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Functionals on function and sequence spaces connected with the exponential stability of evolutionary processes. (English) Zbl 1164.34432
The main results are interesting generalizations of a well-known result obtained by R. Datko [J. Math. Anal. Appl. 32, 610–616 (1970; Zbl 0211.16802)]. This is done by generalizing the improper integral \(\int_0^{\infty}\) and the series \(\sum_{j=0}^{\infty}\) to obtain the specific class \(\mathcal F\) (\(\mathcal G\)) of non-negative functionals defined on non-negative sequences (functions) on \(\mathbb N\) (\(\mathbb R_{+}\)). They are introduced by M. Megan and A. Pogan [Nonlinear Anal., Theory Methods Appl. 52, No. 2, A, 545–556 (2003; Zbl 1035.47047)] for the first time. In addition, some discrete versions of Datko theorem are presented.

34D05 Asymptotic properties of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
34D20 Stability of solutions to ordinary differential equations
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