Fixed point theorems and stability results for fixed point iteration procedures. (English) Zbl 0692.54027

Rhoades, B. E.

Indian J. Pure Appl. Math. 21, No. 1, 1-9 (1990).

Summary: Let \((X,d)\) be a complete metric space, \(T\) a selfmap of \(X\). Let \(x_ 0\) be a point of \(X\) and let \(x_{n+1}=f(T,x_ n)\) denote an iteration procedure which yields a sequence of points \((x_ n)\). Suppose that \(\{x_ n\}\) converges to a fixed point \(p\) of \(T\). Let \(\{y_ n\}\) denote an arbitrary sequence in \(X\), and set \(\epsilon_ n=d(y_{n+1},f(Ty_ n))\). If \(\lim_ n\epsilon_ n=0\) implies that \(\lim_ ny_ n=p\), then the iteration procedure defined by \(x_{n+1}=f(T,x_ n)\) is said to be \(T\)-stable. We show that several iteration procedures are \(T\)-stable for maps \(T\) satisfying a fairly general contractive condition. The results are extensions and generalizations of some of the work of A. M. Harder and T. H. Hicks [Math. Jap. 33, No.5, 693-706 (1988; Zbl 0655.47045)].

MSC:

54H25

Fixed-point and coincidence theorems (topological aspects)

Keywords:

complete metric space; iteration

Citations:

Zbl 0655.47045