Fixed point theorems on complete and compact spaces (Q1058185)

This paper contains two fixed point theorems for mappings on compact Hausdorff spaces or complete metric spaces. Using the techniques of \textit{M.-H. Shih} and \textit{C.-C. Yeh} [Proc. Am. Math. Soc. 85, 465-468 (1982; Zbl 0488.54034)] the authors establish the first theorem, which generalizes the following theorem of \textit{B. Fisher} [Bull. Inst. Math., Acad. Sin. 9, 399-406 (1981; Zbl 0466.54037)]: Let f and g be two self- mappings on a compact metric space (X,d) such that \(gf=fg\), g(X)\(\subset f(X)\), f,g are continuous and \[ d(gx,gy)<\max \{d(fx,fy),d(fx,gx),d(fy,gy),d(fx,gy),d(fy,gx)\} \] for all x,y\(\in X\) for which the right is \(>0\). Then S and T have a unique common fixed point. The second theorem extends a result of \textit{H. Chatterjee} [Indian J. Pure Appl. Math. 10, 400-403 (1979; Zbl 0405.54037)].