On the approximation of the first eigenvalue of some boundary value problems

A two-point $(n-1,1)$-type boundary value problem is investigated for the representation of eigenfunctions in the form of scalar series under the assumption that there is a functional $\tilde{l}$, concentrated at one point, such that the first $n-1$ original boundary conditions and $\tilde{l}x=1$ turn into Cauchy conditions at this point. The eigenfunction of the boundary value problem under consideration, corresponding to the eigenvalue $\lambda_*$, is presented by an expansion in powers of $\lambda_*$. The equation $\Phi(\lambda) = 0$, where $\Phi(\lambda)$ is the sum of the power series in $\lambda$, for finding the eigenvalues of the original problem is considered. Examples of calculating the first eigenvalue of some boundary value problems are given. Various estimates for the coefficients of such power series are obtained. A function of two variables $t$ and $\lambda$ is determined, and a partial differential equation with conditions for this function are obtained. The zeros of the “section” of this function coincide with the eigenvalues of the original boundary value problem, which can be used for their approximate calculation.