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Zbl 1246.54043
Karapinar, Erdal
Edelstein type fixed point theorems. (English)
[J] Ann. Funct. Anal. AFA 2, No. 1, 51-58, electronic only (2011). ISSN 2008-8752/e

The author proves, among others results, the following theorem.\par Theorem 2.1. Let $T$ be a self-mapping on a compact metric space $(X,d)$. Assume that $${1\over 2} d(x,Tx)< d(x,y)\Rightarrow d(Tx,Ty)< M(x,y)\quad\text{for all }x,y\in X,$$ where $M(x,y)= \max\{d(x,y), d(y,Ty),{1\over 2} d(Tx,y),{1\over 2} d(x,Ty)\}$. Then $T$ has a unique fixed point $z\in X$, that is, $Tz= z$.\par This result is some generalization of the fixed point theorem due to {\it T. Suzuki} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, A, 5313--5317 (2009; Zbl 1179.54071)].
[Jarosław Górnicki (Rzeszów)]

MSC 2000:: *54H25 Fixed-point theorems in topological spaces
54E50 Complete metric spaces

Keywords: fixed point theorem; contraction; compact metric space

Citations: Zbl 1179.54071

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