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 \(E\).


54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems