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Classification and criteria of the limit cases for singular second-order linear equations on time scales

Chao Zhang1* and Yuming Shi2

Author Affiliations

1 School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, P.R. China

2 Department of Mathematics, Shandong University, Jinan, Shandong, 250100, P.R. China

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Boundary Value Problems 2012, 2012:103  doi:10.1186/1687-2770-2012-103

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/103


Received:10 April 2012
Accepted:4 September 2012
Published:19 September 2012

© 2012 Zhang and Shi; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with classification and criteria of the limit cases for singular second-order linear equations on time scales. By the different cases of the limiting set, the equations are divided into two cases: the limit-point and limit-circle cases just like the continuous and discrete cases. Several sufficient conditions for the limit-point cases are established. It is shown that the limit cases are invariant under a bounded perturbation. These results unify the existing ones of second-order singular differential and difference equations.

Keywords:
singular second-order linear equation; time scales; limit-point case; limit-circle case

1 Introduction

In this paper, we consider classification and criteria of the limit cases for the following singular second-order linear equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M2">View MathML</a>, q, and w are real and piecewise continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M5">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M6">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a> is the spectral parameter; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a> is a time scale with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M10">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M12">View MathML</a> are the forward and backward jump operators in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M14">View MathML</a> is the Δ-derivative of y; and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M15">View MathML</a>.

The spectral problems of symmetric linear differential operators and difference operators can both be divided into two cases. Those defined over finite closed intervals with well-behaved coefficients are called regular. Otherwise, they are called singular. In 1910, Weyl [1] gave a dichotomy of the limit-point and limit-circle cases for the following singular second-order linear differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M16">View MathML</a>

(1.2)

where q is a real and continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a> is the spectral parameter. Later, Titchmarsh, Coddington, Levinson etc. developed his results and established the Weyl-Titchmarsh theory [2,3]. Their work has been greatly developed and generalized to higher-order differential equations and Hamiltonian systems, and a classification and some criteria of limit cases were formulated [4-9]. Singular spectral problems of self-adjoint scalar second-order difference equations over infinite intervals were firstly studied by Atkinson [10]. His work was followed by Hinton, Jirari etc.[11,12]. In 2001, some sufficient and necessary conditions and several criteria of the limit-point and limit-circle cases were obtained for the following formally self-adjoint second-order linear difference equations with real coefficients [13]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M19">View MathML</a>

(1.3)

where ∇ and Δ are the backward and forward difference operators respectively, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M21">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M23">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M24">View MathML</a> are real numbers with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M25">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M27">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M28">View MathML</a>; λ is a complex spectral parameter. In 2006, Shi [14] established the Weyl-Titchmarsh theory of discrete linear Hamiltonian systems. Later, several sufficient conditions and sufficient and necessary conditions for the limit-point and limit-circle cases were established for the singular second-order linear difference equation with complex coefficients (see [15]).

In the past twenty years, a lot of effort has been put into the study of regular spectral problems on time scales (see [16-23]). But singular spectral problems have started to be considered only quite recently. In 2007, we employed Weyl’s method to divide the following singular second-order linear equations on time scales into two cases: limit-point and limit-circle cases [24]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M29">View MathML</a>

where q is real and continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a> is the spectral parameter. By using the similar method, Huseynov [25] studied the classification of limit cases for the following singular second-order linear equations on time scales:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M32">View MathML</a>

as well as of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M33">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M34">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M2">View MathML</a>) and q are real and piecewise continuous functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M36">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M37">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M4">View MathML</a> for all t, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M39">View MathML</a> is the spectral parameter. Obviously, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M41">View MathML</a>, then (1.1) is the same as (1.4). In 2010, by using the properties of the Weyl matrix disks, Sun [26] established the Weyl-Titchmarsh theory of Hamiltonian systems on time scales. It has been found that the second-order singular differential and difference equations can be divided into limit-point and limit-circle cases. We wonder whether the classification of the limit cases holds on time scales. In the present paper, we extend these results obtained in [24] to Eq. (1.1) and establish several sufficient conditions and sufficient and necessary conditions for the limit-point and limit-circle cases for Eq. (1.1).

This paper is organized as follows. In Section 2, some basic concepts and a fundamental theory about time scales are introduced. In Section 3, a family of nested circles which converge to a limiting set is constructed. The dichotomy of the limit-point and limit-circle cases for singular second-order linear equations on time scales is given by the geometric properties of the limiting set. Finally, several criteria of the limit-point case are established, and the invariance of the limit cases is shown under a bounded perturbation for the potential function q in Section 4.

2 Preliminaries

In this section, some basic concepts and fundamental results on time scales are introduced.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M42">View MathML</a> be a non-empty closed set. The forward and backward jump operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M43">View MathML</a> are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M44">View MathML</a>

respectively, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M46">View MathML</a>. A point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M47">View MathML</a> is called right-scattered, right-dense, left-scattered, and left-dense if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M48">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M49">View MathML</a> separately. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M50">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M51">View MathML</a> is unbounded above and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M52">View MathML</a> otherwise. The graininess <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M53">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M54">View MathML</a>

Let f be a function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a>. f is said to be Δ-differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56">View MathML</a> provided there exists a constant a such that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M57">View MathML</a>, there is a neighborhood U of t (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M58">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M59">View MathML</a>) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M60">View MathML</a>

In this case, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M61">View MathML</a>. If f is Δ-differentiable for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56">View MathML</a>, then f is said to be Δ-differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a>. If f is Δ-differentiable at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M65">View MathML</a>

(2.1)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M66">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M68">View MathML</a> is called an anti-derivative of f on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a>. In this case, define the Δ-integral by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M70">View MathML</a>

For convenience, we introduce the following results ([[27], Chapter 1] and [[28], Chapter 1]), which are useful in this paper.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M56">View MathML</a>.

(i) Iffis Δ-differentiable att, thenfis continuous att.

(ii) Iffandgare Δ-differentiable att, thenfgis Δ-differentiable attand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M73">View MathML</a>

(iii) Iffandgare Δ-differentiable att, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M74">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M75">View MathML</a>is Δ-differentiable attand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M76">View MathML</a>

A function f defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a> is said to be rd-continuous if it is continuous at every right-dense point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a> and its left-sided limit exists at every left-dense point in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M8">View MathML</a>. The set of rd-continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M80">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M81">View MathML</a>. The set of kth Δ-differentiable functions with rd-continuous kth derivative is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M82">View MathML</a>.

Lemma 2.2Iff, gare rd-continuous functions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M83">View MathML</a>, then

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M84">View MathML</a>is rd-continuous andfhas an anti-derivative on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M83">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M86">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M87">View MathML</a>.

(iii) (Integration by parts) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M88">View MathML</a>.

(iv) (Hölder’s inequality [[29], Lemma 2.2(iv)]) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M89">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M90">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M91">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M93">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M94">View MathML</a>

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M95">View MathML</a> is called regressive if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M96">View MathML</a>

Higer [30] showed that for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M97">View MathML</a> and for any given rd-continuous and regressive g, the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M98">View MathML</a>

has a unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M99">View MathML</a>

(2.2)

Lemma 2.3 ([[27], Theorem 6.1])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M101">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M102">View MathML</a>

implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M103">View MathML</a>

We define the Wronskian by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M104">View MathML</a>

(2.3)

The following result is a direct consequence of the Lagrange identity [[27], Theorem 4.30].

Lemma 2.4Letxandybe any two solutions of (1.1). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M105">View MathML</a>is a constant in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M106">View MathML</a>.

3 Classification

In this section, we focus on the classification of the limit cases for singular second-order linear equations on time scales.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M108">View MathML</a> be the two solutions of (1.1) satisfying the following initial conditions:

respectively. Since their Wronskian is identically equal to 1, these two solutions form a fundamental solution system of (1.1). We form a linear combination of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M108">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M112">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M115">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M116">View MathML</a>, and let (3.1) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M117">View MathML</a>

(3.2)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M118">View MathML</a>

(3.3)

It can be verified that the integral identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M119">View MathML</a>

(3.4)

holds for any solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M120">View MathML</a> of (1.1) and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M121">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M124">View MathML</a> in (3.4) and taking its imaginary part, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M125">View MathML</a>

(3.5)

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M126">View MathML</a>

It follows from (3.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M114">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M128">View MathML</a>. Hence, the denominator in (3.3) is not equal to zero, and consequently, m is well defined.

Next, we will show that (3.3) describes a circle for any fixed b. It follows from (3.1) and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M129">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M130">View MathML</a>

By (3.4) and the above two relations, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M131">View MathML</a>

(3.6)

which implies that m lies in the upper half-plane if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M132">View MathML</a>. It follows from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M133">View MathML</a>

which, together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M114">View MathML</a>, yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M135">View MathML</a>

It is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M136">View MathML</a>

(3.7)

By using (3.1), (3.7) can be expanded as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M137">View MathML</a>

(3.8)

Moreover, setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M138">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M139">View MathML</a>

(3.9)

It follows from the last relation in (3.9) that we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M140">View MathML</a>. By using (2.3) and (3.9), it can be verified that

(3.10)

It follows from the first relation in (3.9) and (3.5) that we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M142">View MathML</a>. Then (3.8) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M143">View MathML</a>

(3.11)

which implies that (3.3) forms a circle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144">View MathML</a> as k varies. It is evident that the center of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M146">View MathML</a>

It follows from Lemma 2.4 and (3.10) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M147">View MathML</a>

From (3.11), (3.9), (2.3), and (3.5) we have that the radius of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M149">View MathML</a>

(3.12)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M150">View MathML</a> denote the closed disk bounded by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M144">View MathML</a>. We are going to show that the circle sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M152">View MathML</a> is nested.

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M153">View MathML</a>

From the first relation in (3.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M154">View MathML</a>

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M155">View MathML</a>

So, it follows from (3.6) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M156">View MathML</a>

(3.13)

In the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M132">View MathML</a>, the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M138">View MathML</a> is interior to the circle if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M159">View MathML</a>. This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M160">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M161">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M162">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M163">View MathML</a> and consider the corresponding disks <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M164">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M165">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M166">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M167">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M168">View MathML</a>. This yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M169">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M170">View MathML</a> is nested. Consequently, there are the following two alternatives:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M171">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M172">View MathML</a>. In this case there is one point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M173">View MathML</a> which is common to all the disks <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M175">View MathML</a>. This is called the limit-point case. It follows from (3.12) that this case occurs if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M176">View MathML</a>

(3.14)

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M177">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M172">View MathML</a>. In this case there is a disk <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M179">View MathML</a> contained in all the disks <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M175">View MathML</a>. This is called the limit-circle case. It follows from (3.12) that this case occurs if and only if the integral in (3.14) is convergent, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M182">View MathML</a>.

Theorem 3.1For every non-real<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a>, Eq. (1.1) has at least one non-trivial solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>.

Proof In the limit-circle case, it follows from the above discussion that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M182">View MathML</a>.

Next, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M186">View MathML</a> in the limit-point case. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M187">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M188">View MathML</a> and choose any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M189">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M190">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M191">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M192">View MathML</a> uniformly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M193">View MathML</a> on any finite interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M195">View MathML</a>. Since the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M196">View MathML</a> is bounded from above and its upper bound is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M197">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M198">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M199">View MathML</a>

Hence, by the uniform convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M192">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M201">View MathML</a>

for all ω. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M202">View MathML</a>. This completes the proof. □

Remark 3.1 Similar to the proof of Theorem 3.1, it can be easily verified that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M203">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M204">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M205">View MathML</a> in the limit-circle case. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M108">View MathML</a> are linearly independent. Hence, all the solutions of Eq. (1.1) belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M210">View MathML</a> in the limit-circle case.

Remark 3.2 It follows from (3.14) and Theorem 3.1 that Eq. (1.1) has exactly one linearly independent solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M211">View MathML</a> in the limit point case for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M213">View MathML</a>.

Theorem 3.2If Eq. (1.1) has two linearly independent solutions in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M215">View MathML</a>, then this property holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a>.

Proof Suppose that Eq. (1.1) has two linearly independent solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M218">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M219">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M220">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. For briefness, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M222">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M7">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M224">View MathML</a> be an arbitrary non-trivial solution of (1.1), and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M225">View MathML</a> be the solution of (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M226">View MathML</a> and with the initial values

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M227">View MathML</a>

From the variation of constants [[27], Theorem 3.73], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M228">View MathML</a>

(3.15)

Replacing t with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M11">View MathML</a> in (3.15) and using (ii) of Lemma 2.2, we obtain

which implies by the Hölder inequality in Lemma 2.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M231">View MathML</a>

It follows from the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M232">View MathML</a>

where A, B, C are non-negative numbers, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M233">View MathML</a>

Integrating the two sides of the above inequality with respect to t from a to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M234">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M235">View MathML</a>

which yields that

Hence,

(3.16)

The constant a can be chosen in advance so large that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M238">View MathML</a>

It follows from (3.16) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M239">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M240">View MathML</a>. Therefore, all the solutions of Eq. (1.1) are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. The proof is complete. □

At the end of this section, from the above discussions we present the classification of the limit cases for singular second-order linear equations over the infinite interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a> on time scales.

Definition 3.1 If Eq. (1.1) has only one linear independent solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M244">View MathML</a>, then Eq. (1.1) is said to be in the limit-point case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>. If Eq. (1.1) has two linear independent solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M39">View MathML</a>, then Eq. (1.1) is said to be in the limit-circle case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>.

4 Several criteria of the limit-point and limit-circle cases

In this section, we establish several criteria of the limit-point and limit-circle cases for Eq. (1.1).

We first give two criteria of the limit-point case.

Theorem 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M40">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M250">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251">View MathML</a>. If there exists a positive Δ-differentiable function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M252">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M37">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M254">View MathML</a>and two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M255">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M256">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M257">View MathML</a>,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M258">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M259">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M260">View MathML</a>,

then Eq. (1.1) is in the limit-point case at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>.

Proof Suppose that Eq. (1.1) is in the limit-circle case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>. By Theorem 3.2, all the solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M263">View MathML</a>

(4.1)

are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266">View MathML</a> be the solutions of (4.1) satisfying the following initial conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M267">View MathML</a>

(4.2)

It is evident that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266">View MathML</a> are two linearly independent solutions of (4.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. By Lemma 2.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M271">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M6">View MathML</a>. Hence, we have

It follows from the Hölder inequality and assumption (iii) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M274">View MathML</a>

are divergent. Suppose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M275">View MathML</a>

From (4.1) and assumption (i), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M276">View MathML</a>

(4.3)

Applying integration by parts in Lemma 2.2, by (iii) in Lemma 2.1, we get

(4.4)

Again applying the Hölder inequality, from condition (ii), we have

(4.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M279">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M280">View MathML</a>

it follows from (4.3)-(4.5) that

It follows from the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M282">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M283">View MathML</a>. From the above relation and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M250">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M285">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M286">View MathML</a> is ultimately positive. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M287">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M283">View MathML</a>; and consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a> does not belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. This contradicts the assumption that all the solutions of (4.1) are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. Then Eq. (4.1) has at least one non-trivial solution outside of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a>. It follows from Theorem 3.2 that Eq. (1.1) is in the limit-point case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>. This completes the proof. □

Remark 4.1 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M295">View MathML</a> are two special time scales, Theorem 4.1 not only contains the criterion of the limit-point case for second-order differential equations [[5], Chapter 9, Theorem 2.4], but also the criterion of the limit-point case for second-order difference equation (1.3) [[15], Theorem 3.3].

The following corollary is a direct consequence of Theorem 4.1 by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M296">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251">View MathML</a>.

Corollary 4.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M300">View MathML</a>is bounded below in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M302">View MathML</a>, then Eq. (1.1) is in the limit-point case at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>.

Theorem 4.2If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M304">View MathML</a>

(4.6)

then Eq. (1.1) is in the limit-point case at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>.

Proof On the contrary, suppose that Eq. (1.1) is in the limit-circle case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266">View MathML</a> be two linearly independent solutions of (1.1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> satisfying the initial conditions (4.2). By Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M310">View MathML</a>

which, together with (2.1), implies that

So, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M312">View MathML</a>

which implies

(4.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M314">View MathML</a>. By the Hölder inequality and the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M315">View MathML</a>, one has

Hence, it follows from (4.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M317">View MathML</a>

which is a contradiction to the assumption (4.6). Therefore, Eq. (1.1) is in the limit-point case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>. This completes the proof. □

Remark 4.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M319">View MathML</a>, Theorem 4.2 is the same as that obtained by Chen and Shi for second-order difference equations [[13], Corollary 3.1].

Next, we study the invariance of the limit cases under a bounded perturbation for the potential function q. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M320">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M321">View MathML</a> in [[27], Theorem 2.4(i)]. It follows from [[27], Theorem 2.36(i)], [[27], Theorem 2.39(i)], and [[27], Theorem 2.4(i)] that we have the following lemma, which is useful in the subsequent discussion.

Lemma 4.1 (Gronwall’s inequality)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M100">View MathML</a>be two non-negative functions on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a>andMbe a non-negative constant. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M324">View MathML</a>

(4.8)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M325">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M326">View MathML</a>is defined as in (2.2).

The following result shows that if Eq. (1.1) is in the limit-circle case, so is it under a bounded perturbation for the potential function q.

Lemma 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M327">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M329">View MathML</a>be bounded with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M330">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a>; that is, there exists a positive constantMsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M332">View MathML</a>

(4.9)

Then Eq. (1.1) is in the limit-circle case at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>if and only if the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M334">View MathML</a>

(4.10)

is in the limit-circle case at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>.

Proof Suppose that (4.10) is in the limit-circle case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>. To show that Eq. (1.1) is in the limit-circle case, it suffices to show that each solution (4.1) is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> by Theorem 3.2.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266">View MathML</a> be two solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M340">View MathML</a>

(4.11)

satisfying the initial conditions (4.2). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266">View MathML</a> are two linearly independent solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M184">View MathML</a> by Theorem 3.2.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M344">View MathML</a> be any solution of (4.1). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M345">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M346">View MathML</a>. By the variation of constants [[27], Theorem 3.73] there exist two constants α and β such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M347">View MathML</a>

Hence, replacing t by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M11">View MathML</a> and by (ii) in Lemma 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M349">View MathML</a>

(4.12)

From (4.9) and (4.12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M350">View MathML</a>

(4.13)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M266">View MathML</a> are solutions of Eq. (4.11), which satisfy the initial conditions (4.2), it follows from the existence-uniqueness theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M353">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M354">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M355">View MathML</a>

From (4.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M356">View MathML</a>

It follows from (i) of Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M357">View MathML</a>. By Lemma 4.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M358">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M359">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M360">View MathML</a>; and consequently, Eq. (1.1) is in the limit-circle case at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M245">View MathML</a>.

On the other hand, using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M362">View MathML</a>

one can easily conclude that if Eq. (1.1) is in the limit-circle case, then Eq. (4.10) is in the limit-circle case. This completes the proof. □

Theorem 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M327">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M251">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M329">View MathML</a>be bounded with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M330">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M3">View MathML</a>. Then the limit cases for Eq. (1.1) are invariant.

Remark 4.3 Lemma 4.2 extends the related result [[13], Lemma 2.4] for the singular second-order difference equation to the time scales. In addition, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/103/mathml/M368">View MathML</a> in Lemma 4.2, then we can directly prove [[31], Theorem 6.1] with the similar method.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YS supervised the study and helped the revision. CZ carried out the main results of this article and drafted the manuscript. All the authors have read and approved the final manuscript.

Acknowledgements

Many thanks are due to M Victoria Otero-Espinar (the editor) and the anonymous reviewers for helpful comments and suggestions. This research was supported by the NNSF of China (Grant 11071143 and 11101241), the NNSF of Shandong Province (Grant ZR2009AL003, ZR2010AL016, and ZR2011AL007), the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01), and the NSF of University of Jinan (XKY0918).

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