# On quadratically integrable solutions of the second order linear equation

Archivum Mathematicum (2001)

• Volume: 037, Issue: 1, page 57-62
• ISSN: 0044-8753

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## Abstract

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Integral criteria are established for $dim{V}_{i}\left(p\right)=0$ and $dim{V}_{i}\left(p\right)=1,i\in \left\{0,1\right\}$, where ${V}_{i}\left(p\right)$ is the space of solutions $u$ of the equation ${u}^{\text{'}\text{'}}+p\left(t\right)u=0$ satisfying the condition ${\int }^{+\infty }\frac{{u}^{2}\left(s\right)}{{s}^{i}}ds<+\infty \phantom{\rule{0.166667em}{0ex}}.$

## How to cite

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Chantladze, T., Kandelaki, Nodar, and Lomtatidze, Alexander. "On quadratically integrable solutions of the second order linear equation." Archivum Mathematicum 037.1 (2001): 57-62. <http://eudml.org/doc/248743>.

abstract = {Integral criteria are established for $\dim V_i(p)=0$ and $\dim V_i(p)=1, i\in \lbrace 0,1\rbrace$, where $V_i(p)$ is the space of solutions $u$ of the equation $u^\{\prime \prime \}+p(t)u=0$ satisfying the condition $\int ^\{+\infty \}\frac\{u^2(s)\}\{s^i\}ds<+\infty \,.$},
author = {Chantladze, T., Kandelaki, Nodar, Lomtatidze, Alexander},
journal = {Archivum Mathematicum},
keywords = {second order linear equation; quadratically integrable solutions; vanishing at infinity solutions; second order linear equation; quadratically integrable solutions; vanishing at infinity solutions},
language = {eng},
number = {1},
pages = {57-62},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On quadratically integrable solutions of the second order linear equation},
url = {http://eudml.org/doc/248743},
volume = {037},
year = {2001},
}

TY - JOUR
AU - Kandelaki, Nodar
AU - Lomtatidze, Alexander
TI - On quadratically integrable solutions of the second order linear equation
JO - Archivum Mathematicum
PY - 2001
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 037
IS - 1
SP - 57
EP - 62
AB - Integral criteria are established for $\dim V_i(p)=0$ and $\dim V_i(p)=1, i\in \lbrace 0,1\rbrace$, where $V_i(p)$ is the space of solutions $u$ of the equation $u^{\prime \prime }+p(t)u=0$ satisfying the condition $\int ^{+\infty }\frac{u^2(s)}{s^i}ds<+\infty \,.$
LA - eng
KW - second order linear equation; quadratically integrable solutions; vanishing at infinity solutions; second order linear equation; quadratically integrable solutions; vanishing at infinity solutions
UR - http://eudml.org/doc/248743
ER -

## References

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1. Wintner A., On the non-existence of conjugate points, Amer. J. Math. 73 (1951), 368–380. (1951) MR0042005
2. Kneser A., Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen reelen Werten des Arguments, J. Reine Angew. Math. 116 (1896), 178–212.
3. Kiguradze I.T., Chanturia T.A., Asymptotic properties of solutions of nanautonomous ordinary differential equations, Kluwer Academic Publishers, Dordrecht–Boston–London, 1992. (1992)
4. Kiguradze I.T., Shekhter B.L., Singular boundary value problems for second order differential equations, in “Current Problems in Mathematics: Newest Results,” vol. 30, pp. 105–201, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyzn. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987. (1987) MR0925830
5. Chantladze T., Kandelaki N., Lomtatidze A., Oscillation and nonoscillation criteria for second order linear equations, Georgian Math. J. 6 (1999), No 5, 401–414. (1999) MR1692963

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