Fixed point theorems by altering distances between the points. (English) Zbl 0553.54023

Main result. Let (X,d) be a complete metric space, \(\phi\) : \({\mathbb{R}}_+\to {\mathbb{R}}_+\) an increasing continuous function such that \(\phi (t)=0\) if and only if \(t=0\). Let a, b, c be three decreasing functions from \({\mathbb{R}}_+\setminus \{0\}\) into [0,1) such that \(a(t)+2b(t)+c(t)<1\) for every \(t>0\). Suppose that mapping \(T: X\to X\) satisfies the following condition: \[ \phi (d(Tx,Ty))\leq a(d(x,y))\cdot \phi (d(x,y))+b(d(x,y))\cdot \]
\[ \cdot \{\phi (d(x,Tx))+\phi (d(y,Ty))\}+c(d(x,y))\cdot \min \{\phi (d(x,Ty)),\phi (d(y,Tx))\}, \] for x,y\(\in X\), \(x\neq y\). Then T has a unique fixed point.
Reviewer: V.V.Obuhovskii


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