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Convergence to attractors of nonexpansive set-valued mappings. (English) Zbl 1476.54109

In this paper, the authors first give an alternative proof of the main result from the paper [E. Pustylnik et al., Fixed Point Theory 13, No. 1, 165–172 (2012; Zbl 1329.54052)].
The main result of the paper is the following.
Theorem. Let \((X,d)\) be a metric space, \(F\) a nonempty subset of it and \(T:X\multimap X\) be a set-valued nonexpansive mapping. Assume that
(a)
\(z\in T(z)\) for every \(z\in F\),
(b)
for each \(x\in X\) there exists a sequence \((x_i)\) such that \(x_0=x, x_{i+1}\in T(x_i)\), for \(i\in \mathbb{N}\) and \(\displaystyle\liminf_{i\to \infty}\rho(x_i,F)=0\).
Then, for each \(\delta>0\) and each \(x\in X\), there exist \(z\in F\) and a sequence \((x_i)\) in \(X\) such that \(x_0=x, x_{i+1}\in T(x_i)\), for \(i\in \mathbb{N}\) and \(\rho(x_i,z)<\delta\) for all sufficiently large integers \(i\ge 0\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
54E40 Special maps on metric spaces
54E50 Complete metric spaces
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Full Text: Euclid