Abstract:
The following boundary-value problem is considered:
\begin{gather*}
y^{(4)}(x)-(q(x){y'}(x))'=\lambda y(x),\qquad 0<x<l,
\\
y(0)=y'(0)=y''(l)=0, \qquad
(a\lambda+b)y(l)=(c\lambda+d)Ty(l),
\end{gather*}
where $\lambda$ is the spectral parameter;
$Ty\equiv y'''-qy'$; $q(x)$ is a strictly positive absolutely
continuous function on $[0,l]$; $a$, $b$, $c$, and $d$ are
real constants such that
$bc-ad>0$. The oscillation properties of eigenfunctions are
studied and asymptotic formulae for eigenvalues and
eigenfunctions are deduced. The basis properties in $L_p(0,l)$, $1<p<\infty$, of the
system of eigenfunctions are investigated.
Bibliography: 20 titles.
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