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Sbornik: Mathematics, 2009, Volume 200, Issue 10, Pages 1565–1574
DOI: https://doi.org/10.1070/SM2009v200n10ABEH004050
(Mi sm1499)
 

Differential equations whose solution of the Cauchy problem displays nonclassical behaviour with respect to the parameter $\lambda$

V. Ya. Yakubov

Moscow State Institute of Electronics and Mathematics (Technical University)
References:
Abstract: The behaviour of solutions of the equation $y''+\lambda\rho(x,\lambda)y=0$ with respect to the spectral parameter $\lambda$ is investigated under the assumption that the function $\rho(x,\lambda)$ does not satisfy the classical conditions. Two types of equations are considered: the Sturm-Liouville equation $y''+\lambda\rho(x)y=0$, whose solutions grow like $c(\rho)\lambda^m$ in the norm of $C[0,l]$ (where $m>0$ is arbitrary), and equations of the form $y''+\lambda\rho(x,\lambda)y=0$, $\lim_{\lambda\to+\infty}\rho(x,\lambda)=1$, whose solutions can grow like $c\lambda^m$ in the norm of $C[0,l]$ (where $m>0$ is arbitrary) and even like $\exp\{m\lambda^{1-\gamma}\}$ where $0<\gamma<1$.
Bibliography: 3 titles.
Keywords: Sturm-Liouville problem, eigenfunctions, nonclassical estimates for eigenfunctions, Cauchy problem.
Received: 26.10.2005 and 25.05.2009
Bibliographic databases:
UDC: 517.984
MSC: 34B05, 34B07, 34C11
Language: English
Original paper language: Russian
Citation: V. Ya. Yakubov, “Differential equations whose solution of the Cauchy problem displays nonclassical behaviour with respect to the parameter $\lambda$”, Sb. Math., 200:10 (2009), 1565–1574
Citation in format AMSBIB
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\by V.~Ya.~Yakubov
\paper Differential equations whose solution of the Cauchy problem displays nonclassical behaviour with respect to the parameter~$\lambda$
\jour Sb. Math.
\yr 2009
\vol 200
\issue 10
\pages 1565--1574
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  • https://www.mathnet.ru/eng/sm/v200/i10/p151
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