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

 Title Structure of solutions of differential equations in a Banach space on an infinite interval Publication Type Journal Article Year of Publication 2016 Authors Gorbachuk, VM Abbreviated Key Title Dopov. Nac. akad. nauk Ukr. DOI 10.15407/dopovidi2016.02.007 Issue 2 Section Mathematics Pagination 7-12 Date Published 2/2016 Language Ukrainian Abstract 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. Keywords analytic 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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