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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 1, Pages 21–28 (Mi vuu461)  

MATHEMATICS

Disconjugacy of solutions of a second order differential equation with Colombeau generalized functions in coefficients

I. G. Kim

Department of Mathematical Analysis, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: We consider a differential equation
\begin{equation} Lx\doteq x"+P(t)x'+Q(t)x=0,\qquad t\in[a, b]\subset\mathcal I\doteq(\alpha,\beta)\subset\mathbb R, \end{equation}
where $P,Q$ are $C$-generalized functions defined on $\mathcal I$ and are known as equivalence classes of Colombeau algebra. Let $\mathcal R_P$ and $\mathcal R_Q$ be representatives of $P$ and $Q$ respectively, $\mathcal A_N$ are classes of functions with compact support used to define Colombeau algebra. We obtain new sufficient conditions for disconjugacy of the equation (1). We prove that if the condition
\begin{equation*} (\exists N\in\mathbb N) (\forall\varphi\in\mathcal A_N) (\exists\mu_0<1) \int_a^b|\mathcal R_P(\varphi_\mu,t)| dt+\int_a^b|\mathcal R_Q(\varphi_\mu,t)| dt<\frac4{b-a+4}\quad(0<\mu<\mu_0), \end{equation*}
is satisfied, where $\varphi_\mu\doteq\frac1\mu\varphi(\frac t\mu)$, then the equation (1) is disconjugate on $[a,b]$. We prove the separation theorem and its corollary.

Keywords: $C$-generalized function, $C$-generalized number, weak equation, disconjugacy.

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UDC: 517.917
MSC: 46F30
Received: 18.01.2015

Citation: I. G. Kim, “Disconjugacy of solutions of a second order differential equation with Colombeau generalized functions in coefficients”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:1 (2015), 21–28

Citation in format AMSBIB
\Bibitem{Kim15}
\by I.~G.~Kim
\paper Disconjugacy of solutions of a~second order differential equation with Colombeau generalized functions in coefficients
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2015
\vol 25
\issue 1
\pages 21--28
\mathnet{http://mi.mathnet.ru/vuu461}
\elib{http://elibrary.ru/item.asp?id=23142047}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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