2) for all t 1 ; t 2 2 R + ; (ii) f is upper semi-continuous; (iii) f(t) < t for each t > 0. The following theorem which generalizes the results from 1], 2], 5], 6] and 8] is proved in 10].
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