Optimal Conditions for Maximum and Antimaximum Principles of the Periodic Solution Problem

Boundary Value Problems20102010:410986

DOI: 10.1155/2010/410986

Received: 18 September 2009

Accepted: 11 April 2010

Published: 17 May 2010

Abstract

Given a periodic, integrable potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq1_HTML.gif , we will study conditions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq2_HTML.gif so that the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq3_HTML.gif admits the maximum principle or the antimaximum principle with respect to the periodic boundary condition. By exploiting Green functions, eigenvalues, rotation numbers, and their estimates, we will give several optimal conditions.

1. Introduction and Main Results

Maximum Principle (MP) and AntiMaximum Principle (AMP) are fundamental tools in many problems. Generally speaking, criteria for MP and AMP are related to the location of relevant eigenvalues. See, for example, [15]. We also refer the reader to Campos et al. [6] for a recent abstract setting of MP and AMP.

In this paper we are studying criteria of MP and AMP for the periodic solution problem of ODEs. For such a problem, MP and AMP are not only related to periodic eigenvalues, but also to antiperiodic eigenvalues. Though there exist several sufficient conditions of MP and AMP for the periodic solution problem in literature like [79] (for a brief explanation to these conditions, see Section 4.3), an optimal characterization on MP and AMP is not available. The main aim of this paper is to give several optimal criteria of MP and AMP of the periodic solution problem of ODEs which are expressed using eigenvalues, Green functions, or rotation numbers.

Mathematically, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq4_HTML.gif be the circle of length http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq5_HTML.gif . Given a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq6_HTML.gif -periodic potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq7_HTML.gif , which defines a linear differential operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq8_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ1_HTML.gif
(1.1)

we say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq9_HTML.gif admits the antimaximum principle if

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq10_HTML.gif is invertible, and, moreover,

(ii)for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq11_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq12_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq13_HTML.gif . Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq14_HTML.gif means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq15_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq17_HTML.gif on a subset of positive measure.

In an abstract setting, these mean that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq18_HTML.gif is a strictly positive operator with respect to the ordering http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq19_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq20_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq21_HTML.gif .

In terminology of differential equations, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq22_HTML.gif admits AMP if and only if

(i)for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq23_HTML.gif , the following equation:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ2_HTML.gif
(1.2)

has a unique http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq24_HTML.gif -periodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq25_HTML.gif , and, moreover,

(ii)if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq26_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq27_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq28_HTML.gif .

We say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq29_HTML.gif admits the maximum principle if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq30_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq31_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq32_HTML.gif .

Using periodic and antiperiodic eigenvalues of Hill's equations [10, 11], we will obtain the following complete characterizations on MP and AMP.

Theorem 1.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq33_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq34_HTML.gif admits MP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq35_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq36_HTML.gif admits AMP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq37_HTML.gif .

Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq39_HTML.gif are the smallest http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq40_HTML.gif -periodic and the smallest http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq41_HTML.gif -antiperiodic eigenvalues of
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ3_HTML.gif
(1.3)

respectively. For the precise meaning of these eigenvalues, see Section 2.2.

Given an arbitrary potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq42_HTML.gif , by introducing the parameterized potentials http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq44_HTML.gif , Theorem 1.1 can be stated as follows.

Theorem 1.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq45_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq46_HTML.gif admits MP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq47_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq48_HTML.gif admits AMP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq49_HTML.gif .

We will also use Green functions to give complete characterizations on MP and AMP of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq50_HTML.gif . See Theorem 4.1 and Corollary 4.4.

The paper is organized as follows. In Section 2, we will briefly introduce some concepts on Hill's equations [10, 12, 13], including the Poincaré matrixes http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq51_HTML.gif , eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq52_HTML.gif and rotation numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq53_HTML.gif and oscillation of solutions. In Section 3, we will use the Poincaré matrixes and fundamental matrix solutions to give the formula of the Green functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq54_HTML.gif of the periodic solution problem (1.2). We will introduce for each potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq55_HTML.gif two matrixes, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq57_HTML.gif , and two functions, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq59_HTML.gif . They are related with the Poincaré matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq60_HTML.gif and the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq61_HTML.gif , respectively. Some remarkable properties on these new objects will be established.

Section 4 is composed of three subsections. At first, in Section 4.1, we will use the sign of Green functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq62_HTML.gif to establish in Theorem 4.1 and Corollary 4.4 optimal conditions for MP and AMP. Then, in Section 4.2, we will use eigenvalues to give a complete description for the sign of Green functions. The proofs of Theorems 1.1 and 1.2 will be given. One may notice that in the deduction of the sign of Green functions, besides eigenvalues, rotation numbers, and oscillation of solutions, some important estimates on Poincaré matrixes in [10, 12] will be used. Moreover, in the deduction of AMP, a very remarkable reduction for elliptic Hill's equations by Ortega [14, 15] is effectively used to simplify the argument. Note that such a reduction is originally used to deduce the formula for the first Birkhoff twist coefficient of periodic solutions of nonlinear, scalar Newtonian equations. Finally, in Section 4.3, we will outline how the known sufficient conditions on AMP can be easily deduced from Theorem 1.1.

2. Basic Facts on Hill's Equations

2.1. Fundamental Solutions and Poincaré Matrixes

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq63_HTML.gif , let us introduce some basic concepts on the Hill's equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ4_HTML.gif
(2.1)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq65_HTML.gif , be the fundamental solutions of (2.1), that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq66_HTML.gif are solutions satisfying the initial values
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ5_HTML.gif
(2.2)
The fundamental matrix solution of (2.1) is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ6_HTML.gif
(2.3)
The Liouville theorem asserts that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq67_HTML.gif . That is,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ7_HTML.gif
(2.4)

the symplectic group of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq68_HTML.gif .

The Poincaré matrix of (2.1) is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ8_HTML.gif
(2.5)
In particular,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ9_HTML.gif
(2.6)

The Floquet multipliers of (2.1) are eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq69_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq70_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq71_HTML.gif , following from (2.6).

We say that (2.1) is elliptic, hyperbolic or parabolic, respectively, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq73_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq74_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq75_HTML.gif , respectively. We write the sets of those potentials as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq78_HTML.gif , respectively.

By introducing the trace
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ10_HTML.gif
(2.7)

we have the following classification.

Lemma 2.1 (see [10]).

Equaqtion (2.1) is elliptic, hyperbolic, or parabolic, iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq79_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq80_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq81_HTML.gif , respectively. In particular, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq82_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq83_HTML.gif .

Proof.

We need to prove the last conclusion. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq84_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq85_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq87_HTML.gif . These are impossible.

2.2. Eigenvalues, Rotation Numbers, and Oscillation of Solutions

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq88_HTML.gif , consider eigenvalue problems of (1.3) with respect to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq89_HTML.gif -periodic boundary condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ11_HTML.gif
(2.8)
or with respect to the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq90_HTML.gif -antiperiodic boundary condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ12_HTML.gif
(2.9)
It is well known that one has (real) sequences
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ13_HTML.gif
(2.10)

such that

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq91_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq92_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq93_HTML.gif ;

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq94_HTML.gif is an eigenvalue of problem (1.3)–(2.8) (of problem (1.3)–(2.9), resp.) iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq95_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq96_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq97_HTML.gif is even ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq98_HTML.gif is odd, resp.). Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq99_HTML.gif is void;

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq100_HTML.gif is a periodic (an antiperiodic, resp.) eigenvalue of (1.3) iff
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ14_HTML.gif
(2.11)

For these general results, one can refer to [10, 11]. Note that in [10] only piecewise continuous potentials are considered. However, these are also true for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq101_HTML.gif potentials. See [12, 16].

Denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ15_HTML.gif
(2.12)
Using periodic eigenvalues or traces of Poincaré matrixes, the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq102_HTML.gif can be characterized as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ16_HTML.gif
(2.13)

Here the equivalence of (2.13) follows from (2.11).

Let us introduce the rotation number for (2.1). Under the transformation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq103_HTML.gif , we know from (2.1) that the argument http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq104_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ17_HTML.gif
(2.14)

Definition 2.2 (see [1719]).

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq105_HTML.gif . Define
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ18_HTML.gif
(2.15)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq106_HTML.gif is any solution of (2.14). The limit (2.15) does exist and is independent of the choice of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq107_HTML.gif . Such a number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq108_HTML.gif is called the rotation number of (2.1). An alternative definition for (2.15) is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ19_HTML.gif
(2.16)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq109_HTML.gif is any nonzero solution of (2.1).

The connection between eigenvalues and oscillation of solutions is as follows.

Lemma 2.3.

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq110_HTML.gif , consider the parameterized Hill's equations (1.3) where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq111_HTML.gif . Then

(i)in case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq112_HTML.gif , any nonzero solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq113_HTML.gif of (1.3) is nonoscillatory. More precisely, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq114_HTML.gif has at most one zero in the whole line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq115_HTML.gif

(ii)in case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq116_HTML.gif , any nonzero solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq117_HTML.gif of (1.3) is oscillatory. More precisely, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq118_HTML.gif has infinitely many zeros.

2.3. Continuous Dependence on Potentials

Associated with the Hill's equation (2.1), we have the objects http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq120_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq121_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq122_HTML.gif . All are determined by the potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq123_HTML.gif . It is a classical result that all of these objects are continuously dependent on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq124_HTML.gif when the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq125_HTML.gif topology http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq126_HTML.gif is considered. For the fundamental matrix solutions, this can be stated as follows.

Lemma 2.4 (see [12, 13]).

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq127_HTML.gif , the following mapping:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ20_HTML.gif
(2.17)

is continuously Frechét differentiable. Moreover, the Frechét derivatives can be expressed using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq128_HTML.gif .

In the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq129_HTML.gif , one has also the weak topology http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq130_HTML.gif which is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ21_HTML.gif
(2.18)

In a recent paper [20], Zhang has proved that these objects have stronger dependence on potentials http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq131_HTML.gif . Some statements of these facts are as follows.

Lemma 2.5 (Zhang [20]).

The following mapping is continuous:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ22_HTML.gif
(2.19)
Moreover, the following (nonlinear) functionals:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ23_HTML.gif
(2.20)

are also continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq132_HTML.gif .

From this lemma, the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq133_HTML.gif is open in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq134_HTML.gif and in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq135_HTML.gif .

3. Green Functions and Their Variants

3.1. Green Functions

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq136_HTML.gif . Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq137_HTML.gif , (1.2) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq138_HTML.gif satisfying the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq139_HTML.gif -periodic boundary condition (2.8). From the Fredholm principle, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq140_HTML.gif can be represented as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ24_HTML.gif
(3.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ25_HTML.gif
(3.2)
is the so-called Green function of the periodic solution problem (1.2)–(2.8). Another definition of the Green function is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ26_HTML.gif
(3.3)

Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq141_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq142_HTML.gif -periodic unit Dirac measure located at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq143_HTML.gif . The Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq144_HTML.gif can be expressed using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq145_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq146_HTML.gif as follows.

Lemma 3.1.

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq147_HTML.gif , we have the following results.

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq148_HTML.gif is given by

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ27_HTML.gif
(3.4)

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq149_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq150_HTML.gif and is symmetric

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ28_HTML.gif
(3.5)

Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq151_HTML.gif can be extended to a continuous http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq152_HTML.gif -periodic function in both arguments, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq153_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq154_HTML.gif .

Proof.
  1. (i)

    Formula (3.4) can be found from related references. For completeness, let us give the proof.

     
Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq155_HTML.gif . By the constant-of-variant formula, solutions of (1.2) are given by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ29_HTML.gif
(3.6)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq156_HTML.gif are constants. In order that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq157_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq158_HTML.gif -periodic, it is necessary and sufficient that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq159_HTML.gif satisfies (2.8), that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq160_HTML.gif satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ30_HTML.gif
(3.7)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq161_HTML.gif , we know that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ31_HTML.gif
(3.8)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ32_HTML.gif
(3.9)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq162_HTML.gif has the form of (3.4).
  1. (ii)
    From formula (3.4), the symmetry (3.5) is obvious. Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq163_HTML.gif . Finally, let us show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq164_HTML.gif can be extended to a continuous function on the torus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq165_HTML.gif . By using (2.2), (2.5), and (2.6), one has from (3.4)
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ33_HTML.gif
    (3.10)
     
By the symmetry (3.5), one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ34_HTML.gif
(3.11)

Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq166_HTML.gif can be understood as a function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq167_HTML.gif .

In general, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq168_HTML.gif is not differentiable at the diagonal http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq169_HTML.gif .

3.2. Two Matrixes and Two Functions

Let us introduce, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq170_HTML.gif , the following two matrixes:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ35_HTML.gif
(3.12)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ36_HTML.gif
(3.13)
Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq171_HTML.gif is a symmetric matrix. Using the Poincaré matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq174_HTML.gif can be rewritten as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ37_HTML.gif
(3.14)
Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq175_HTML.gif denotes the transpose of matrixes, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq176_HTML.gif is the identity matrix, and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ38_HTML.gif
(3.15)

Some results on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq178_HTML.gif and their connections with the Poincaré matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq179_HTML.gif are as follows. All of them can be verified directly.

Lemma 3.2.

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq180_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq182_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq183_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ39_HTML.gif
(3.16)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ40_HTML.gif
(3.17)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ41_HTML.gif
(3.18)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ42_HTML.gif
(3.19)

From equalities in Lemma 3.2, we have the following statements.

Lemma 3.3.

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq184_HTML.gif , then

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq185_HTML.gif is nonsingular iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq186_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq187_HTML.gif is nonsingular iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq188_HTML.gif ;

(ii)Equation (2.1) is elliptic, hyperbolic, or parabolic, iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq190_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq191_HTML.gif , respectively.

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq192_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq193_HTML.gif -periodic, one has the following equality for the fundamental matrix solution
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ43_HTML.gif
(3.20)
Let us introduce the vector-valued function
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ44_HTML.gif
(3.21)
which is composed by the fundamental solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq194_HTML.gif of (2.1). Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ45_HTML.gif
(3.22)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ46_HTML.gif
(3.23)
In the following, we use http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq195_HTML.gif to denote the Euclidean inner product on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq196_HTML.gif . In case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq197_HTML.gif , the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq198_HTML.gif in (3.4) can be rewritten as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ47_HTML.gif
(3.24)

Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq199_HTML.gif is as in (3.12). Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq200_HTML.gif .

Suggested by (3.24), let us introduce for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq201_HTML.gif two functions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ48_HTML.gif
(3.25)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ49_HTML.gif
(3.26)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq203_HTML.gif are as in (3.12) and (3.13). Note that these functions are well defined on the whole plane and the whole line, respectively. Some properties are as follows.

Lemma 3.4.

For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq204_HTML.gif , one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ50_HTML.gif
(3.27)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ51_HTML.gif
(3.28)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ52_HTML.gif
(3.29)

Proof.

We need only to verify (3.27) for the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq205_HTML.gif . To this end, one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ53_HTML.gif
(3.30)
For (3.28), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ54_HTML.gif
(3.31)

Finally, equality (3.29) follows simply from (3.26) and (3.27).

We remark that, in general, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq206_HTML.gif is not true for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq207_HTML.gif . Note that (3.29) asserts that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq208_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq209_HTML.gif -periodic. Some further properties on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq210_HTML.gif are as follows.

Lemma 3.5.
  1. (i)

    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq211_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq212_HTML.gif does not have any zero and therefore does not change sign.

     
  2. (ii)

    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq213_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq214_HTML.gif has only nondegenerate zeros, if they exist.

     
  3. (iii)
    Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq215_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq216_HTML.gif has a constant sign. Moreover,
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ55_HTML.gif
    (3.32)
     
Proof.
  1. (i)

    Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq217_HTML.gif is elliptic. We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq218_HTML.gif from Lemma 2.1. By (3.17), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq219_HTML.gif . Hence the symmetric matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq220_HTML.gif is either positive definite or negative definite, according to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq221_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq222_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq223_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq224_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq225_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq226_HTML.gif .

     
  2. (ii)
    Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq227_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq228_HTML.gif . Thus there exists an orthogonal transformation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq229_HTML.gif such that
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ56_HTML.gif
    (3.33)
     
Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq230_HTML.gif are eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq231_HTML.gif and satisfy http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq232_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ57_HTML.gif
(3.34)
Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq233_HTML.gif is also a system of fundamental solutions of (2.1). As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq234_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ58_HTML.gif
(3.35)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ59_HTML.gif
(3.36)
Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq235_HTML.gif is a linearly independent system of solutions of (2.1). From (3.35), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq236_HTML.gif has only nondegenerate zeros, if they exist. In fact, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq237_HTML.gif , say http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq238_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq240_HTML.gif . Thus
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ60_HTML.gif
(3.37)
  1. (iii)
    Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq241_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq242_HTML.gif . Then one eigenvalue of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq243_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq244_HTML.gif and another is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq245_HTML.gif . In this case,
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ61_HTML.gif
    (3.38)
     

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq246_HTML.gif is a nonzero solution of (2.1). This shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq247_HTML.gif does not change sign.

We distinguish two cases.

(i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq248_HTML.gif is stable-parabolic, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq249_HTML.gif . In this case, one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq250_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq251_HTML.gif .

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq252_HTML.gif is unstable-parabolic, that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq253_HTML.gif . In this case, we assert that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq254_HTML.gif .

Otherwise, assume http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq255_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ62_HTML.gif
(3.39)

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq256_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq257_HTML.gif , we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq258_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq259_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq260_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq261_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq262_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq263_HTML.gif is stable-parabolic. In conclusion, for unstable-parabolic case, we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq264_HTML.gif . Now it follows from (3.38) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq265_HTML.gif . As proved before, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq266_HTML.gif does not change sign. Moreover, it is easy to see from (3.38) that all zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq267_HTML.gif must be degenerate, if they exist.

From these, (3.32) is clear.

4. Optimal Conditions for MP and AMP

4.1. Complete Characterizations of MP and AMP Using Green Functions

Using Green functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq268_HTML.gif , we have the following characterizations on MP and AMP.

Theorem 4.1.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq269_HTML.gif with the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq270_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq271_HTML.gif admits MP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq272_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq273_HTML.gif

Proof.

We give only the proof for AMP.

The sufficiency is as follows. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq274_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq275_HTML.gif . Then, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq276_HTML.gif , it is easy to see from (3.1) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq277_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq278_HTML.gif . We will show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq279_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq280_HTML.gif and consequently (1.2) admits AMP.

Otherwise, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq281_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq282_HTML.gif , that is,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ63_HTML.gif
(4.1)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq283_HTML.gif , we have necessarily
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ64_HTML.gif
(4.2)

From (3.24), we know that

(i)on the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq284_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ65_HTML.gif
(4.3)

is a solution of (2.1);

(ii)on the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq285_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ66_HTML.gif
(4.4)

is also a solution of (2.1).

We assert that these solutions are nonzero when the corresponding intervals are nontrivial. As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq286_HTML.gif is composed of two linearly independent solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq287_HTML.gif , the nontriviality of these solutions is the same as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ67_HTML.gif
(4.5)

which are evident because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq288_HTML.gif and (3.16) shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq289_HTML.gif .

From the above assertion, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq290_HTML.gif (= http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq291_HTML.gif has only isolated zeros for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq292_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq293_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq294_HTML.gif , a contradiction with (4.2).

For the necessity, let us assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq295_HTML.gif . Then one has some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq296_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq297_HTML.gif . Hence one has some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq298_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ68_HTML.gif
(4.6)
Let us choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq299_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ69_HTML.gif
(4.7)
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq300_HTML.gif . However, the corresponding periodic solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq301_HTML.gif of (1.2) satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ70_HTML.gif
(4.8)

Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq302_HTML.gif does not admit AMP.

In order to apply Theorem 4.1, it is important to compute the signs of the following nonlinear functionals of potentials:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ71_HTML.gif
(4.9)

To this end, let us establish some relation between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq303_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq304_HTML.gif .

For general http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq305_HTML.gif , denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ72_HTML.gif
(4.10)
Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq306_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq307_HTML.gif is meaningful. We assert that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ73_HTML.gif
(4.11)
In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq308_HTML.gif , the first case of (4.11) follows immediately from the defining equalities (3.24), (3.25), and (4.10). On the other hand, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq309_HTML.gif , from the second case of (3.24), one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ74_HTML.gif
(4.12)

Hence (4.11) is also true for this case.

By introducing the domain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ75_HTML.gif
(4.13)
and the following nonlinear functionals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq310_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ76_HTML.gif
(4.14)

we have the following statements.

Lemma 4.2.

There hold, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq311_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ77_HTML.gif
(4.15)

Proof.

We only prove the first equality of (4.15) because the second one is similar. By (4.11), for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq312_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ78_HTML.gif
(4.16)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ79_HTML.gif
(4.17)
Consequently,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ80_HTML.gif
(4.18)

This is just (4.15) because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq313_HTML.gif .

Remark 4.3.
  1. (i)

    The functionals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq314_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq315_HTML.gif are well defined for all potentials http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq316_HTML.gif . Moreover, by (4.15), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq317_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq318_HTML.gif have the same signs with the functionals in (4.9).

     
  2. (ii)

    Compared with the defining formulas in (4.9), the novelty of formulas in (4.14) is that when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq319_HTML.gif is fixed, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq320_HTML.gif is a solution of (2.1), while when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq321_HTML.gif is fixed, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq322_HTML.gif is also a solution of (2.1). A similar observation is used in [8] as well.

     
  3. (iii)

    Due to the factor http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq323_HTML.gif which is zero at those http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq324_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq325_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq326_HTML.gif are in general discontinuous at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq327_HTML.gif . However, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq328_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq329_HTML.gif are continuous at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq330_HTML.gif in the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq331_HTML.gif topology http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq332_HTML.gif or even in the weak topology http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq333_HTML.gif . See Lemmas 2.4 and 2.5.

     

By Lemma 4.2, Theorem 4.1 can be restated as follows.

Corollary 4.4.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq334_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq335_HTML.gif admits MP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq336_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq337_HTML.gif admits AMP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq338_HTML.gif .

4.2. Complete Characterizations of MP and AMP Using Eigenvalues

Lemma 4.5.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq339_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq340_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq341_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq342_HTML.gif admits MP.

Proof.

For simplicity, denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ81_HTML.gif
(4.19)

For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq343_HTML.gif , one has http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq344_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq345_HTML.gif . See [10]. Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq346_HTML.gif . In the following let us fix any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq347_HTML.gif .

Step 1.

We assert that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ82_HTML.gif
(4.20)

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq348_HTML.gif , we can use the representation (3.35) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq349_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq350_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq351_HTML.gif are nonzero solutions of (2.1). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq352_HTML.gif , both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq353_HTML.gif have at most one zero. See Lemma 2.3. Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq354_HTML.gif has at most two zeros. However, as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq355_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq356_HTML.gif -periodic, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq357_HTML.gif does not have any zero. This proves (4.20).

Step 2.

We assert that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ83_HTML.gif
(4.21)
If (4.21) is false, there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq358_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq359_HTML.gif . By introducing
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ84_HTML.gif
(4.22)
one has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ85_HTML.gif
(4.23)
We know from (3.28) and (4.20) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq360_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ86_HTML.gif
(4.24)
This shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq361_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq362_HTML.gif is a nonzero solution of (2.1), (4.23) implies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ87_HTML.gif
(4.25)

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq363_HTML.gif has the same nonzero value at the end-points of the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq364_HTML.gif , it is easy to see from (4.24) and (4.25) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq365_HTML.gif must have another zero http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq366_HTML.gif which is different from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq367_HTML.gif . Consequently, the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq368_HTML.gif of (2.1) has at least zeros http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq369_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq370_HTML.gif . This is impossible because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq371_HTML.gif . See Lemma 2.3.

Step 3.

Let us notice that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ88_HTML.gif
(4.26)
We assert that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ89_HTML.gif
(4.27)
To prove (4.27), let us fix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq372_HTML.gif and consider http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq373_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq374_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq375_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq376_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq377_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq378_HTML.gif . When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq379_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq380_HTML.gif can be estimated. The basic idea is to consider (1.3) as a perturbation of the equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ90_HTML.gif
(4.28)
for which
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ91_HTML.gif
(4.29)
It is well known that the difference http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq381_HTML.gif can be controlled by the norm of the potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq382_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq383_HTML.gif . For piecewise continuous and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq384_HTML.gif potentials, see [10] and [12], respectively. Similar estimates are also true for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq385_HTML.gif potentials. In fact, these can be generalized to Hill's equations with coefficients being measures [16]. We quote from [12, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq386_HTML.gif ] the following result:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ92_HTML.gif
(4.30)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ93_HTML.gif
(4.31)
as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq387_HTML.gif . We conclude
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ94_HTML.gif
(4.32)
On the other hand, by (4.21) and (4.26),
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ95_HTML.gif
(4.33)

Moreover, it follows from Lemma 2.4 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq388_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq389_HTML.gif . Thus (4.27) follows simply from (4.32) and (4.33).

Step 4.

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq390_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq391_HTML.gif . It follows from (4.21), (4.26), and (4.27) that, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq392_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq393_HTML.gif has the same sign with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq394_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq395_HTML.gif . By Corollary 4.4, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq396_HTML.gif admits MP.

Lemma 4.6.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq397_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq398_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq399_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq400_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq401_HTML.gif admits AMP.

Proof.

For simplicity, denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ96_HTML.gif
(4.34)
Recall from [11] that eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq402_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq403_HTML.gif can be characterized using rotation numbers by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ97_HTML.gif
(4.35)
Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq404_HTML.gif is arbitrary. Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ98_HTML.gif
(4.36)
In the following, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq405_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq406_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq407_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq408_HTML.gif . Now we argue as in the proof of Lemma 4.5. In this case, result (4.20) can be obtained from Lemma 3.5(i) because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq409_HTML.gif . If (4.21) is false at some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq410_HTML.gif , we have also http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq411_HTML.gif . By letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq412_HTML.gif be as in (4.22), one has also some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq413_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq414_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq415_HTML.gif . With loss of generality, let us assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq416_HTML.gif . Notice that the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq417_HTML.gif of (2.1) has zeros http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq418_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq419_HTML.gif . By the Sturm comparison theorem, any nonzero solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq420_HTML.gif of (2.1) has at least one zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq421_HTML.gif . In particular, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq422_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq423_HTML.gif is a solution of (2.1). Hence there exists some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq424_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ99_HTML.gif
(4.37)
By equality (3.27),
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ100_HTML.gif
(4.38)
Thus
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ101_HTML.gif
(4.39)
From these, the distribution of zeros of the specific solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq425_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ102_HTML.gif
(4.40)
By definition (2.16) for the rotation number, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ103_HTML.gif
(4.41)

a contradiction with the characterization of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq426_HTML.gif . Thus (4.21) is also true for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq427_HTML.gif .

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq428_HTML.gif , we have from (4.21) and (4.26) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq429_HTML.gif , because we will prove in Lemma 4.7 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq430_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq431_HTML.gif .

Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq432_HTML.gif is the set of potentials which are in the first ellipticity zone. By Lemmas 2.1 or 3.5, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq433_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq434_HTML.gif . It seems that there are several ways to deduce that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq435_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq436_HTML.gif . However, some remarkable result on elliptic Hill's equations by Ortega [14, 15] can simplify the argument. Let us describe the result. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq437_HTML.gif . Consider the temporal-spatial transformation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ104_HTML.gif
(4.42)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq438_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq439_HTML.gif . Then (2.1) is transformed into a new Hill's equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ105_HTML.gif
(4.43)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq440_HTML.gif is now http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq441_HTML.gif periodic. The result of Ortega shows that it is always possible to choose some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq442_HTML.gif such that the Poincaré matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq443_HTML.gif (of the period http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq444_HTML.gif ) of (4.43) is a rigid rotation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ106_HTML.gif
(4.44)

See [15, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq445_HTML.gif ] and [21]. We remark that such a result is very important to study the twist character and Lyapunov stability of periodic solutions of nonlinear Newtonian equations and planar Hamiltonian systems. See, for example, [14, 15, 21].

Note that the transformation (4.42) does not change rotation numbers. Recall that the polar coordinates to define rotation numbers are
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ107_HTML.gif
(4.45)
We see from (4.44) that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq446_HTML.gif is related with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq447_HTML.gif via
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ108_HTML.gif
(4.46)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ109_HTML.gif
(4.47)

Lemma 4.7.

We assert that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ110_HTML.gif
(4.48)

Proof.

We first prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq448_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq449_HTML.gif , is invariant under transformations (4.42). In fact, it is well known that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq450_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq451_HTML.gif are conjugate
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ111_HTML.gif
(4.49)
for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq452_HTML.gif . Denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ112_HTML.gif
(4.50)
From (4.49), one has the explicit relation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ113_HTML.gif
(4.51)
Note that the quadratic form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq453_HTML.gif is definite. See the proof of Lemma 3.5(i). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq454_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ114_HTML.gif
(4.52)

Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq455_HTML.gif is invariant under transformations (4.42).

Now (4.48) can be obtained as follows. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq456_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq457_HTML.gif . By (4.47), the transformed potential http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq458_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq459_HTML.gif . By the invariance, we have the desired result (4.48).

Lemma 4.8.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq460_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq461_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq462_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq463_HTML.gif admits AMP.

Proof.

Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq464_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq465_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq466_HTML.gif . See (2.11). Moreover, by (2.10), we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq467_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq468_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq469_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq470_HTML.gif . We know from Lemma 4.6 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq471_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq472_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq473_HTML.gif and noticing that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq474_HTML.gif is continuous at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq475_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ115_HTML.gif
(4.53)
On the other hand, let us take an antiperiodic eigen function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq476_HTML.gif of (2.1) associated with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq477_HTML.gif . Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq478_HTML.gif the smallest nonnegative zero of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq479_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq480_HTML.gif . Moreover, both http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq481_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq482_HTML.gif are zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq483_HTML.gif because of the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq484_HTML.gif -antiperiodicity of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq485_HTML.gif . By the Sturm comparison theorem, the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq486_HTML.gif of (2.1) must have some zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq487_HTML.gif . Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq488_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq489_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ116_HTML.gif
(4.54)

In conclusion we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq490_HTML.gif .

Lemma 4.9.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq491_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq492_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq493_HTML.gif does not admit neither MP nor AMP.

Proof.

We need not to consider the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq494_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq495_HTML.gif is not invertible.

In the following let us assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq496_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq497_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq498_HTML.gif . The following is a modification of the last part of the proof of Lemma 4.8.

Let us take an antiperiodic eigenfunction http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq499_HTML.gif associated with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq500_HTML.gif . Then the set of all zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq501_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq502_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq503_HTML.gif . Denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ117_HTML.gif
(4.55)
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq504_HTML.gif is a nonzero solution of (2.1). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq505_HTML.gif , by applying the Sturm comparison theorem to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq506_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq507_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq508_HTML.gif must have some zero http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq509_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq510_HTML.gif , the interior of the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq511_HTML.gif because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq512_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq513_HTML.gif are consecutive zeros of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq514_HTML.gif . As http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq515_HTML.gif , one must have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ118_HTML.gif
(4.56)
Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq516_HTML.gif changes sign near http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq517_HTML.gif . Consequently,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ119_HTML.gif
(4.57)
Now Corollary 4.4 shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq518_HTML.gif does not admit AMP. We have also
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ120_HTML.gif
(4.58)

Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq519_HTML.gif does not admit MP.

Due to the ordering (2.10) of eigenvalues, the statements in Theorems 1.1 and 1.2 are equivalent. Now let us give the proof of Theorem 1.2. Recall that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq520_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq521_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq522_HTML.gif . By Lemma 4.5, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq523_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq524_HTML.gif admits MP. By Lemmas 4.6 and 4.8, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq525_HTML.gif admits AMP for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq526_HTML.gif . By Lemma 4.9, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq527_HTML.gif does not admit MP nor AMP for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq528_HTML.gif . Using the ordering (2.10) for eigenvalues, we complete the proof of Theorem 1.2.

From Lemmas 4.5, 4.6, 4.8, and 4.9, the sign of Green functions is clear in all cases.

Definition 4.10.

Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq529_HTML.gif , we say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq530_HTML.gif admits strong antimaximum principle (SAMP) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq531_HTML.gif admits AMP and, moreover, there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq532_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ121_HTML.gif
(4.59)

Then we have the following complete characterizations for SAMP.

Theorem 4.11.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq533_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq534_HTML.gif admits SAMP iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq535_HTML.gif iff http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq536_HTML.gif .

4.3. Explicit Conditions for AMP

Let us recall some known sufficient conditions for AMP.

Lemma 4.12 (Torres and Zhang [9]).

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq537_HTML.gif satisfies the following two conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ122_HTML.gif
(4.60)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ123_HTML.gif
(4.61)

Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq538_HTML.gif admits AMP.

In the proof there, the positiveness condition (4.60) is technically used extensively. Some optimal estimates on condition (4.61) can be found in Zhang and Li [22]. For an exponent http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq539_HTML.gif , let us introduce the following Sobolev constant:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ124_HTML.gif
(4.62)
Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq540_HTML.gif . These constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq541_HTML.gif can be explicitly expressed using the Gamma function of Euler. The following lower bound for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq542_HTML.gif is established in [22]:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ125_HTML.gif
(4.63)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq543_HTML.gif . Hence one sufficient condition for (4.61) is
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ126_HTML.gif
(4.64)
Now such an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq544_HTML.gif condition (4.64) is quite standard in literature like [8, 23], because in case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq545_HTML.gif , (4.64) reads as the classical condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ127_HTML.gif
(4.65)

In order to overcome the technical assumption (4.60) on positiveness of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq546_HTML.gif , one observation is as follows.

Lemma 4.13 (Torres [8, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq547_HTML.gif ]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq548_HTML.gif . Suppose that all gaps of consecutive zeros of all nonzero solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq549_HTML.gif of (2.1) are strictly greater than the period 1. Then the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq550_HTML.gif has a constant sign.

By Theorem 4.1 of this paper, one sees that the hypothesis in Lemma 4.13 on solutions of (2.1) can yield MP or AMP. Combining ideas from [8, 9, 22], Cabada and Cid have overcome the positiveness condition (4.60) to obtain the following criteria.

Lemma 4.14 (Cabada and Cid [7, Theorem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq551_HTML.gif ]).

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq552_HTML.gif satisfies the following two conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ128_HTML.gif
(4.66)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ129_HTML.gif
(4.67)

Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq553_HTML.gif admits AMP.

Very recently, Cabada et al. [24, 25] have generalized criteria (4.66)-(4.67) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq554_HTML.gif to AMP of the periodic solutions of the so-called http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq555_HTML.gif -Laplacian problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ130_HTML.gif
(4.68)

with the constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq556_HTML.gif being replaced by more general Sobolev constants [26].

We end the paper with some remarks.
  1. (i)
    Recall the following trivial upper bound:
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ131_HTML.gif
    (4.69)
     
See, for example, [26]. Criteria (4.66)-(4.67) can be deduced from Theorem 1.1 with the help of estimates (4.63) and (4.69). In fact, by Theorem 4.11, conditions (4.66) and (4.67) guarantee that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq557_HTML.gif admits SAMP. For AMP of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq558_HTML.gif , condition (4.67) can be improved as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ132_HTML.gif
(4.70)
Theorem 1.1 shows that condition (4.61) is optimal, while the complete generalization of condition (4.60) is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq559_HTML.gif .
  1. (ii)
    It is also possible to construct many potentials http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq560_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq561_HTML.gif admits AMP, while (4.70) is violated. For example, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq562_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq563_HTML.gif be defined by
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ133_HTML.gif
    (4.71)
     
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq564_HTML.gif and the Riemann-Lebesgue lemma shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq565_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq566_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq567_HTML.gif is arbitrarily fixed. In particular, it follows from Lemma 2.5 that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ134_HTML.gif
(4.72)
Since
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ135_HTML.gif
(4.73)
we conclude that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq568_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq569_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq570_HTML.gif admits AMP. However, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq571_HTML.gif is large and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq572_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ136_HTML.gif
(4.74)
is also large. Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq573_HTML.gif does not satisfy (4.70).
  1. (iii)

    Notice that the lower bound (4.63) has actually shown that, under (4.67) ((4.70), resp.), the gaps of consecutive zeros of all nonzero solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq574_HTML.gif of (2.1) are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq575_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq576_HTML.gif , resp.). However, for those potentials as in Theorem 1.1, zeros of solutions of (2.1) may not be so evenly distributed. This is the difference between the sufficient conditions in this subsection and our optimal conditions given in Theorem 1.1.

     

Declarations

Acknowledgments

The author is supported by the Major State Basic Research Development Program (973 Program) of China (no. 2006CB805903), the Doctoral Fund of Ministry of Education of China (no. 20090002110079), the Program of Introducing Talents of Discipline to Universities (111 Program) of Ministry of Education and State Administration of Foreign Experts Affairs of China (2007), and the National Natural Science Foundation of China (no. 10531010).

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Tsinghua University
(2)
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University

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© Meirong Zhang. 2010

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