Asymptotic behavior of individual orbits of discrete systems. (English) Zbl 1170.47004
We consider the asymptotic behavior of bounded solutions of the difference equations of the form \(x(n+1)=Bx(n) + y(n)\) in a Banach space \(\mathbb X\), where \(n=1,2,\ldots\), \(B\) is a linear continuous operator in \(\mathbb X\), and \((y(n))\) is a sequence in \(\mathbb X\) converging to \(0\) as \(n\to\infty\). An obtained result with an elementary proof says that if \(\sigma (B)\cap\{|z|=1\}\subset\{1\}\), then every bounded solution \(x(n)\) has the property that \(\lim_{n\to\infty}(x(n+1)-x(n)) =0\). This result extends a theorem due to Y.Katznelson and L.Tzafriri [J. Funct.Anal.68, No.3, 313–328 (1986; Zbl 0611.47005)]. Moreover, the techniques of the proof are furthered to study the individual stability of solutions of the discrete system. A discussion on further extensions is also given.
MSC:
| 47A35 | Ergodic theory of linear operators |
| 39A30 | Stability theory for difference equations |
| 47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
Katznelson-Tzafriri theorem; discrete system; individual orbit; stability; asymptotically almost periodicCitations:
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