Open Access

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

Boundary Value Problems20122012:103

DOI: 10.1186/1687-2770-2012-103

Received: 10 April 2012

Accepted: 4 September 2012

Published: 19 September 2012

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:
( p ( t ) y Δ ( t ) ) Δ + q ( t ) y σ ( t ) = λ w ( t ) y σ ( t ) , t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ1_HTML.gif
(1.1)

where p Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq1_HTML.gif, q, and w are real and piecewise continuous functions on [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif, p ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq3_HTML.gif and w ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq4_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq5_HTML.gif; λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif is the spectral parameter; T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif is a time scale with ρ ( 0 ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq8_HTML.gif and sup T = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq9_HTML.gif; σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq10_HTML.gif and ρ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq11_HTML.gif are the forward and backward jump operators in T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif; y Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq12_HTML.gif is the Δ-derivative of y; and y σ ( t ) : = y ( σ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq13_HTML.gif.

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:
y ( t ) + q ( t ) y ( t ) = λ y ( t ) , t [ 0 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ2_HTML.gif
(1.2)
where q is a real and continuous function on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq14_HTML.gif, λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif 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 [49]. 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]:
( p ( n ) Δ y ( n ) ) + q ( n ) y ( n ) = λ w ( n ) y ( n ) , n { n } n = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ3_HTML.gif
(1.3)

where and Δ are the backward and forward difference operators respectively, namely y ( n ) : = y ( n ) y ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq15_HTML.gif and Δ y ( n ) : = y ( n + 1 ) y ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq16_HTML.gif; p ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq17_HTML.gif, q ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq18_HTML.gif, and w ( n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq19_HTML.gif are real numbers with w ( n ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq20_HTML.gif for n [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq21_HTML.gif and  p ( n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq22_HTML.gif for n [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq23_HTML.gif; 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 [1623]). 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]:
y Δ Δ ( t ) + q ( t ) y σ ( t ) = λ y σ ( t ) , t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equa_HTML.gif
where q is real and continuous on [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif, λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif 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:
( p ( t ) y Δ ( t ) ) + q ( t ) y ( t ) = λ y ( t ) , t ( a , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equb_HTML.gif
as well as of the form
( p ( t ) y Δ ( t ) ) Δ + q ( t ) y σ ( t ) = λ y σ ( t ) , t [ a , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ4_HTML.gif
(1.4)

where p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq24_HTML.gif (or p Δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq1_HTML.gif) and q are real and piecewise continuous functions in ( a , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq25_HTML.gif (or [ a , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq26_HTML.gif), p ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq3_HTML.gif for all t, and λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq27_HTML.gif is the spectral parameter. Obviously, let w ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq28_HTML.gif and ρ ( 0 ) = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq29_HTML.gif, 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 T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq30_HTML.gif be a non-empty closed set. The forward and backward jump operators σ , ρ : T T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq31_HTML.gif are defined by
σ ( t ) : = inf { s T : s > t } , ρ ( t ) : = sup { s T : s < t } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equc_HTML.gif
respectively, where inf = sup T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq32_HTML.gif, sup = inf T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq33_HTML.gif. A point t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq34_HTML.gif is called right-scattered, right-dense, left-scattered, and left-dense if σ ( t ) > t , σ ( t ) = t , ρ ( t ) < t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq35_HTML.gif, and ρ ( t ) = t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq36_HTML.gif separately. Denote T k : = T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq37_HTML.gif if T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq38_HTML.gif is unbounded above and T k : = T ( ρ ( max T ) , max T ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq39_HTML.gif otherwise. The graininess μ : T [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq40_HTML.gif is defined by
μ ( t ) : = σ ( t ) t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equd_HTML.gif
Let f be a function defined on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif. f is said to be Δ-differentiable at t T k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq41_HTML.gif provided there exists a constant a such that, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq42_HTML.gif, there is a neighborhood U of t (i.e., U = ( t δ , t + δ ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq43_HTML.gif for some δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq44_HTML.gif) with
| f ( σ ( t ) ) f ( s ) a ( σ ( t ) s ) | ε | σ ( t ) s | for all  s U . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Eque_HTML.gif
In this case, denote f Δ ( t ) : = a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq45_HTML.gif. If f is Δ-differentiable for every t T k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq41_HTML.gif, then f is said to be Δ-differentiable on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif. If f is Δ-differentiable at t T k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq41_HTML.gif, then
f Δ ( t ) = { lim s T s t f ( t ) f ( s ) t s , if  μ ( t ) = 0 , f ( σ ( t ) ) f ( t ) μ ( t ) , if  μ ( t ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ5_HTML.gif
(2.1)
If F Δ ( t ) = f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq46_HTML.gif for all t T k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq41_HTML.gif, then F ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq47_HTML.gif is called an anti-derivative of f on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif. In this case, define the Δ-integral by
s t f ( τ ) Δ τ = F ( t ) F ( s ) for all  s , t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equf_HTML.gif

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

Lemma 2.1 Let f , g : T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq48_HTML.gif and t T k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq41_HTML.gif.
  1. (i)

    If f is Δ-differentiable at t, then f is continuous at t.

     
  2. (ii)
    If f and g are Δ-differentiable at t, then fg is Δ-differentiable at t and
    ( f g ) Δ ( t ) = f σ ( t ) g Δ ( t ) + f Δ ( t ) g ( t ) = f Δ ( t ) g σ ( t ) + f ( t ) g Δ ( t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equg_HTML.gif
     
  3. (iii)
    If f and g are Δ-differentiable at t, and f ( t ) f σ ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq49_HTML.gif, then f 1 g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq50_HTML.gif is Δ-differentiable at t and
    ( g f 1 ) Δ ( t ) = ( g Δ ( t ) f ( t ) g ( t ) f Δ ( t ) ) ( f σ ( t ) f ( t ) ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equh_HTML.gif
     

A function f defined on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif is said to be rd-continuous if it is continuous at every right-dense point in T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif and its left-sided limit exists at every left-dense point in T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq7_HTML.gif. The set of rd-continuous functions f : T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq51_HTML.gif is denoted by C r d ( T ) = C r d ( T , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq52_HTML.gif. The set of k th Δ-differentiable functions with rd-continuous k th derivative is denoted by C r d k ( T ) = C r d k ( T , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq53_HTML.gif.

Lemma 2.2 If f, g are rd-continuous functions on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq54_HTML.gif, then
  1. (i)

    f σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq55_HTML.gif is rd-continuous and f has an anti-derivative on T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq54_HTML.gif;

     
  2. (ii)

    t σ ( t ) f ( τ ) Δ τ = μ ( t ) f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq56_HTML.gif for all t T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq57_HTML.gif.

     
  3. (iii)

    (Integration by parts) a b f σ ( τ ) g Δ ( τ ) Δ τ = f ( b ) g ( b ) f ( a ) g ( a ) a b f Δ ( τ ) g ( τ ) Δ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq58_HTML.gif.

     
  4. (iv)
    (Hölder’s inequality [[29], Lemma 2.2(iv)]) Let r , s T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq59_HTML.gif with r s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq60_HTML.gif, then
    r s | f ( τ ) g ( τ ) | Δ τ { r s | f ( τ ) | p Δ τ } 1 p { r s | g ( τ ) | q Δ τ } 1 q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equi_HTML.gif
     

where p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq61_HTML.gif and q = p / ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq62_HTML.gif.

Let
L w 2 ( ρ ( 0 ) , ) : = { y σ : [ ρ ( 0 ) , ) C | ρ ( 0 ) w ( t ) | y σ ( t ) | 2 Δ t < } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equj_HTML.gif
A function g : T R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq63_HTML.gif is called regressive if
1 + μ ( t ) g ( t ) 0 for all  t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equk_HTML.gif
Higer [30] showed that for any given t 0 T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq64_HTML.gif and for any given rd-continuous and regressive g, the initial value problem
y Δ ( t ) = g ( t ) y ( t ) , y ( t 0 ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equl_HTML.gif
has a unique solution
e g ( t , t 0 ) = exp { t 0 t ξ μ ( τ ) ( g ( τ ) ) Δ τ } , ξ h ( z ) = { Log ( 1 + h z ) h , if  h 0 , z , if  h = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ6_HTML.gif
(2.2)

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

Let y , f C r d ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq65_HTML.gif and g R + : = { g C r d ( T ) : 1 + μ ( t ) g ( t ) > 0 , t T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq66_HTML.gif. Then
y Δ ( t ) g ( t ) y ( t ) + f ( t ) , t T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equm_HTML.gif
implies
y ( t ) y ( t 0 ) e g ( t , t 0 ) + t 0 t e g ( t , σ ( τ ) ) f ( τ ) Δ τ , t T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equn_HTML.gif
We define the Wronskian by
W [ x , y ] ( t ) = p ( t ) [ x ( t ) y Δ ( t ) x Δ ( t ) y ( t ) ] , x , y C r d 2 ( T ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ7_HTML.gif
(2.3)

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

Lemma 2.4 Let x and y be any two solutions of (1.1). Then W [ x , y ] ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq67_HTML.gif is a constant in [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq68_HTML.gif.

3 Classification

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

Let y 1 ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq69_HTML.gif and y 2 ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq70_HTML.gif be the two solutions of (1.1) satisfying the following initial conditions:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equo_HTML.gif
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 y 1 ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq69_HTML.gif and y 2 ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq70_HTML.gif
y ( t , λ , m ) : = y 1 ( t , λ ) + m y 2 ( t , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ8_HTML.gif
(3.1)
Let b ( ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq71_HTML.gif, k R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq72_HTML.gif, λ = μ + i ν https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq73_HTML.gif with ν 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq74_HTML.gif, and let (3.1) satisfy
p ( b ) y Δ ( b , λ , m ) + k y ( b , λ , m ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ9_HTML.gif
(3.2)
Then
m = p ( b ) y 1 Δ ( b , λ ) + k y 1 ( b , λ ) p ( b ) y 2 Δ ( b , λ ) + k y 2 ( b , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ10_HTML.gif
(3.3)
It can be verified that the integral identity
[ y ( t , λ ) ¯ p ( t ) y Δ ( t , λ ) ] | t 1 t 2 t 1 t 2 p ( t ) | y Δ ( t , λ ) | 2 Δ t + t 1 t 2 [ λ w ( t ) q ( t ) ] | y σ ( t , λ ) | 2 Δ t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ11_HTML.gif
(3.4)
holds for any solution y ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq75_HTML.gif of (1.1) and for any t 1 , t 2 [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq76_HTML.gif. Setting y ( t , λ ) = y 2 ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq77_HTML.gif, t 1 = ρ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq78_HTML.gif, t 2 = b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq79_HTML.gif in (3.4) and taking its imaginary part, we obtain
[ y 2 ( b , λ ) ¯ p ( b ) y 2 Δ ( b , λ ) ] = ν ρ ( 0 ) b w ( t ) | y 2 σ ( t , λ ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ12_HTML.gif
(3.5)
So
( p ( b ) y 2 Δ ( b , λ ) y 2 ( b , λ ) ) = [ y 2 ( b , λ ) ¯ p ( b ) y 2 Δ ( b , λ ) ] | y 2 ( b , λ ) | 2 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equp_HTML.gif

It follows from (3.2) and k R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq72_HTML.gif that k p ( b ) y 2 Δ ( b , λ ) y 2 ( b , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq80_HTML.gif. 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
y ( ρ ( 0 ) , λ , m ) ¯ p ( ρ ( 0 ) ) y Δ ( ρ ( 0 ) , λ , m ) = m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equq_HTML.gif
and
y ( b , λ , m ) ¯ p ( b ) y Δ ( b , λ , m ) = k | y ( b , λ , m ) | 2 R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equr_HTML.gif
By (3.4) and the above two relations, we have
( m ) = ν ρ ( 0 ) b w ( t ) | y σ ( t , λ , m ) | 2 Δ t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ13_HTML.gif
(3.6)
which implies that m lies in the upper half-plane if ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq81_HTML.gif. It follows from (3.2) that
k = p ( b ) y Δ ( b , λ , m ) y ( b , λ , m ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equs_HTML.gif
which, together with k R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq72_HTML.gif, yields that
p ( b ) [ y Δ ( b , λ , m ) y ( b , λ , m ) ¯ y Δ ( b , λ , m ) ¯ y ( b , λ , m ) ] = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equt_HTML.gif
It is equivalent to
W [ y , y ¯ ] ( b , λ , m ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ14_HTML.gif
(3.7)
By using (3.1), (3.7) can be expanded as
| m | 2 W [ y 2 , y 2 ¯ ] ( b , λ ) + m W [ y 2 , y 1 ¯ ] ( b , λ ) + m ¯ W [ y 1 , y 2 ¯ ] ( b , λ ) + W [ y 1 , y 1 ¯ ] ( b , λ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ15_HTML.gif
(3.8)
Moreover, setting m = u + i v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq82_HTML.gif, we have
W [ y 2 , y 2 ¯ ] ( b , λ ) = 2 i A , W [ y 1 , y 1 ¯ ] ( b , λ ) = 2 i D , W [ y 2 , y 1 ¯ ] ( b , λ ) = B + i C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ16_HTML.gif
(3.9)
It follows from the last relation in (3.9) that we have W [ y 1 , y 2 ¯ ] ( b , λ ) = B i C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq83_HTML.gif. By using (2.3) and (3.9), it can be verified that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ17_HTML.gif
(3.10)
It follows from the first relation in (3.9) and (3.5) that we have A = ν ρ ( 0 ) b w ( t ) | y 2 σ ( t , λ ) | 2 Δ t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq84_HTML.gif. Then (3.8) becomes
( u C 2 A ) 2 + ( v B 2 A ) 2 = B 2 + C 2 4 A D 4 A 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ18_HTML.gif
(3.11)
which implies that (3.3) forms a circle C b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq85_HTML.gif as k varies. It is evident that the center of C b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq85_HTML.gif is
z 0 = C + i B 2 A = B i C 2 i A = W [ y 1 , y 2 ¯ ] ( b , λ ) W [ y 2 , y 2 ¯ ] ( b , λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equu_HTML.gif
It follows from Lemma 2.4 and (3.10) that
B 2 + C 2 4 A D = | W [ y 1 , y 2 ] ( b , λ ) | 2 = | W [ y 1 , y 2 ] ( ρ ( 0 ) , λ ) | 2 = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equv_HTML.gif
From (3.11), (3.9), (2.3), and (3.5) we have that the radius of C b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq85_HTML.gif is
r b = | B 2 + C 2 4 A D 4 A 2 | 1 2 = | 2 i A | 1 = | W [ y 2 , y 2 ¯ ] ( b , λ ) | 1 = [ 2 | ν | ρ ( 0 ) b w ( t ) | y 2 σ ( t , λ ) | 2 Δ t ] 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ19_HTML.gif
(3.12)

Let C b ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq86_HTML.gif denote the closed disk bounded by C b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq85_HTML.gif. We are going to show that the circle sequence { C b ¯ } ( ρ ( 0 ) < b < ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq87_HTML.gif is nested.

Set
U + i V = ν ρ ( 0 ) b w ( t ) y 1 σ ( t , λ ) y 2 σ ( t , λ ) ¯ Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equw_HTML.gif
From the first relation in (3.9), we have
A = ν ρ ( 0 ) b w ( t ) | y 2 σ ( t , λ ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equx_HTML.gif
Similarly,
D = ν ρ ( 0 ) b w ( t ) | y 1 σ ( t , λ ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equy_HTML.gif
So, it follows from (3.6) that
v = A ( u 2 + v 2 ) + 2 U u + 2 V v + D . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ20_HTML.gif
(3.13)
In the case of ν > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq81_HTML.gif, the point m = u + i v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq82_HTML.gif is interior to the circle if v > A ( u 2 + v 2 ) + 2 U u + 2 V v + D https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq88_HTML.gif. This shows that m C b ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq89_HTML.gif if and only if
( m ) ν ρ ( 0 ) b w ( t ) | y σ ( t , λ , m ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equz_HTML.gif
Let b 1 , b 2 [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq90_HTML.gif with b 1 < b 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq91_HTML.gif and consider the corresponding disks C b 1 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq92_HTML.gif and C b 2 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq93_HTML.gif. For any m C b 2 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq94_HTML.gif, we have
( m ) ν ρ ( 0 ) b 2 w ( t ) | y σ ( t , λ , m ) | 2 Δ t ν ρ ( 0 ) b 1 w ( t ) | y σ ( t , λ , m ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equaa_HTML.gif
Hence, m C b 1 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq95_HTML.gif. This yields that C b 2 ¯ C b 1 ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq96_HTML.gif. Therefore, { C b ¯ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq97_HTML.gif is nested. Consequently, there are the following two alternatives:
  1. (1)
    r b 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq98_HTML.gif as b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq99_HTML.gif. In this case there is one point m = m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq100_HTML.gif which is common to all the disks C b ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq86_HTML.gif, b [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq101_HTML.gif. This is called the limit-point case. It follows from (3.12) that this case occurs if and only if
    ρ ( 0 ) w ( t ) | y 2 σ ( t , λ ) | 2 Δ t = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ21_HTML.gif
    (3.14)
     
  2. (2)

    r b r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq102_HTML.gif as b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq99_HTML.gif. In this case there is a disk C ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq103_HTML.gif contained in all the disks C b ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq86_HTML.gif, b [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq101_HTML.gif. 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., y 2 ( , λ ) L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq104_HTML.gif.

     

Theorem 3.1 For every non-real λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif, Eq. (1.1) has at least one non-trivial solution in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif.

Proof In the limit-circle case, it follows from the above discussion that y 2 ( , λ ) L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq104_HTML.gif.

Next, we will show that y 1 ( , λ ) + m ( λ ) y 2 ( , λ ) L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq106_HTML.gif in the limit-point case. Let { b n } T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq107_HTML.gif with 0 < b n < b n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq108_HTML.gif and choose any m n C b n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq109_HTML.gif. Then m n m ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq110_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq111_HTML.gif and y σ ( t , λ , m n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq112_HTML.gif uniformly converges to y σ ( t , λ , m ( λ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq113_HTML.gif on any finite interval [ ρ ( 0 ) , ω ] T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq114_HTML.gif, ω T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq115_HTML.gif. Since the sequence { ( m n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq116_HTML.gif is bounded from above and its upper bound is denoted by y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq117_HTML.gif, then for b n > ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq118_HTML.gif,
y 0 ( m n ) = ν ρ ( 0 ) b n w ( t ) | y σ ( t , λ , m n ) | 2 Δ t ν ρ ( 0 ) ω w ( t ) | y σ ( t , λ , m n ) | 2 Δ t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equab_HTML.gif
Hence, by the uniform convergence of y σ ( t , λ , m n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq112_HTML.gif, we have
y 0 ν ρ ( 0 ) ω w ( t ) | y σ ( t , λ , m ( λ ) ) | 2 Δ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equac_HTML.gif

for all ω. Therefore, y ( , λ , m ( λ ) ) = y 1 ( , λ ) + m ( λ ) y 2 ( , λ ) L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq119_HTML.gif. This completes the proof. □

Remark 3.1 Similar to the proof of Theorem 3.1, it can be easily verified that y ( , λ , m ) L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq120_HTML.gif for any m C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq121_HTML.gif with ( λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq122_HTML.gif in the limit-circle case. Clearly, y ( t , λ , m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq123_HTML.gif and y 2 ( t , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq70_HTML.gif are linearly independent. Hence, all the solutions of Eq. (1.1) belong to L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif for any λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif with ( λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq124_HTML.gif 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 L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq125_HTML.gif in the limit point case for any λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif with ( λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq126_HTML.gif.

Theorem 3.2 If Eq. (1.1) has two linearly independent solutions in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif for some λ 0 C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq127_HTML.gif, then this property holds for all λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif.

Proof Suppose that Eq. (1.1) has two linearly independent solutions in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif for λ = λ 0 C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq128_HTML.gif. Then y 1 ( t , λ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq129_HTML.gif and y 2 ( t , λ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq130_HTML.gif are in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. For briefness, denote
u 1 ( t ) = y 1 ( t , λ 0 ) , u 2 ( t ) = y 2 ( t , λ 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equad_HTML.gif
For any λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq6_HTML.gif, let v ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq131_HTML.gif be an arbitrary non-trivial solution of (1.1), and let u ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq132_HTML.gif be the solution of (1.1) with λ = λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq133_HTML.gif and with the initial values
u ( a ) = v ( a ) , u Δ ( a ) = v Δ ( a ) , a ( 0 , ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equae_HTML.gif
From the variation of constants [[27], Theorem 3.73], we have
v ( t ) = u ( t ) + ( λ λ 0 ) a t [ u 1 ( t ) u 2 σ ( s ) u 2 ( t ) u 1 σ ( s ) ] w ( s ) v σ ( s ) Δ s , t [ a , ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ22_HTML.gif
(3.15)
Replacing t with σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq10_HTML.gif in (3.15) and using (ii) of Lemma 2.2, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equaf_HTML.gif
which implies by the Hölder inequality in Lemma 2.2 that
| w 1 2 ( t ) v σ ( t ) | | w 1 2 ( t ) u σ ( t ) | + | λ λ 0 | | w 1 2 ( t ) u 1 σ ( t ) | × [ a t w ( s ) | u 2 σ ( s ) | 2 Δ s a t w ( s ) | v σ ( s ) | 2 Δ s ] 1 2 + | λ λ 0 | | w 1 2 ( t ) u 2 σ ( t ) | [ a t w ( s ) | u 1 σ ( s ) | 2 Δ s a t w ( s ) | v σ ( s ) | 2 Δ s ] 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equag_HTML.gif
It follows from the inequality
( A + B + C ) 2 3 ( A 2 + B 2 + C 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equah_HTML.gif
where A, B, C are non-negative numbers, that
1 3 w ( t ) | v σ ( t ) | 2 w ( t ) | u σ ( t ) | 2 + | λ λ 0 | 2 [ w ( t ) | u 1 σ ( t ) | 2 a t w ( s ) | u 2 σ ( s ) | 2 Δ s + w ( t ) | u 2 σ ( t ) | 2 a t w ( s ) | u 1 σ ( s ) | 2 Δ s ] a t w ( s ) | v σ ( s ) | 2 Δ s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equai_HTML.gif
Integrating the two sides of the above inequality with respect to t from a to τ ( a , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq134_HTML.gif, we get
1 3 a τ w ( t ) | v σ ( t ) | 2 Δ t a τ w ( t ) | u σ ( t ) | 2 Δ t + | λ λ 0 | 2 a τ [ w ( t ) | u 1 σ ( t ) | 2 a t w ( s ) | u 2 σ ( s ) | 2 Δ s + w ( t ) | u 2 σ ( t ) | 2 a t w ( s ) | u 1 σ ( s ) | 2 Δ s ] a t w ( s ) | v σ ( s ) | 2 Δ s Δ t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equaj_HTML.gif
which yields that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equak_HTML.gif
Hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ23_HTML.gif
(3.16)
The constant a can be chosen in advance so large that
6 | λ λ 0 | 2 a w ( t ) | u 1 σ ( t ) | 2 Δ t a w ( t ) | u 2 σ ( t ) | 2 Δ t < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equal_HTML.gif

It follows from (3.16) that v L w 2 ( a , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq135_HTML.gif and hence v L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq136_HTML.gif. Therefore, all the solutions of Eq. (1.1) are in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. 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 [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif on time scales.

Definition 3.1 If Eq. (1.1) has only one linear independent solution in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif for some λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq137_HTML.gif, then Eq. (1.1) is said to be in the limit-point case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif. If Eq. (1.1) has two linear independent solutions in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif for some λ C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq27_HTML.gif, then Eq. (1.1) is said to be in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif.

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 4.1 Let w ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq28_HTML.gif and p ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq139_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq140_HTML.gif. If there exists a positive Δ-differentiable function M ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq141_HTML.gif on [ a , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq26_HTML.gif for some a ρ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq142_HTML.gif and two positive constants k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq143_HTML.gif and k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq144_HTML.gif such that for all t [ a , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq145_HTML.gif,
  1. (i)

    q ( t ) k 1 M σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq146_HTML.gif,

     
  2. (ii)

    p 1 2 ( t ) | M Δ ( t ) | ( M ( t ) ) 1 ( M σ ( t ) ) 1 2 k 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq147_HTML.gif,

     
  3. (iii)

    a ( p ( t ) M σ ( t ) ) 1 2 Δ t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq148_HTML.gif,

     

then Eq. (1.1) is in the limit-point case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif.

Proof Suppose that Eq. (1.1) is in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif. By Theorem 3.2, all the solutions of
( p ( t ) y Δ ( t ) ) Δ + q ( t ) y σ ( t ) = 0 , t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ24_HTML.gif
(4.1)
are in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. Let y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif and y 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq150_HTML.gif be the solutions of (4.1) satisfying the following initial conditions:
y 1 ( ρ ( 0 ) ) = p ( ρ ( 0 ) ) y 2 Δ ( ρ ( 0 ) ) = 0 , p ( ρ ( 0 ) ) y 1 Δ ( ρ ( 0 ) ) = y 2 ( ρ ( 0 ) ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ25_HTML.gif
(4.2)
It is evident that y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif and y 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq150_HTML.gif are two linearly independent solutions of (4.1) in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. By Lemma 2.4, W [ y 1 , y 2 ] ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq151_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq5_HTML.gif. Hence, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equam_HTML.gif
It follows from the Hölder inequality and assumption (iii) that
a p ( τ ) ( y 1 Δ ( τ ) ) 2 M σ ( τ ) Δ τ or a p ( τ ) ( y 2 Δ ( τ ) ) 2 M σ ( τ ) Δ τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equan_HTML.gif
are divergent. Suppose
a p ( τ ) ( y 1 Δ ( τ ) ) 2 M σ ( τ ) Δ τ = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equao_HTML.gif
From (4.1) and assumption (i), we have
a t y 1 σ ( τ ) [ p ( τ ) y 1 Δ ( τ ) ] Δ M σ ( τ ) Δ τ = a t q ( τ ) ( y 1 σ ( τ ) ) 2 M σ ( τ ) Δ τ k 1 a t ( y 1 σ ( τ ) ) 2 Δ τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ26_HTML.gif
(4.3)
Applying integration by parts in Lemma 2.2, by (iii) in Lemma 2.1, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ27_HTML.gif
(4.4)
Again applying the Hölder inequality, from condition (ii), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ28_HTML.gif
(4.5)
where
H ( t ) : = a t p ( τ ) ( y 1 Δ ( τ ) ) 2 M σ ( τ ) Δ τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equap_HTML.gif
Since
a ( y 1 σ ( τ ) ) 2 Δ τ > a t ( y 1 σ ( τ ) ) 2 Δ τ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equaq_HTML.gif
it follows from (4.3)-(4.5) that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equar_HTML.gif

It follows from the assumption that H ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq152_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq153_HTML.gif. From the above relation and p ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq139_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq154_HTML.gif, we have that y 1 ( t ) y 1 Δ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq155_HTML.gif is ultimately positive. Therefore, y 1 ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq156_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq153_HTML.gif; and consequently, y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif does not belong to L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. This contradicts the assumption that all the solutions of (4.1) are in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. Then Eq. (4.1) has at least one non-trivial solution outside of L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif. It follows from Theorem 3.2 that Eq. (1.1) is in the limit-point case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif. This completes the proof. □

Remark 4.1 Since R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq157_HTML.gif and N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq158_HTML.gif 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 M ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq159_HTML.gif for t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq140_HTML.gif.

Corollary 4.1 If w ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq28_HTML.gif, p ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq139_HTML.gif, q ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq160_HTML.gif is bounded below in [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif, and ρ ( 0 ) ( p ( t ) ) 1 2 Δ t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq161_HTML.gif, then Eq. (1.1) is in the limit-point case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif.

Theorem 4.2 If
ρ ( 0 ) μ σ ( t ) [ w ( t ) w σ ( t ) ] 1 2 | p σ ( t ) | Δ t = , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ29_HTML.gif
(4.6)

then Eq. (1.1) is in the limit-point case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif.

Proof On the contrary, suppose that Eq. (1.1) is in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif. Let y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif and y 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq150_HTML.gif be two linearly independent solutions of (1.1) in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif satisfying the initial conditions (4.2). By Lemma 2.4, we have
W [ y 1 , y 2 ] ( t ) = W [ y 1 , y 2 ] ( ρ ( 0 ) ) 1 , t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equas_HTML.gif
which, together with (2.1), implies that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equat_HTML.gif
So, we get
| y 1 ( t ) | | y 2 σ ( t ) | + | y 2 ( t ) | | y 1 σ ( t ) | μ ( t ) | p ( t ) | , t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equau_HTML.gif
which implies
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ30_HTML.gif
(4.7)
where y σ 2 ( t ) = y σ ( σ ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq162_HTML.gif. By the Hölder inequality and the assumption that y 1 , y 2 L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq163_HTML.gif, one has
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equav_HTML.gif
Hence, it follows from (4.7) that
ρ ( 0 ) μ σ ( t ) [ w ( t ) w σ ( t ) ] 1 2 | p σ ( t ) | Δ t < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equaw_HTML.gif

which is a contradiction to the assumption (4.6). Therefore, Eq. (1.1) is in the limit-point case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif. This completes the proof. □

Remark 4.2 Let T = N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq164_HTML.gif, 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 f ( t ) = M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq165_HTML.gif and p ( t ) = f ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq166_HTML.gif 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 y , f C r d ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq65_HTML.gif be two non-negative functions on [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif and M be a non-negative constant. If
y ( t ) M + ρ ( 0 ) t f ( τ ) y ( τ ) Δ τ for all t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ31_HTML.gif
(4.8)
then
y ( t ) M e f ( t , ρ ( 0 ) ) for all t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equax_HTML.gif

where e f ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq167_HTML.gif 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 4.2 Let q ( t ) = d ( t ) + e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq168_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq140_HTML.gif and e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq169_HTML.gif be bounded with respect to w ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq170_HTML.gif on [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif; that is, there exists a positive constant M such that
| e ( t ) | M w ( t ) , t [ ρ ( 0 ) , ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ32_HTML.gif
(4.9)
Then Eq. (1.1) is in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif if and only if the equation
( p ( t ) y Δ ( t ) ) Δ + d ( t ) y σ ( t ) = λ w ( t ) y σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ33_HTML.gif
(4.10)

is in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif.

Proof Suppose that (4.10) is in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif. To show that Eq. (1.1) is in the limit-circle case, it suffices to show that each solution (4.1) is in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif by Theorem 3.2.

Let y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif and y 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq150_HTML.gif be two solutions of the equation
( p ( t ) y Δ ( t ) ) Δ + d ( t ) y σ ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ34_HTML.gif
(4.11)

satisfying the initial conditions (4.2). Then y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif, y 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq150_HTML.gif are two linearly independent solutions in L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq105_HTML.gif by Theorem 3.2.

Let y ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq171_HTML.gif be any solution of (4.1). Then
( p ( t ) y Δ ( t ) ) Δ + d ( t ) y σ ( t ) = r ( t ) for all  t [ ρ ( 0 ) , ) T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equay_HTML.gif
where r ( t ) : = e ( t ) y σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq172_HTML.gif. By the variation of constants [[27], Theorem 3.73] there exist two constants α and β such that
y ( t ) = α y 1 ( t ) + β y 2 ( t ) + ρ ( 0 ) t r ( τ ) ( y 1 σ ( τ ) y 2 ( t ) y 2 σ ( τ ) y 1 ( t ) ) Δ τ for all  t [ ρ ( 0 ) , ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equaz_HTML.gif
Hence, replacing t by σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq10_HTML.gif and by (ii) in Lemma 2.2, we get
y σ ( t ) = α y 1 σ ( t ) + β y 2 σ ( t ) + ρ ( 0 ) t r ( τ ) ( y 1 σ ( τ ) y 2 σ ( t ) y 2 σ ( τ ) y 1 σ ( t ) ) Δ τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ35_HTML.gif
(4.12)
From (4.9) and (4.12), we have
| y σ ( t ) | | α | | y 1 σ ( t ) | + | β | | y 2 σ ( t ) | + M ρ ( 0 ) t ( | y 1 σ ( τ ) | | y 2 σ ( t ) | + | y 2 σ ( τ ) | | y 1 σ ( t ) | ) w ( τ ) | y σ ( τ ) | Δ τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equ36_HTML.gif
(4.13)
Since y 1 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq149_HTML.gif, y 2 ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq150_HTML.gif are solutions of Eq. (4.11), which satisfy the initial conditions (4.2), it follows from the existence-uniqueness theorem that | y 1 σ ( t ) | + | y 2 σ ( t ) | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq173_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq174_HTML.gif. Let
y 0 σ ( t ) : = | y σ ( t ) | | y 1 σ ( t ) | + | y 2 σ ( t ) | for all  t [ ρ ( 0 ) , ) T . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equba_HTML.gif
From (4.13), we have
y 0 σ ( t ) | α | | y 1 σ ( t ) | + | β | | y 2 σ ( t ) | | y 1 σ ( t ) | + | y 2 σ ( t ) | + M ρ ( 0 ) t ( | y 1 σ ( τ ) | | y 2 σ ( t ) | + | y 2 σ ( τ ) | | y 1 σ ( t ) | ) w ( τ ) | y σ ( τ ) | | y 1 σ ( t ) | + | y 2 σ ( t ) | Δ τ | α | + | β | + M ρ ( 0 ) t ( | y 1 σ ( τ ) | + | y 2 σ ( τ ) | ) w ( τ ) | y σ ( τ ) | Δ τ = | α | + | β | + M ρ ( 0 ) t ( | y 1 σ ( τ ) | + | y 2 σ ( τ ) | ) 2 w ( τ ) | y 0 σ ( τ ) | Δ τ | α | + | β | + 2 M ρ ( 0 ) t ( | y 1 σ ( τ ) | 2 + | y 2 σ ( τ ) | 2 ) w ( τ ) | y 0 σ ( τ ) | Δ τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equbb_HTML.gif
It follows from (i) of Lemma 2.2 that y 0 σ ( ) C r d ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq175_HTML.gif. By Lemma 4.1, we have
y 0 σ ( t ) ( | α | + | β | ) e [ 2 M w ( | y 1 σ | 2 + | y 2 σ | 2 ) ] ( t , ρ ( 0 ) ) = ( | α | + | β | ) exp [ ρ ( 0 ) t ξ μ ( τ ) ( 2 M w ( τ ) ( | y 1 σ ( τ ) | 2 + | y 2 σ ( τ ) | 2 ) ) Δ τ ] = ( | α | + | β | ) exp [ ρ ( 0 ) t 1 μ ( τ ) Log ( 1 + μ ( τ ) 2 M w ( τ ) ( | y 1 σ ( τ ) | 2 + | y 2 σ ( τ ) | 2 ) ) Δ τ ] ( | α | + | β | ) exp [ ρ ( 0 ) t 2 M w ( τ ) ( | y 1 σ ( τ ) | 2 + | y 2 σ ( τ ) | 2 ) Δ τ ] ( | α | + | β | ) exp [ ρ ( 0 ) 2 M w ( τ ) ( | y 1 σ ( τ ) | 2 + | y 2 σ ( τ ) | 2 ) Δ τ ] = : C < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equbc_HTML.gif

which implies that | y σ ( t ) | C ( | y 1 σ ( t ) | + | y 2 σ ( t ) | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq176_HTML.gif. Hence, y ( ) L w 2 ( ρ ( 0 ) , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq177_HTML.gif; and consequently, Eq. (1.1) is in the limit-circle case at t = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq138_HTML.gif.

On the other hand, using
( p ( t ) y Δ ( t ) ) Δ + d ( t ) y σ ( t ) = ( p ( t ) y Δ ( t ) ) Δ + ( q ( t ) e ( t ) ) y σ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_Equbd_HTML.gif

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 4.3 Let q ( t ) = d ( t ) + e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq168_HTML.gif for all t [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq140_HTML.gif and e ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq169_HTML.gif be bounded with respect to w ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq170_HTML.gif on [ ρ ( 0 ) , ) T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq2_HTML.gif. 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 T = R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-103/MediaObjects/13661_2012_Article_212_IEq178_HTML.gif in Lemma 4.2, then we can directly prove [[31], Theorem 6.1] with the similar method.

Declarations

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).

Authors’ Affiliations

(1)
School of Mathematical Sciences, University of Jinan
(2)
Department of Mathematics, Shandong University

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