Esimation of Eigenvalues and Zeros of Eigenfunction for a Class of Discontinuous Sturm−Liouville Problems

For discontinuous Sturm-Liouville problems (SLPs) with periodic boundary conditions, utilizing the theory of initial value problems for ordinary differential equations, asymptotic estimates for two linearly independent solutions of the aforementioned discontinuous SLPs were calculated. By leveraging Gronwall’s inequality, properties of eigenvalues in differential equations, and the asymptotic estimation formula of the solutions, an asymptotic estimation form for the eigenvalues of the discontinuous SLPs was derived.Using the Prufer transformation, it was shown that the nth eigenfunction corresponding to the nth eigenvalue of the discontinuous SLPs has n zeros within the interval (0,c)∪(c,π).These findings are significant for calculating the indices of eigenvalues in discontinuous SLPs and for studying the oscillatory behavior of their solutions.