Existence of Eigenvalues and Boundedness of the Eigenfunctions and the Orthogonality of the Eigenfunctions of a Type of Spectral Problem

Abstract

Background:

This research aims to investigate certain requirements for the presence of eigenvalues, as well as the boundaries of eigenfunctions and their derivatives, specifically, the eigenfunctions' first as well as second derivatives.

Materials and Methods:

In this study, we use the spectral problem of second-order differential equations:

,

with mixed boundary conditions                                              

        , where  is a spectral parameter.

and the normalized condition , where .

Results:

We get that the spectral parameter of a second-order differential operator is real. And we obtain Lagrange’s identity for a spectral problem. Also, we prove that the spectral problem is self-adjoint, and the property of orthogonality of eigenfunctions is shown.

Conclusions:

In this research, we studied the existence of eigenvalues and the estimation of the norm of eigenfunctions for problem (1) - (3). Furthermore, we investigated the self-adjoint nature of the problem, and we proved that the eigenfunctions are orthogonal.