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# On behavior at infinity of solutions of elliptic differential equations in a Banach space

### Abstract

For a differential equation of the form $y''(t) - By(t) = 0, \ t \in (0, \infty)$, where $B$ is a weakly positive linear operator in a Banach space $\mathfrak{B}$, the conditions on the operator $B$, under which this equation is uniformly or uniformly exponentially stable are given. As distinguished from earlier works dealing only with continuous at 0 solutions, in this paper no conditions on behavior of a solution near 0 are imposed.

Key words: Elliptic differential equation in a Banach space, uniformly and uniformly exponentially stable equation, stable solution, Dirichlet problem, $C_0$-semigroup of linear operators, bounded analytic $C_0$-semigroup, infinitely differentiable, entire, entire of exponential type vectors of a closed operator.

### Article Information

 Title On behavior at infinity of solutions of elliptic differential equations in a Banach space Source Methods Funct. Anal. Topology, Vol. 23 (2017), no. 2, 108-122 MathSciNet MR3668808 zbMATH 06810671 Milestones Received 28/12/2016 Copyright The Author(s) 2017 (CC BY-SA)

### Authors Information

M. L. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine; National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 37 Prospect Peremogy, Kyiv, 03056, Ukraine

V. M. Gorbachuk
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 37 Prospect Peremogy, Kyiv, 03056, Ukraine

### Citation Example

M. L. Gorbachuk and V. M. Gorbachuk, On behavior at infinity of solutions of elliptic differential equations in a Banach space, Methods Funct. Anal. Topology 23 (2017), no. 2, 108-122.

### BibTex

@article {MFAT966,
AUTHOR = {M. L. Gorbachuk and V. M. Gorbachuk},
TITLE = {On behavior at infinity of solutions of elliptic differential equations in a Banach space},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {23},
YEAR = {2017},
NUMBER = {2},
PAGES = {108-122},
ISSN = {1029-3531},
MRNUMBER = {MR3668808},
ZBLNUMBER = {06810671},
URL = {http://mfat.imath.kiev.ua/article/?id=966},
}

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