Uniform convergence of spectral expansions for a problem with a boundary condition depending on a spectral parameter (Q2094821)

The authors study the fourth-order eigenvatue problem \[ y^{(4)}(x) - (q(x) y'(x))' = \lambda y(x) \] on a compact interval \([0,l]\) with a positive function \(q \in AC[0,l]\) and equipped with boundary conditions involving the (derivatives of) the function at \(0\) and \(l\). It is the essential feature of the paper that one of the four linear boundary conditions depends linearly on the eigenvalue parameter \(\lambda\). In an earlier paper [\textit{Z. S. Aliyev}, Cent. Eur. J. Math. 8, No. 2, 378--388 (2010; Zbl 1207.34112)] a self-adjoint and lower semibounded operator \(L\) with discrete spectrum in the Hilbert space \(L_2(0,l) \oplus {\mathbb C}\) was constructed such that the eigenvalue problem has the linearization \(L\hat{y} = \lambda \hat{y}\). The current focus is at first on asymptotic estimates for the eigenvalues and eigenfunctions. To this end modified problems are considered without \(\lambda\)-dependency. Then, the location of the original eigenvalues are estimated by those of the simpler problems and also oscillation properties of the corresponding eigenfunctions are obtained. In the next step, an asymptotic result on a simpler problem is proved and transferred to the \(\lambda\)-dependent problem. Finally, the basis property of the eigenfunctions in \(L_p(0,l)\) is obtained for \(1 < p < \infty\) where one of the eigenfunctions is omitted. Furthermore, a condion is formulated such that the corresponding Fourier series (i.e. eigenfunction expansion) for a continuous function converges uniformly.