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Russia Izhevsk
Section Mathematics
Title Disconjugacy of solutions of a second order differential equation with Colombeau generalized functions in coefficients
Author(-s) Kim I.G.a
Affiliations Udmurt State Universitya
Abstract 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.
Keywords $C$-generalized function, $C$-generalized number, weak equation, disconjugacy
UDC 517.917
MSC 46F30
DOI 10.20537/vm150103
Received 18 January 2015
Language Russian
Citation 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.
  1. Derr V.Ya. Disconjugacy of solutions of linear differential equations, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 1, pp. 46-89 (in Russian).
  2. Colombeau J.F. Elementary introduction to new generalized functions, Amsterdam: North Holland Math. Studies, 1985, 300 p.
  3. Derr V.Ya., Dizendorf K.I. On differential equations in $C$-generalized functions, Izv. Vyssh. Uchebn. Zaved. Mat., 1996, no. 11 (414), pp. 39-49 (in Russian).
  4. Derr V.Ya., Kim I.G. The spaces of regulated functions and differential equations with generalized functions in coefficients, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 1, pp. 3-18 (in Russian).
  5. Derr V.Ya. Teoriya funktsii deistvitel'noi peremennoi. Lektsii i uprazhneniya (Theory of functions of a real variable. Lectures and exercises), Moscow: Vyssh. Shkola, 2008, 384 p.
  6. Tolstonogov A.A. Properties of the space of proper functions, Mathematical Notes, 1984, vol. 35, issue 6, pp. 422-427.
  7. Dieudonne J. Foundations of modern analysis, New York: Academic Press, 2006, 408 p. Translated under the title Osnovy sovremennogo analiza, Moscow: Mir, 1964, 430 p.
  8. Schwartz L. Analyse mathematique, Paris: Hermann, 1967, vol. 1, 554 p.
  9. Ligeza J. Remarks on generalized solutions of some ordinary nonlinear differential equations of second order in the Colombeau algebra, Annales Mathematicae Silesianae, 1996, vol. 10, pp. 87-101.
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