Eigenvalues of the Sturm-Liouville problem with a frozen argument on time scales

TL;DR

This paper investigates the eigenvalues of a Sturm-Liouville boundary value problem with a frozen argument on various time scales, providing solutions, properties, and asymptotic formulas for eigenvalues.

Contribution

It introduces a new analysis of Sturm-Liouville problems with frozen arguments on arbitrary time scales, including eigenvalue properties and asymptotic behavior.

Findings

Derived solutions and characteristic functions for the problem

Established properties of eigenvalues on finite time scales

Provided asymptotic formulas for eigenvalues on specific time scales

Abstract

In this study, we consider a boundary value problem generated by the Sturm-Liouville problem with a frozen argument and with non-separated boundary conditions on a time scale. Firstly, we present some solutions and characteristic function of the problem on an arbitrary bounded time scale. Secondly, we prove some properties of eigenvalues and obtain a formulation for the eigenvalues-number on a finite time scale. Finally, we give an asymptotic formula for eigenvalues of the problem on another special time scale.