A generalization of Brian Fisherâ€™s theorem. (English)

Math. Educ. 38, No. 2, 92-94 (2004).

Suppose $S$ and $T$ are self-maps of a complete metric space $(X, d)$ satisfying the inequality:$d(Sx, TSy)\lec. max \{d(x, Sx), d(x, Sy), d(Sy, TSy), ((x, TSy)+d(Sx, Sy))/2\}$ for all $x, y\inX$, where $0\lec<1$. Brian Fisher has proved that if either $S$ or $T$ is continuous then $S$ and $T$ have a unique common fixed point. In this paper, we obtain a generalization of this result by relaxing the completeness of $X$ and dropping the condition of continuity. (Authorâ€™s abstract)