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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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