×

zbMATH — the first resource for mathematics

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

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
PDF BibTeX XML Cite
Full Text: DOI EuDML