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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)

47H10 Fixed-point theorems
47H04 Set-valued operators
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