## Some stability results in complete metric space.(English)Zbl 1199.54232

Let $$(E,d)$$ be a metric and $$T:E\rightarrow E$$ a self map with the set of fixed points nonempty. Consider a fixed point iteration procedure $$\{x_{n}\}_{n=1}^{\infty }$$ defined by a general relation of the form $x_{n+1}=f(T,x_{n}),\;\;n=0,1,2,\dots, \tag{1}$ where $$x_{0}\in E$$ is the initial guess and $$f$$ is given.
Let us assume that the sequence $$\{x_{n}\}_{n=1}^{\infty }$$ produced by (1) converges to a fixed point $$p$$ of $$T.$$ Let also $$\{y_{n}\}_{n=0}^{\infty }$$ be an arbitrary sequence in $$E$$ and set $\varepsilon _{n}=d(y_{n+1},\,f(T,y_{n})),\;\;\;\text{for}\;\;\;n=0,1,2,\dots$ The fixed point iteration procedure (1) is said to be $$T$$-stable or stable with respect to $$T$$ if and only if $\lim_{n\rightarrow \infty }\varepsilon _{n}=0\Leftrightarrow \lim_{n\rightarrow \infty }y_{n}=p.$ In the paper the author obtains stability results for mappings satisfying various general contractive conditions, in the presence of one metric and respectively two metrics on the set $$E$$. The stability results in the case of a set endowed with two metrics appear to be given in this paper for the first time.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects)
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### References:

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