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Compatible mappings of type (B) and common fixed point theorems of Greguš type. (English) Zbl 0848.54030
Let \(S\) and \(T\) be two maps from a normed space \((X,d)\) into itself. Then \(S\) and \(T\) are called compatible of type \((B)\) if \[ \lim_{n \to \infty} d(STx_n, TTx_n) \leq 1/2 \Bigl[ \lim_{n \to \infty} d(STx_n, St) + \lim_{n \to \infty} d(St, SSx_n) \Bigr] \] and \[ \lim_{n \to \infty} d(TSx_n, SSx_n) \leq 1/2 \Bigl[ \lim_{n \to \infty} d(TSx_n, Tt) + \lim_{n \to \infty} d(Tt, TTx_n) \Bigr] \] whenever \(\{x_n\}\) is a sequence such that \(Sx_n\), \(Tx_n \to t\) for some point \(t\) in \(X\). Under this definition, the authors prove a common fixed point theorem of Greguš type. Suitable examples prove that the compatibility of type (B) is more general than analogous definitions valid in literature, e.g., see the paper of Y. J. Cho, S. M. Kang and G. Jungck [Int. J. Math. Math. Sci. 13, No. 1, 61-66 (1990; Zbl 0711.54029)].
Reviewer: S.Sessa (Napoli)

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
Citations:
Zbl 0711.54029
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References:
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