Eigenvalues of a Linear Fourth-Order Differential Operator with Squared Spectral Parameter in a Boundary Condition

Abstract

We consider the spectrum of the following linear fourth-order eigenvalue problem: $$\begin{aligned} (py'')''(x)-(q(x)y'(x))'= & {} \lambda y(x),\quad x\in (0,l),\\ y'(0)\cos \alpha -y''(0)\sin \alpha= & {} 0,\\ y(0)\cos \beta +Ty(0)\sin \beta= & {} 0,\\ y'(l)\cos \gamma +y''(l)\sin \gamma= & {} 0,\\ (c_0+c_1\lambda +c_2\lambda ^2) y(l)= & {} (d_0+d_1\lambda +d_2\lambda ^2)Ty(l), \end{aligned}$$ where $$\lambda $$ is a spectral parameter, $$Ty=(py'')'-qy'$$ , p(x) has absolutely continuous derivative, q(x) is absolutely continuous on [0, l]; $$\alpha ,$$ $$\beta $$ , $$\gamma $$ , $$c_i$$ and $$d_i$$ ( $$i=0,1,2$$ ) are real constants; $$0\le \alpha ,\beta ,\gamma \le \pi /2$$ . By giving a new condition to guarantee the self-definiteness of the corresponding operator L, we obtain the simplicity and interlacing properties of the eigenvalues and the oscillation properties of the corresponding eigenfunctions. Meanwhile, some exceptional cases are also discussed when the self-definiteness condition does not hold. These results extend some existing results of the linear fourth-order eigenvalue problems with linear parameter in the boundary conditions and some existing results of the classical eigenvalue problems.