Electron. J. Diff. Equ., Vol. 2015 (2015), No. 278, pp. 1-9.

Basicity in Lp of root functions for differential equations with involution

Leonid V. Kritskov, Abdizhahan M. Sarsenbi

Abstract:
We consider the differential equation
$$ \alpha u''(-x)-u''(x)=\lambda u(x), \quad -1<x<1, $$
with the nonlocal boundary conditions $u(-1)=0$, $u'(-1)=u'(1)$ where $\alpha\in (-1,1)$. We prove that if $r=\sqrt{(1-\alpha)/(1+\alpha)}$ is irrational then the system of its eigenfunctions is complete and minimal in $L_p(-1,1)$ for any $p>1$, but does not constitute a basis. In the case of a rational value of r we specify the way of choosing the associated functions which provides the system of all root functions of the problem forms a basis in $L_p(-1,1)$.

Submitted October 17, 2015. Published November 4, 2015.
Math Subject Classifications: 34K08, 34L10, 46B15.
Key Words: ODE with involution; nonlocal boundary-value problem; basicity of root functions.

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Leonid V. Kritskov
Lomonosov Moscow State University
Faculty of Computational Mathematics and Cybernetics
119899 Moscow, Russia
email: kritskov@cs.msu.ru
Abdizhahan M. Sarsenbi
Auezov South-Kazakhstan State University
Department of Mathematical Methods and Modeling
160012 Shymkent Kazakhstan
email: abzhahan@mail.ru

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