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Zbl 1152.54031
Berinde, Vasile; Păcurar, Mădălina
Fixed points and continuity of almost contractions.
(English)
[J] Fixed Point Theory 9, No. 1, 23-34 (2008). ISSN 1583-5022

Let $(X,d)$ be a metric space, $CB(X)$ the family of nonempty, closed, bounded subsets of $X$, $H$ the Hausdorff metric on $CB(X)$ induced by $d,a\in (0,1)$, $b\ge 0$. If $T: X\to X$ is a map such that for all $x,y\in X$, $$d(Tx,Ty)\le ad(x,y)+ bd(y, Tx),$$ then $T$ is continuous at its fixed points. The same result holds for multivalued mappings $T: X\to CB(X)$ such that $$H(Tx,Ty)\le ad(x,y)+ bd(y,Tx).$$
[Delfina Roux (Milano)]
MSC 2000:
*54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: almost contractions; multivalued almost contractions; continuity at fixed points

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