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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.

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