## Stability results for fixed point iteration procedures.(English)Zbl 0655.47045

Let $$(X,d)$$ be a metric spaces and $$T:X\to X$$ a mapping. Let $$x_{n+1}=f(T,x_ n)$$, $$x_ 0\in X$$ be an iteration procedure such that $$(x_ n)_{n\geq 0}$$ converges to a fixed point $$p$$ of $$T$$. Let $$(y_ n)_{n\geq 0}$$ be an arbitrary sequence in $$X$$ and set $$a_ n=d(y_{n+1},f(T,y_ n))$$ for $$n\in \mathbb N$$. By definition if $$\lim_{n\to \infty}a_ n=0$$ implies that $$\lim_{n\to \infty}y_ n=p$$, then the iteration procedure $$x_{n+1}=f(T,x_ n)$$ is said to be stable with respect to $$T$$. Stability results are established for three iteration procedures with respect to some kind of generalized contractions. Some examples are given.
Reviewer: I.A.Rus

### MSC:

 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 54H25 Fixed-point and coincidence theorems (topological aspects) 65D15 Algorithms for approximation of functions