Principal solution in Weyl-Titchmarsh theory for second order Sturm-Liouville equation on time scales

Petr Zemánek

doi:10.14232/ejqtde.2017.1.2

Abstract

A connection between the oscillation theory and the Weyl--Titchmarsh theory for the second order Sturm--Liouville equation on time scales is established by using the principal solution. In particular, it is shown that the Weyl solution coincides with the principal solution in the limit point case, and consequently the square integrability of the Weyl solution is obtained. Moreover, both limit point and oscillatory criteria are derived in the case of real-valued coefficients, while a~generalization of the invariance of the limit circle case is proven for complex-valued coefficients. Several of these results are new even in the discrete time case. Finally, some illustrative examples are provided.

Highlights

In this paper we continue in the development of the Weyl–Titchmarsh theory for the second order Sturm–Liouville dynamic equation
The history of the Weyl–Titchmarsh theory goes back to the celebrated paper [23] devoted to the second order Sturm–Liouville differential equation
We utilize the principal solution of (Eλ) for a development of a limit point criterion and we discuss its connection with the Weyl solution and its square integrability in the limit point case

Summary

Introduction

We utilize the principal solution of (Eλ) for a development of a limit point criterion (see Theorem 3.1) and we discuss its connection with the Weyl solution and its square integrability in the limit point case (see Theorem 3.5). These results are new in the case T = Z, while in the case T = R they can be found in [5, Section 2]. We derive a generalization of the invariance of the limit circle case, recall several results from the Weyl–Titchmarsh theory equation (Eλ), and present basic properties of the principal solution.

Preliminaries

Main results

Let condition