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On a common fixed point theorem of a Greguš type. (English) Zbl 0753.54023
Summary: It is proved that if $$T$$ and $$E$$ ($$E$$ continuous) are two compatible self mappings of a closed subset $$K$$ of a complete convex metric space $$X$$ such that the condition: $d(Tx,Ty)\leq \text{ad}(Ex,Ey)+(1- a)\max\{d(Ex,Tx),d(Ey,Ty)\}$ holds for all $$x$$, $$y$$ in $$K$$, where $$0<a<1$$, and $$\text{Co}[T(K)]\subseteq E(K)$$, then $$T$$ and $$E$$ have a unique common fixed point. This result generalizes a theorem of B. Fisher and S. Sessa [Int. J. Math. Math. Sci. 9, 23-28 (1986; Zbl 0597.47036)] and a theorem of R. N. Mukherjee and V. Verma [Math. Jap. 33, No. 5, 745-749 (1988; Zbl 0655.47047)] and shows that these theorems remain true when the hypotheses of linearity and non- expansivity of $$E$$ are reduced to the continuity of $$E$$.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
##### Keywords:
common fixed point