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Approximate fixed point theorems. (English) Zbl 1164.54028

This article deals with ε-fixed points of operators on metric spaces (a point x 0 from a metric space X is called an ε-fixed one if ρ(x 0 ,f(x 0 ,f(x 0 )))<ε, the set of such points is denoted by F ε (f)). The first statement of this article is an evident conclusion

lim n d(f n (x),f n+1 (x))=0,xX,impliesF ε (f),ε>0·

The second one estimates diamF ε (f) provided that f satisfies the condition

d(x,y)φ(d(x,y)-d(f(x),f(y)),x,yF ε (f);((*))

in this case, the inequality diamF ε (f)φ(2ε) 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))a[d(x,f(x))+d(y,f(y)], a<1 2, when f satisfies the Chatterjea condition d(f(x),f(y))a[d(x,f(y))+d(y,f(x)], a<1 2, when f satisfies the Zamfirescu condition d(f(x),f(y)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<1 2, and at last, when f satisfies condition d(f(x),f(y))ad(x,y)+Ld(y,f(x)), a<1, L0 (Theorems 2.5 and 3.5); this last condition was offered by V. Berinde. In all these cases, the function φ is calculated.

54H25Fixed-point and coincidence theorems in topological spaces