Common fixed point theorems in Menger spaces (Q885619)

Let \(X\) be a complete Menger space endowed with the triangular norm ``min''. Let \(B(X)\) denote the family of nonempty bounded subsets of \(X\) and \(S, T:X \rightarrow B(X)\). Consider four single-valued maps \(f, g, \eta, \xi: X \rightarrow X\) such that (i) \(S(X) \subseteq \xi g(X), T(X) \subseteq \eta f(X)\); (ii) \(\eta f = f\eta, \xi g = g \xi, Sf = fS, Tg = gT\); (iii) one of \(\eta f\) or \(\xi g\) is continuous; (iv) \((S, \eta f)\) and \((T, \xi g)\) are pairwise compatible. Then the main result of this paper states that these six maps satisfying a general contraction type condition [cf. \textit{S. N. Mishra}, Math. Jap. 36, No. 2, 283--289 (1991; Zbl 0731.54037)] have a unique common fixed point. Some special cases are discussed.