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An extension of Gregus fixed point theorem. (English) Zbl 1158.47043
Let $$C$$ be a closed convex subset of a complete metrizable space equipped with an $$F$$-norm. Let $$T$$ be a self-map of $$C$$ that satisfies $F(Tx - Ty) \leq aF(x - y) + bF(x - Tx) + cF(y - Ty) + eF(y - Tx) + fF(x - Ty)$ for all $$x, y$$ in $$C$$, where $$0< a < 1, b \geq 0, c \geq 0, f\geq 0$$ and $$a + b + c + e + f = 1$$. Then, extending a result of M. Greguš, Jr. [Boll. Unione Mat. Ital., V. Ser. A 17, 193–198 (1980; Zbl 0538.47035)], the authors show that $$T$$ has a unique fixed point.
Editor’s remark: a counterexample has been published in [Zbl 1215.47046]

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