# Asymptotic relationship between solutions of two linear differential systems

Mathematica Bohemica (1998)

• Volume: 123, Issue: 2, page 163-175
• ISSN: 0862-7959

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## Abstract

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In this paper new generalized notions are defined: $\Psi$-boundedness and $\Psi$-asymptotic equivalence, where $\Psi$ is a complex continuous nonsingular $n×n$ matrix. The $\Psi$-asymptotic equivalence of linear differential systems ${y}^{\text{'}}=A\left(t\right)y$ and ${x}^{\text{'}}=A\left(t\right)x+B\left(t\right)x$ is proved when the fundamental matrix of ${y}^{\text{'}}=A\left(t\right)y$ is $\Psi$-bounded.

## How to cite

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Miklo, Jozef. "Asymptotic relationship between solutions of two linear differential systems." Mathematica Bohemica 123.2 (1998): 163-175. <http://eudml.org/doc/248305>.

@article{Miklo1998,
abstract = {In this paper new generalized notions are defined: $\{\mathbf \{\Psi \}\}$-boundedness and $\{\mathbf \{\Psi \}\}$-asymptotic equivalence, where $\{\mathbf \{\Psi \}\}$ is a complex continuous nonsingular $n\times n$ matrix. The $\{\mathbf \{\Psi \}\}$-asymptotic equivalence of linear differential systems $y^\{\prime \}= A(t) y$ and $x^\{\prime \}= A(t) x+ B(t) x$ is proved when the fundamental matrix of $y^\{\prime \}= A(t) y$ is $\{\mathbf \{\Psi \}\}$-bounded.},
author = {Miklo, Jozef},
journal = {Mathematica Bohemica},
keywords = {$\{\mathbf \{\Psi \}\}$-boundedness; $\{\mathbf \{\Psi \}\}$-asymptotic equivalence; -boundedness; -asymptotic equivalence},
language = {eng},
number = {2},
pages = {163-175},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic relationship between solutions of two linear differential systems},
url = {http://eudml.org/doc/248305},
volume = {123},
year = {1998},
}

TY - JOUR
AU - Miklo, Jozef
TI - Asymptotic relationship between solutions of two linear differential systems
JO - Mathematica Bohemica
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 123
IS - 2
SP - 163
EP - 175
AB - In this paper new generalized notions are defined: ${\mathbf {\Psi }}$-boundedness and ${\mathbf {\Psi }}$-asymptotic equivalence, where ${\mathbf {\Psi }}$ is a complex continuous nonsingular $n\times n$ matrix. The ${\mathbf {\Psi }}$-asymptotic equivalence of linear differential systems $y^{\prime }= A(t) y$ and $x^{\prime }= A(t) x+ B(t) x$ is proved when the fundamental matrix of $y^{\prime }= A(t) y$ is ${\mathbf {\Psi }}$-bounded.
LA - eng
KW - ${\mathbf {\Psi }}$-boundedness; ${\mathbf {\Psi }}$-asymptotic equivalence; -boundedness; -asymptotic equivalence
UR - http://eudml.org/doc/248305
ER -

## References

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1. R. Bellman, Stability Theory of Differential Equations, New York, 1953. (1953) Zbl0053.24705MR0061235
2. E. A. Coddington N. Levinson, Theory of Ordinary Differential Equations, New York, 1955. (1955) MR0069338
3. M. Greguš M. Švec V. Šeda, Ordinary Differential Equations, Bratislava, 1985. (In Slovak.) (1985)
4. A. Haščák, Asymptotic and integral equivalence of multivalued differential systems, Hiroshima Math. J. 20 (1990), no. 2, 425-442. (1990) MR1063376
5. A. Haščák M. Švec, Integral equivalence of two systems of differential equations, Czechoslovak Math. J. 32 (1982), 423-436. (1982) MR0669785
6. M. Švec, Asymptotic relationship between solutions of two systems of differential equations, Czechoslovak Math. J. 2J, (1974), 44-58. (1974) MR0348202
7. M. Švec, Integral and asymptotic equivalence of two systems of differential equations, Equadiff 5. Proceedings of the Fifth Czechoslovak Conference on Differential Equations and Their Applications held in Bratislava 1981. Teubner, Leipzig, 1982, pp. 329-338. (1981) MR0716002

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