## 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

### Citations:

Zbl 1235.54052; Zbl 1329.54052
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