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An extension of multi-valued contraction mappings and fixed points. (English) Zbl 0948.47058
Let $$X$$ be a complete metric space and $$CB(X)$$ the set of all non-empty bounded and closed subsets of $$X$$ together with a Hausdorff metric defined on it. In this interesting paper the authors have proved the following generalization of the well-known Banach contraction principle: Let $$F$$: $$X\to CB(X)$$ be a multivalued and upper semi-continuous mapping, $$x_0\in X$$ a given point and $$\sigma\in (0,1]$$ a constant. Let $$h:[0,+\infty)\to [0,+\infty)$$ be a continuous non-decreasing function satisfying $\int^{+\infty}_0 {dr\over 1+ h(r)}=+\infty,$ such that for any $$x\in X$$, if $$x\not\in F(x)$$, then for any $$y\in F(x)$$ $\sup_{x\in F(x)} d(z, F(y))\leq \Biggl(1-{\sigma\over 1+ h(d(x_0, x))}\Biggr) d(x,y).$ Then $$F$$ has a fixed point. Several consequences of the above result and examples are also given.
Reviewer: Ismat Beg (Kuwait)

##### MSC:
 47H10 Fixed-point theorems 47H04 Set-valued operators
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##### References:
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