On fixed points in Hausdorff spaces (Q1066507)

The author presents fixed point theorems of the following type: Let X be a Hausdorff space, \(T: X\to X\) continuous function, and \(F: X\times X\to R^+\) be a continuous function such that (a) F(x,y)\(\neq 0\), for all \(x\neq y\), (b) F(Tx,Ty)\(\leq F(x,y)-\omega (F(x,y))\), for all x,y\(\in X\), where \(\omega\) : \(R^+\to R^+\) is continuous function, with \(Q<\omega (r)<r\) for all \(r\in R_+-\{0\}\). If for some \(x_ 0\in X\) the sequence \(x_ n=(T^ nx_ 0)\) has a convergent subsequence, then T has a unique fixed point.