Common fixed points for nonself compatible maps on compacta (Q1803351)

The authors prove the following result: Let \((M,d)\) be a complete convex metric space, \(K\) be a nonempty compact subset of \(M\); \(f\), \(T: K\to M\) be continuous maps such that \(\partial K\subset TK\), \(fK\cap K\subset TK\) and \(Tx\in\partial K\Rightarrow fx\in K\). If \(f\) and \(T\) are compatible and \(d(fx,fy)< d(Tx,Ty)\) for all \(x\), \(y\in K\) with \(Tx\neq Ty\), then \(f\) and \(T\) have a unique common fixed point in \(K\). This theorem generalizes some earlier results. It has been mentioned that a theorem due to \textit{O. Hadžić} [Zb. Rad., Prir.-Mat. Fak., Univ. Novom Sadu, Ser. Mat. 15, No. 2, 1-13 (1985; Zbl 0639.54035)] cannot be applied in the present situation.