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Fixed point theorems for set-valued contractions in complete metric spaces. (English) Zbl 1133.54025
Let $$(M,d)$$ be a metric space and let $$H(A,B)$$ denote the Pompeiu-Hausdorff distance between the sets $$A,B\subset M$$. The main results of this paper are fixed point theorems for set-valued contractions in complete metric spaces which are obtained by considering, instead of the classical contraction conditions of the form
$H(Tx,Ty)\leq \varphi(d(x,y))d(x,y),\,x,y\in M,$ a more general condition: for each $$x\in M$$, there exists
$y\in I_{b}^{x}:=\left\{y\in Tx:\,bd(x,y)\leq d(x,Tx)\right\},$ for a certain $$b\in (0,1]$$, such that $d(y,Ty)\leq \varphi(d(x,y))d(x,y).$ Several related results in literature are thus extended or generalized.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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##### References:
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