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Approximate fixed point theorems. (English) Zbl 1164.54028
This article deals with $$\varepsilon$$-fixed points of operators on metric spaces (a point $$x_0$$ from a metric space $$X$$ is called an $$\varepsilon$$-fixed one if $$\rho(x_0,f(x_0,f(x_0))) < \varepsilon$$, the set of such points is denoted by $$F_\varepsilon(f)$$). The first statement of this article is an evident conclusion $\lim_{n \to \infty} \;d(f^n(x),f^{n+1}(x)) = 0, \;x \in X, \;\;\text{implies} \;\;F_\varepsilon(f) \neq \emptyset, \;\varepsilon > 0.$ The second one estimates $$\text{diam} \, F_\varepsilon(f)$$ provided that $$f$$ satisfies the condition $d(x,y) \leq \varphi(d(x,y) - d(f(x),f(y)), \quad x, y \in F_\varepsilon(f);\tag{(*)}$ in this case, the inequality $$\text{diam} \, F_\varepsilon(f) \leq \varphi(2\varepsilon)$$ holds. In the main part of the article, these simple statements apply when $$f$$ is a usual contraction, when $$f$$ satisfies the Kannan condition $$d(f(x),f(y)) \leq a[d(x,f(x)) + d(y,f(y)]$$, $$a < \frac12$$, when $$f$$ satisfies the Chatterjea condition $$d(f(x),f(y)) \leq a[d(x,f(y)) + d(y,f(x)]$$, $$a < \frac12$$, when $$f$$ satisfies the Zamfirescu condition $$d(f(x),f(y) \leq \min \;\{ad(x,y),b[d(x,f(x)) + d(y,f(y)],c[d(x,f(y)) + d(y,f(x)]\}$$, $$a < 1$$, $$b, c < \frac12$$, and at last, when $$f$$ satisfies condition $$d(f(x),f(y)) \leq ad(x,y) + Ld(y,f(x))$$, $$a < 1$$, $$L \geq 0$$ (Theorems 2.5 and 3.5); this last condition was offered by V. Berinde. In all these cases, the function $$\varphi$$ is calculated.

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