On the spectrum of a multipoint boundary value problem for a fourth-order equation. (English. Russian original) Zbl 1366.34031

Differ. Equ. 52, No. 3, 316-326 (2016); translation from Differ. Uravn. 52, No. 3, 324-333 (2016).
The paper is concerned with the spectrum of a multipoint boundary value problem of a fourth order equation that models small deformations of a chain of rigidity connected rods with elastic support. More precisely, let \(\Gamma=(a,b)/\{a_i\}_{i=1}^n\) and let \(L_{\delta_1,\dots,\delta_n}\) be the differential operator defined by \[ L_{\delta_1,\dots,\delta_n}u := (pu'')''-(qu')'=\lambda u \qquad (x \in \Gamma) \] for \(p \in C^2(\Gamma)\), \(q \in C^1(\Gamma)\) and \(p>0\), \(q \geq 0\) with the boundary conditions \[ \begin{gathered} u(a) + \alpha(a) Du(a) = 0, \quad \beta(a)u''(a) - \gamma(a) u'(a) =0, \\ u(b) - \alpha(b) Du(b) = 0, \quad \beta(b)u''(b) + \gamma(b) u'(b) =0, \end{gathered} \] where \(Du=(pu'')' - q u'\) and \(\alpha, \beta, \gamma\) are positive functions with \(\beta+\gamma \neq 0\). The following matching conditions are also imposed \[ \begin{gathered} u(a_i-0) - u(a_i+0)=0, \\ u'(a_i-0)-u'(a_i+0)=0, \\ (pu'')(a_i-0) -(pu'')(a_i+0) =0, \\ Du(a_i-0)-Du(a_i+0)-\delta_iu(a_i)=0. \end{gathered} \] The non-negative numbers \(\delta_i\) specify the elasticity of the support.
In the main result of the paper, the author obtains a separation result for the eigenvalues \(\lambda_j(\delta_1,\dots,\delta_n)\) of \(L_{\delta_1,\dots,\delta_n}\). First, the author considers the simpler situation of a single elastic support, say \(\delta_1=\delta>0\) and \(\delta_i=0\) for \(i=2,\dots,n\). In that case, the spectrum \(\Lambda_\delta\) of \(L_\delta\) consists of infinitely many real eigenvalues with no finite accumulation point. For \(0 \leq \delta<\delta' \leq \infty\), the author first proves that the spectra \(\Lambda_{\delta}\) and \(\Lambda_{\delta'} \) interlace, i.e. \[ \lambda_j(\delta) \leq \lambda_j(\delta') \leq \lambda_{j+1}(\delta) \qquad (j=0,1,2,\dots). \] This result easily extends to the more general case of several elastic supports, namely \[ \lambda_j(\delta_{i_1},\dots,\delta_{i_m}) \leq \lambda_j(\delta_{i_1},\dots,\delta_{i_{m-1}}, \delta_{i_m'}) \leq \lambda_{j+1} (\delta_{i_1},\dots,\delta_{i_m}) \qquad (j=0,1,2,\dots) \] for \(0 \leq \delta_{i_m} \leq \delta_{i_m}' \leq \infty\). The proofs of these results are based on a series of several interesting lemmas about the eigenvalues and eigenvectors of the operator \(L_{\delta_1,\dots,\delta_n}\).


34B09 Boundary eigenvalue problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34L05 General spectral theory of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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