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Section
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Mathematics
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Title
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Disconjugacy of solutions of a second order differential equation with Colombeau generalized functions in coefficients
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Author(-s)
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Kim I.G.a
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Affiliations
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Udmurt State Universitya
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Abstract
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We consider a differential equation
$$
Lx\doteq x''+P(t)x'+Q(t)x=0,\quad t\in[a, b]\subset \mathcal{I}\doteq(\alpha,\beta)\subset\mathbb{R}, \qquad(1)
$$
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
$$(\exists\, N\in\mathbb{N}) (\forall\, \varphi\in \mathcal{A}_N) (\exists\, \mu_0<1)\
\int_a^b|\mathcal{R}_P(\varphi_\mu,t)|\,dt+\int_a^b|\mathcal{R}_Q(\varphi_\mu,t)|\,dt<\frac{4}{b-a+4}\quad (0<\mu<\mu_0)
$$
is satisfied, where $\varphi_{\mu}\doteq \frac{1}{\mu}\varphi \left(\frac{t}{\mu}\right)$, then the equation $(1)$ is disconjugate on $[a, b]$. We prove the separation theorem and its corollary.
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Keywords
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$C$-generalized function, $C$-generalized number, weak equation, disconjugacy
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UDC
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517.917
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MSC
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46F30
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DOI
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10.20537/vm150103
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Received
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18 January 2015
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Language
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Russian
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Citation
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Kim I.G. Disconjugacy of solutions of a second order differential equation with Colombeau generalized functions in coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 21-28.
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References
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