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On the structure of solutions of operator-differential equations on the whole real axis

Abstract

We consider differential equations of the form $\left(\frac{d^{2}}{dt^{2}} - B\right)^{m}y(t) = f(t)$, $m \in \mathbb{N}, \ t \in (-\infty, \infty)$, where $B$ is a positive operator in a Banach space $\mathfrak{B}, \ f(t)$ is a bounded continuous vector-valued function on $(-\infty, \infty)$ with values in $\mathfrak{B}$, and describe all their solutions. In the case, where $f(t) \equiv 0$, we prove that every solution of such an equation can be extended to an entire $\mathfrak{B}$-valued function for which the Phragmen-Lindel\"{o}f principle is fulfilled. It is also shown that there always exists a unique bounded on $\mathbb{R}^{1}$ solution, and if $f(t)$ is periodic or almost periodic, then this solution is the same as $f(t)$.

Key words: Positive operator, differential equation in a Banach space, classic and generalized solutions, C0-semigroup of linear operators, bounded analytic C0-semigroup, entire vector of a closed operator, entire vector-valued function, Phragmen-Lindel¨of princip

Article Information

 Title On the structure of solutions of operator-differential equations on the whole real axis Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 2, 170–178 MathSciNet 3407908 zbMATH 06533474 Milestones Received 29/01/2015 Copyright The Author(s) 2015 (CC BY-SA)

Authors Information

V. M. Gorbachuk
National Technical University "KPI", 37 Peremogy Prosp., Kyiv, 06256, Ukraine

Citation Example

V. M. Gorbachuk, On the structure of solutions of operator-differential equations on the whole real axis, Methods Funct. Anal. Topology 21 (2015), no. 2, 170–178.

BibTex

@article {MFAT775,
AUTHOR = {Gorbachuk, V. M.},
TITLE = {On the structure of solutions of operator-differential equations on the whole real axis},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {2},
PAGES = {170–178},
ISSN = {1029-3531},
MRNUMBER = {3407908},
ZBLNUMBER = {06533474},
URL = {http://mfat.imath.kiev.ua/article/?id=775},
}

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