On spectral properties of a Sturm-Liouville problem with transmission conditions and eigenparameter dependent boundary conditions with retarded argument

TL;DR

This paper investigates the spectral properties of a Sturm-Liouville problem featuring transmission conditions, eigenparameter-dependent boundary conditions, and retarded arguments, deriving asymptotic formulas for eigenvalues and eigenfunctions.

Contribution

It introduces a novel analysis of a Sturm-Liouville problem with multiple complex features, including transmission conditions and retarded arguments, providing new asymptotic formulas.

Findings

Derived asymptotic formulas for eigenvalues.

Analyzed eigenfunctions' behavior asymptotically.

Extended spectral theory to complex boundary-value problems.

Abstract

In this work, we consider not only a discontinuous boundary-value problem with retarded argument and four supplementary transmission conditions at the two points of discontinuities but also, eigenparameter-dependent boundary conditions and obtain asymptotic formulas for the eigenvalues and corresponding eigenfunctions.