On a~fourth-order problem with spectral and physical parameters in the boundary condition

J. Ben Amara ; A. A. Vladimirov

Izvestiya. Mathematics , Tome 68 (2004) no. 4, pp. 645-658

Résumé

We consider the following fourth-order boundary-value problem: \begin{gather*} [(py'')'-qy']'=\lambda ry, \\ y(0)=y'(0)=y''(1)=[(py'')'-qy'](1)+\lambda my(1)=0 \end{gather*} with spectral parameter $\lambda\in\mathbb C$ and physical parameter $m\in\mathbb R$. We assign to this problem a linear pencil of bounded operators $T_m=T_m(\lambda)$ depending on the physical parameter $m$ and acting from $\mathcal H_2=\{y\mid y\in W_2^2[0,1],\ y(0)=y'(0)=0\}$ to the dual space $\mathcal H_{-2}$. We study the spectral properties of $T_m$ and use the results of this study to describe properties of the eigenvalues of the problem for various values of $m$. In particular, we establish asymptotics of these eigenvalues as $m\nearrow0$.