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

Harder, Alberta M.; Hicks, Troy L.

Math. Jap. 33, No. 5, 693-706 (1988).

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

Keywords:

iteration procedure; Stability results; generalized contractions