A comparison of various types of compatible maps and common fixed points (Q1355689)

The authors split the definition of compatibility of type \((A)\) due to \textit{G. Jungck}, \textit{P. P. Murthy} and \textit{Y. J. Cho} [Math. Jap. 38, No. 2, 381-390 (1993; Zbl 0791.54059)] into two parts defining \(A\)-compatibility and \(S\)-compatibility. Precisely, let \(A,S\) be selfmaps of a metric space \((X,d)\). If \(\{x_n\}\) is a sequence in \(X\) such that \(Ax_n\), \(Sx_n \to t\in X\) implies \(d(ASx_n,SSx_n) \to 0\) (resp. \(d(SAx_n, AAx_n) \to 0)\), then \(A\) and \(S\) are called \(A\)-compatible (resp. \(S\)-compatible). Suitable examples show the generality of this definition with respect to the compatibility of type \((A)\). A common fixed point theorem is given.