An almost-periodicity criterion for solutions of the oscillatory differential equation y ' ' = q ( t ) y and its applications

Staněk, Svatoslav

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 2, page 229-241
  • ISSN: 0044-8753

Abstract

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The linear differential equation ( q ) : y ' ' = q ( t ) y with the uniformly almost-periodic function q is considered. Necessary and sufficient conditions which guarantee that all bounded (on ℝ ) solutions of ( q ) are uniformly almost-periodic functions are presented. The conditions are stated by a phase of ( q ) . Next, a class of equations of the type ( q ) whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations y ' ' = q ( t ) y + f ( t ) are considered. The results are applied to the Appell and Kummer differential equations.

How to cite

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Staněk, Svatoslav. "An almost-periodicity criterion for solutions of the oscillatory differential equation $y^{\prime \prime }=q(t)y$ and its applications." Archivum Mathematicum 041.2 (2005): 229-241. <http://eudml.org/doc/249507>.

@article{Staněk2005,
abstract = {The linear differential equation $(q):y^\{\prime \prime \}=q(t)y$ with the uniformly almost-periodic function $q$ is considered. Necessary and sufficient conditions which guarantee that all bounded (on $\mathbb \{R\}$) solutions of $(q)$ are uniformly almost-periodic functions are presented. The conditions are stated by a phase of $(q)$. Next, a class of equations of the type $(q)$ whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations $y^\{\prime \prime \}=q(t)y+f(t)$ are considered. The results are applied to the Appell and Kummer differential equations.},
author = {Staněk, Svatoslav},
journal = {Archivum Mathematicum},
keywords = {linear second-order differential equation; Appell equation; Kummer equation; uniformly almost-periodic solution; bounded solution; phase; linear second-order differential equation; Appell equation; Kummer equation},
language = {eng},
number = {2},
pages = {229-241},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {An almost-periodicity criterion for solutions of the oscillatory differential equation $y^\{\prime \prime \}=q(t)y$ and its applications},
url = {http://eudml.org/doc/249507},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Staněk, Svatoslav
TI - An almost-periodicity criterion for solutions of the oscillatory differential equation $y^{\prime \prime }=q(t)y$ and its applications
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 2
SP - 229
EP - 241
AB - The linear differential equation $(q):y^{\prime \prime }=q(t)y$ with the uniformly almost-periodic function $q$ is considered. Necessary and sufficient conditions which guarantee that all bounded (on $\mathbb {R}$) solutions of $(q)$ are uniformly almost-periodic functions are presented. The conditions are stated by a phase of $(q)$. Next, a class of equations of the type $(q)$ whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations $y^{\prime \prime }=q(t)y+f(t)$ are considered. The results are applied to the Appell and Kummer differential equations.
LA - eng
KW - linear second-order differential equation; Appell equation; Kummer equation; uniformly almost-periodic solution; bounded solution; phase; linear second-order differential equation; Appell equation; Kummer equation
UR - http://eudml.org/doc/249507
ER -

References

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  16. Mingarelli A. B., Pu P. Q., Zheng L., A Counter-example in the theory of almost periodic differential equations, Rocky Mountain J. Math. 25 (1995), 437–440. (1995) Zbl0833.34041MR1340018
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  18. Staněk S., On some properties of solutions of the disconjugate equation y ' ' = q ( t ) y with an almost periodic coefficient q , Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 25 (1986), 31–56. (1986) Zbl0644.34039MR0918368

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