On coefficients of eigenfunction expansions for boundary-value problems with a parameter in the boundary conditions
Consider \(l(y,\lambda ):=l_0(y)+\lambda l_1(y)+\dots+\lambda ^nl_n(y)\) on \([0,1]\), where the \(l_\nu\) are differential expressions of order \(n-\nu\) with constant leading coeffient (which may be zero for \(\nu\neq0,n\)), together with boundary conditions which depend polynomially on the eigenvalue parameter \(\lambda \). \textit{A. A. Shkalikov} [Tr. Semin. Im. I. G. Petrovskogo 9, 190--229 (1983; Zbl 0553.34014)] proved that this problem is linearized by an operator \(H\) in a finite-codimensional subspace \(\mathcal W_{2,U}^r\times \mathbb C^{N_r}\) of \(W_2^{n+r-1}\times \dots \times W_2^r\times \mathbb C^{N_r}\) with suitable integers \(r\) and \(N_r\) and that the eigenfunctions and associated functions of \(H\) form a Riesz basis. In the present paper, the dual problem is explicitly calculated, and the expansion coefficients of an element \(\tilde v\in \mathcal W_{2,U}^r\times \mathbb C^{N_r}\) with respect to the above mentioned Riesz basis are calculated explicitly using the biorthogonal system of eigenfunctions and associated functions from the dual problem.
spectral parameter in boundary conditions
scientific article; zbMATH DE number 2121380