## Fixed points and continuity of almost contractions.(English)Zbl 1152.54031

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 $$d,a\in (0,1)$$, $$b\geq 0$$. If $$T: X\to X$$ is a map such that for all $$x,y\in X$$,
$d(Tx,Ty)\leq ad(x,y)+ bd(y, Tx),$
then $$T$$ is continuous at its fixed points. The same result holds for multivalued mappings $$T: X\to CB(X)$$ such that
$H(Tx,Ty)\leq ad(x,y)+ bd(y,Tx).$

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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