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Asymptotic recurrence in correspondences with memory. (Russian)
Authors: Gerko A. I.
Abstract
With equations of form $x'=f (t, x)$ it is naturally associated the correspondence $X\subset X_1\times X_2$, where $X_1$ is the functional space of right parts of the equation, $X_2$ is the functional space of the solutions of the equation with right part from $X_1$ and $(f,\varphi) \in X$ $\Longleftrightarrow\, \varphi $ is solution of a equation with the right part $f$. In the article we introduce the concept of correspondence with memory, as abstraction from an equation with uniqueness properties and continuous dependence of the solutions on initial dates and right parts of the equation. To correspondence with memory some results about asymptotically almost periodic solutions of ordinary differential equation are extendet. Except asymptotically almost periodicity it is considered asymptotically recurence and asymptotically distality. In researches a technique of extensions of topological transformation semigroups is used. Participating semigroup dynamic systems of shifts in the functional spaces are not necessary locally compact phase semigroup. It permits to apply the developed theory to equations, where of space-time is infinite-dimensional (equation in complete derivative, equation in $\beta$-derivatives).Universitatea de Stat din Moldova
str. Mateevici 60, Chişinău, MD-2009 Moldova
str. Mateevici 60, Chişinău, MD-2009 Moldova
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