Basicity in L_p of root functions for differential equations with involution
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 s...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/278/abstr.html |
| Summary: | 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 $L_p(-1,1)$. |
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| ISSN: | 1072-6691 |