Structure of solutions of differential equations in a Banach space on an infinite interval

TitleStructure of solutions of differential equations in a Banach space on an infinite interval
Publication TypeJournal Article
Year of Publication2016
AuthorsGorbachuk, VM
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published2/2016
For an equation of the form $({d}/{dt} - A)^{n}({d}/{dt} + A)^{m}y(t)  = 0$, $(n, m \in \mathbb{N}_{0} = \{0\}\textstyle\bigcup \mathbb{N}, n + m \geq 1)$ on the semiaxis or the whole real axis, where $A$ is the infinitesimal generator of a bounded analytic $C_{0}$-semigroup of linear operators on a Banach space, all its solutions are described. It is shown that any solution of the equation under consideration on $(0,\infty)$ is an analytic vector-valued function on this semiaxis, and every its solution on $(-\infty,\infty)$ admits an extension to an entire vectorvalued function. In both cases, an analogue of the Phragmén-Lindelöf principle for the solutions is established.
Keywordsanalytic and entire vectors of a closed operator, bounded analytic semigroup, C0-semigroup of linear operators, differential equation in a Banach space, the Phragmén-Lindelöf principle
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