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Zbl 1133.54025
Klim, D.; Wardowski, D.
Fixed point theorems for set-valued contractions in complete metric spaces.
(English)
[J] J. Math. Anal. Appl. 334, No. 1, 132-139 (2007). ISSN 0022-247X

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.
[Vasile Berinde (Baia Mare)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: metric space; Pompeiu-Hausdorff distance; set-valued mapping; contraction; fixed point theorem

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