On the estimations of the small eigenvalues of Sturm–Liouville operators with periodic and antiperiodic boundary conditions

Abstract

We give a new approach for the estimations of the eigenvalues of non-self-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions. Moreover, we give error estimations, and finally we present some numerical examples.

Two‐linked periodic Sturm–Liouville problems with transmission conditions

“…This implies that W(𝜙(., 𝜆), φ(., 𝜆); a) = W(𝜙(., 𝜆), φ(., 𝜆); b). (13) Further, using the transmission conditions T i 𝜙(., 𝜆) = T i φ(., 𝜆) = 0, i = 1, 2, we calculate…”

“…An interesting approach was given by Dinibütün and Veliev 12 for estimation of the small periodic eigenvalues. In a previous study, 13 a new technique is developed to estimate the eigenvalues of non-selfadjoint Sturm-Liouville problems with periodic and anti-periodic boundary conditions. In recent years, there has been growing interest of Sturm-Liouville problems with additional transmission conditions at some interior points of interaction, because of the appearance of new and interesting applications in mathematical physics (see, e.g., previous studies [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the references therein).…”

“…An interesting approach was given by Dinibütün and Veliev 12 for estimation of the small periodic eigenvalues. In a previous study, 13 a new technique is developed to estimate the eigenvalues of non‐selfadjoint Sturm–Liouville problems with periodic and anti‐periodic boundary conditions.…”

Spectrum of periodic Sturm-Liouville problems involving additional transmission conditions

“…One of the interesting approaches was given by Dinibütün and Veliev [15]. In [24,36] have been given a new approach for the estimations of the eigenvalues of non-self-adjoint Sturm-Liouville operators with periodic and anti periodic boundary conditions.…”