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

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

Zbl 0655.47045