Common fixed points of compatible mappings in PM-spaces. (English) Zbl 0731.54037

The author proves the following result and derives its metric analogue: (Theorem 1) Let A, B, S and T be self-mappings of a complete Menger space (X,F,t) where t is continuous and t(x,x)\(\geq x\) for all \(x\in [0,1]\). Suppose that S and T are continuous, the pairs A, S and B, T are compatible, and that A(X)\(\subset T(X)\) and B(X)\(\subset S(X)\). If there exists a positive number \(k<1\) such that \(F_{Ap,Bq}(kx)\geq t(F_{Ap,Sp}(x),t(F_{Bq,Tq}(x),\quad t(F_{Sp,Tq}(x),\quad t(F_{Ap,Tq}(\alpha x),\quad F_{Bq,Sp}(2x-\alpha x)))))\) for all p,q\(\in X\), \(x>0\) and \(\alpha\in (0,2)\), then A, B, S and T have a unique common fixed point in X. - His results improve certain results of X. P. Ding [Kexue Tongbao, Foreign. Lang. Ed. 29, 147-150 (1984; Zbl 0522.54040)] and others.


54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems