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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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