Compatible mapping and common fixed points. (English) Zbl 0613.54029

The author obtains common fixed point theorems for collections of mappings satisfying certain contractive type conditions in complete metric spaces. These results extend a number of earlier ones primarily by replacing commutativity assumptions with a weaker ”compatibility” assumption. Self-mappings f and g of a metric space (X,d) are compatible if \(\lim_{n}d(g(f(x_ n),f(g(x_ n))=\emptyset\) whenever \(\{x_ n\}\) is a sequence in X such that \(\lim_{n}f(x_ n)=\lim_{n}g(x_ n)=t\) for some \(t\in X\). This definition implies that f and g commute on the set \(\{\) \(x\in X:\) \(f(x)=g(x)\}.\)
A typical corollary of the results obtained is the following: Let S and T be self-maps of a complete metric space (X,d) and let A,B: \(X\to S(X)\cap T(X)\). Suppose that S and T are continuous and that the pairs A, S and B, T are compatible. If there exists \(r\in (0,1)\) such that d(Ax,By)\(\leq rd(Sx,T(y))\), x,y\(\in X\), then A, B, S, and T have a unique common fixed point.
Reviewer: W.A.Kirk


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