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.