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Zbl 0553.54023
Khan, M.S.; Swaleh, M.; Sessa, S.
Fixed point theorems by altering distances between the points.
(English)
[J] Bull. Aust. Math. Soc. 30, 1-9 (1984). ISSN 0004-9727

Main result. Let (X,d) be a complete metric space, $\phi$ : ${\bbfR}\sb+\to {\bbfR}\sb+$ an increasing continuous function such that $\phi (t)=0$ if and only if $t=0$. Let a, b, c be three decreasing functions from ${\bbfR}\sb+\setminus \{0\}$ 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: $$\phi (d(Tx,Ty))\le a(d(x,y))\cdot \phi (d(x,y))+b(d(x,y))\cdot$$ $$\cdot \{\phi (d(x,Tx))+\phi (d(y,Ty))\}+c(d(x,y))\cdot \min \{\phi (d(x,Ty)),\phi (d(y,Tx))\},$$ for x,y$\in X$, $x\ne y$. Then T has a unique fixed point.
[V.V.Obuhovskii]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces

Keywords: complete metric space

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