Remarks on recent fixed point theorems for compatible maps (Q1922864)

Let \(A,B,S,T\) be selfmaps of a metric space \((X,d)\). The pair \(\{A,B\}\) is said to be an \((\varepsilon,\delta)\)-contraction relative to the pair \(\{S,T\}\) if \(A(X) \subseteq T(X)\), \(B(X) \subseteq S(X)\) and there exists a function \(\delta: (0,\infty) \to (0,\infty)\) such that \(\delta (\varepsilon)> \varepsilon\) for all \(\varepsilon\), and for \(x,y\) in \(X\), we have \(\varepsilon \leq m(x,y) < \delta (\varepsilon)\) implies \(d(Ax,By)< \varepsilon\) and \(Ax=By\) whenever \(Sx=Ty\). Here \(m(x,y)= \max \{d(Sx,Ty)\), \(k\cdot d(Sx,Ax)\), \(k\cdot d(Ty,By)\), \({1\over 2} (d(Sx,By) + d(Ax,Ty))\}\), where \(k\in(0,1)\). Under this assumption, the authors prove that if one of \(A(X)\), \(B(X)\), \(S(X)\), \(T(X)\) is a complete subspace of \(X\), then \(A\) and \(S\) (resp. \(B\) and \(T)\) have a coincidence point. \(A,B,S,T\) have a unique common fixed point if \(A\) and \(S\) (resp. \(B\) and \(T)\) commute on their set of coincidence points. A similar conclusion is given if the four maps satisfy generalized contractive conditions.