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Approximate fixed point theorems. (English) Zbl 1164.54028

This article deals with $\epsilon$-fixed points of operators on metric spaces (a point ${x}_{0}$ from a metric space $X$ is called an $\epsilon$-fixed one if $\rho \left({x}_{0},f\left({x}_{0},f\left({x}_{0}\right)\right)\right)<\epsilon$, the set of such points is denoted by ${F}_{\epsilon }\left(f\right)$). The first statement of this article is an evident conclusion

$\underset{n\to \infty }{lim}\phantom{\rule{4pt}{0ex}}d\left({f}^{n}\left(x\right),{f}^{n+1}\left(x\right)\right)=0,\phantom{\rule{4pt}{0ex}}x\in X,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{implies}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{F}_{\epsilon }\left(f\right)\ne \varnothing ,\phantom{\rule{4pt}{0ex}}\epsilon >0·$

The second one estimates $\text{diam}\phantom{\rule{0.166667em}{0ex}}{F}_{\epsilon }\left(f\right)$ provided that $f$ satisfies the condition

$d\left(x,y\right)\le \phi \left(d\left(x,y\right)-d\left(f\left(x\right),f\left(y\right)\right),\phantom{\rule{1.em}{0ex}}x,y\in {F}_{\epsilon }\left(f\right);\phantom{\rule{2.em}{0ex}}\left(\left(*\right)\right)$

in this case, the inequality $\text{diam}\phantom{\rule{0.166667em}{0ex}}{F}_{\epsilon }\left(f\right)\le \phi \left(2\epsilon \right)$ holds. In the main part of the article, these simple statements apply when $f$ is a usual contraction, when $f$ satisfies the Kannan condition $d\left(f\left(x\right),f\left(y\right)\right)\le a\left[d\left(x,f\left(x\right)\right)+d\left(y,f\left(y\right)\right]$, $a<\frac{1}{2}$, when $f$ satisfies the Chatterjea condition $d\left(f\left(x\right),f\left(y\right)\right)\le a\left[d\left(x,f\left(y\right)\right)+d\left(y,f\left(x\right)\right]$, $a<\frac{1}{2}$, when $f$ satisfies the Zamfirescu condition $d\left(f\left(x\right),f\left(y\right)\le min\phantom{\rule{4pt}{0ex}}\left\{ad\left(x,y\right),b\left[d\left(x,f\left(x\right)\right)+d\left(y,f\left(y\right)\right],c\left[d\left(x,f\left(y\right)\right)+d\left(y,f\left(x\right)\right]\right\}$, $a<1$, $b,c<\frac{1}{2}$, and at last, when $f$ satisfies condition $d\left(f\left(x\right),f\left(y\right)\right)\le ad\left(x,y\right)+Ld\left(y,f\left(x\right)\right)$, $a<1$, $L\ge 0$ (Theorems 2.5 and 3.5); this last condition was offered by V. Berinde. In all these cases, the function $\phi$ is calculated.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces