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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.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
##### Keywords:
Menger space; unique common fixed point