Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq1_HTML.gif , we will study conditions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq2_HTML.gif so that the operator https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq4_HTML.gif be the circle of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq5_HTML.gif . Given a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq6_HTML.gif -periodic potential https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq7_HTML.gif , which defines a linear differential operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq8_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ1_HTML.gif
(1.1)

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

(i) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq11_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq12_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq13_HTML.gif . Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq14_HTML.gif means that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq15_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq16_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq19_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq20_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq21_HTML.gif .

In terminology of differential equations, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq23_HTML.gif , the following equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ2_HTML.gif
(1.2)

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

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

We say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq29_HTML.gif admits the maximum principle if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq30_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq31_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq33_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq34_HTML.gif admits MP iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq35_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq36_HTML.gif admits AMP iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq37_HTML.gif .

Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq39_HTML.gif are the smallest https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq40_HTML.gif -periodic and the smallest https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq41_HTML.gif -antiperiodic eigenvalues of
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq42_HTML.gif , by introducing the parameterized potentials https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq43_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq45_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq46_HTML.gif admits MP iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq47_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq48_HTML.gif admits AMP iff https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq51_HTML.gif , eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq52_HTML.gif and rotation numbers https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq55_HTML.gif two matrixes, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq57_HTML.gif , and two functions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq58_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq60_HTML.gif and the Green function https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ4_HTML.gif
(2.1)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq64_HTML.gif , https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq66_HTML.gif are solutions satisfying the initial values
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq67_HTML.gif . That is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ7_HTML.gif
(2.4)

the symplectic group of https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ8_HTML.gif
(2.5)
In particular,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq69_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq70_HTML.gif . Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq73_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq74_HTML.gif , or https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq78_HTML.gif , respectively.

By introducing the trace
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq79_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq80_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq81_HTML.gif , respectively. In particular, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq82_HTML.gif implies that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq84_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq85_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq86_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq89_HTML.gif -periodic boundary condition
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq90_HTML.gif -antiperiodic boundary condition
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ13_HTML.gif
(2.10)

such that

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

(ii) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq95_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq96_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq97_HTML.gif is even ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq98_HTML.gif is odd, resp.). Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq99_HTML.gif is void;

(iii) https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq101_HTML.gif potentials. See [12, 16].

Denote
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq102_HTML.gif can be characterized as
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq104_HTML.gif satisfies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq105_HTML.gif . Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ18_HTML.gif
(2.15)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq107_HTML.gif . Such a number https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ19_HTML.gif
(2.16)

where https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq111_HTML.gif . Then

(i)in case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq112_HTML.gif , any nonzero solution https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq115_HTML.gif

(ii)in case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq116_HTML.gif , any nonzero solution https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq121_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq122_HTML.gif . All are determined by the potential https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq124_HTML.gif when the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq125_HTML.gif topology https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq127_HTML.gif , the following mapping:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq128_HTML.gif .

In the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq129_HTML.gif , one has also the weak topology https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq130_HTML.gif which is defined by
https://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 https://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:
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ23_HTML.gif
(2.20)

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

From this lemma, the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq133_HTML.gif is open in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq134_HTML.gif and in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq136_HTML.gif . Then, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq137_HTML.gif , (1.2) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq138_HTML.gif satisfying the https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq140_HTML.gif can be represented as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ24_HTML.gif
(3.1)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ26_HTML.gif
(3.3)

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

Lemma 3.1.

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

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

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

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

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

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq151_HTML.gif can be extended to a continuous https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq153_HTML.gif , https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ29_HTML.gif
(3.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq156_HTML.gif are constants. In order that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq157_HTML.gif is https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq159_HTML.gif satisfies (2.8), that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq160_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ30_HTML.gif
(3.7)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq161_HTML.gif , we know that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ31_HTML.gif
(3.8)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ32_HTML.gif
(3.9)
where https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq163_HTML.gif . Finally, let us show that https://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 https://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)
    https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ34_HTML.gif
(3.11)

Thus https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq167_HTML.gif .

In general, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq168_HTML.gif is not differentiable at the diagonal https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq170_HTML.gif , the following two matrixes:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ35_HTML.gif
(3.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ36_HTML.gif
(3.13)
Note that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq174_HTML.gif can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ37_HTML.gif
(3.14)
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq175_HTML.gif denotes the transpose of matrixes, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq176_HTML.gif is the identity matrix, and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ38_HTML.gif
(3.15)

Some results on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq177_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq180_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq182_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq183_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ39_HTML.gif
(3.16)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ40_HTML.gif
(3.17)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ41_HTML.gif
(3.18)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq184_HTML.gif , then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq185_HTML.gif is nonsingular iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq186_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq187_HTML.gif is nonsingular iff https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq189_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq190_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq191_HTML.gif , respectively.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq192_HTML.gif is https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq194_HTML.gif of (2.1). Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ45_HTML.gif
(3.22)
Hence
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq196_HTML.gif . In case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq197_HTML.gif , the Green function https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ47_HTML.gif
(3.24)

Here https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq201_HTML.gif two functions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ48_HTML.gif
(3.25)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ49_HTML.gif
(3.26)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq202_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq204_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ50_HTML.gif
(3.27)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ51_HTML.gif
(3.28)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq205_HTML.gif . To this end, one has
https://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
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq206_HTML.gif is not true for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq207_HTML.gif . Note that (3.29) asserts that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq208_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq209_HTML.gif -periodic. Some further properties on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq211_HTML.gif . Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq213_HTML.gif . Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq215_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq216_HTML.gif has a constant sign. Moreover,
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq217_HTML.gif is elliptic. We have https://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), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq219_HTML.gif . Hence the symmetric matrix https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq221_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq222_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq223_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq224_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq225_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq226_HTML.gif .

     
  2. (ii)
    Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq227_HTML.gif . We have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq228_HTML.gif . Thus there exists an orthogonal transformation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq229_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ56_HTML.gif
    (3.33)
     
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq230_HTML.gif are eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq231_HTML.gif and satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq232_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ57_HTML.gif
(3.34)
Note that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq234_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ58_HTML.gif
(3.35)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ59_HTML.gif
(3.36)
Note that https://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), https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq237_HTML.gif , say https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq238_HTML.gif . We have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq239_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq240_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ60_HTML.gif
(3.37)
  1. (iii)
    Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq241_HTML.gif . We have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq242_HTML.gif . Then one eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq243_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq244_HTML.gif and another is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq245_HTML.gif . In this case,
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ61_HTML.gif
    (3.38)
     

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

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

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

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq256_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq257_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq258_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq259_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq260_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq261_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq262_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq265_HTML.gif . As proved before, https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq269_HTML.gif with the Green function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq270_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq271_HTML.gif admits MP iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq272_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq274_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq275_HTML.gif . Then, for any https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq277_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq278_HTML.gif . We will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq279_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq281_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq282_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ63_HTML.gif
(4.1)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq283_HTML.gif , we have necessarily
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq284_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq285_HTML.gif ,
https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ67_HTML.gif
(4.5)

which are evident because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq288_HTML.gif and (3.16) shows that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq290_HTML.gif (= https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq291_HTML.gif has only isolated zeros for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq292_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq293_HTML.gif , we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq295_HTML.gif . Then one has some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq296_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq297_HTML.gif . Hence one has some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq298_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ68_HTML.gif
(4.6)
Let us choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq299_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ69_HTML.gif
(4.7)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq300_HTML.gif . However, the corresponding periodic solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq301_HTML.gif of (1.2) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ70_HTML.gif
(4.8)

Hence https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq303_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq304_HTML.gif .

For general https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq305_HTML.gif , denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ72_HTML.gif
(4.10)
Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq306_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq307_HTML.gif is meaningful. We assert that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ73_HTML.gif
(4.11)
In fact, for https://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 https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq310_HTML.gif
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq311_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq312_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ78_HTML.gif
(4.16)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ79_HTML.gif
(4.17)
Consequently,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq314_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq315_HTML.gif are well defined for all potentials https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq316_HTML.gif . Moreover, by (4.15), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq317_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq319_HTML.gif is fixed, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq321_HTML.gif is fixed, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq323_HTML.gif which is zero at those https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq324_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq325_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq326_HTML.gif are in general discontinuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq327_HTML.gif . However, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq328_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq329_HTML.gif are continuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq330_HTML.gif in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq331_HTML.gif topology https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq332_HTML.gif or even in the weak topology https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq334_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq335_HTML.gif admits MP iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq336_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq337_HTML.gif admits AMP iff https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq339_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq340_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq341_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq342_HTML.gif admits MP.

Proof.

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

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq343_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq344_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq345_HTML.gif . See [10]. Thus https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq347_HTML.gif .

Step 1.

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

Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq349_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq350_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq352_HTML.gif , both https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq355_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq356_HTML.gif -periodic, https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq358_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq359_HTML.gif . By introducing
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ84_HTML.gif
(4.22)
one has
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq360_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ86_HTML.gif
(4.24)
This shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq361_HTML.gif . Since https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ87_HTML.gif
(4.25)

Since https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq365_HTML.gif must have another zero https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq366_HTML.gif which is different from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq367_HTML.gif . Consequently, the solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq369_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq370_HTML.gif . This is impossible because https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ88_HTML.gif
(4.26)
We assert that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq372_HTML.gif and consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq373_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq374_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq375_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq376_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq377_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq378_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq379_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ90_HTML.gif
(4.28)
for which
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq382_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq383_HTML.gif . For piecewise continuous and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq386_HTML.gif ] the following result:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ92_HTML.gif
(4.30)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ93_HTML.gif
(4.31)
as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq387_HTML.gif . We conclude
https://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),
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq388_HTML.gif is continuous in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq390_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq392_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq393_HTML.gif has the same sign with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq394_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq395_HTML.gif . By Corollary 4.4, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq397_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq398_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq399_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq400_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq401_HTML.gif admits AMP.

Proof.

For simplicity, denote
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq402_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ97_HTML.gif
(4.35)
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq404_HTML.gif is arbitrary. Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ98_HTML.gif
(4.36)
In the following, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq405_HTML.gif . We have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq406_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq407_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq410_HTML.gif , we have also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq411_HTML.gif . By letting https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq413_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq414_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq416_HTML.gif . Notice that the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq417_HTML.gif of (2.1) has zeros https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq418_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq421_HTML.gif . In particular, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq422_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq424_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ99_HTML.gif
(4.37)
By equality (3.27),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ100_HTML.gif
(4.38)
Thus
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq425_HTML.gif satisfies
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq427_HTML.gif .

Since https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq430_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq431_HTML.gif .

Note that https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq433_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq435_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq437_HTML.gif . Consider the temporal-spatial transformation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ104_HTML.gif
(4.42)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq438_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ105_HTML.gif
(4.43)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq440_HTML.gif is now https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq442_HTML.gif such that the Poincaré matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq443_HTML.gif (of the period https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ106_HTML.gif
(4.44)

See [15, Lemma https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq446_HTML.gif is related with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq447_HTML.gif via
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ108_HTML.gif
(4.46)
Hence
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq448_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq450_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq451_HTML.gif are conjugate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ111_HTML.gif
(4.49)
for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq452_HTML.gif . Denote
https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq454_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ114_HTML.gif
(4.52)

Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq456_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq457_HTML.gif . By (4.47), the transformed potential https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq458_HTML.gif satisfies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq460_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq461_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq462_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq463_HTML.gif admits AMP.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq464_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq465_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq467_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq468_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq469_HTML.gif for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq471_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq472_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq473_HTML.gif and noticing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq474_HTML.gif is continuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq475_HTML.gif , we get
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq476_HTML.gif of (2.1) associated with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq477_HTML.gif . Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq478_HTML.gif the smallest nonnegative zero of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq479_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq480_HTML.gif . Moreover, both https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq481_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq482_HTML.gif are zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq483_HTML.gif because of the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq484_HTML.gif -antiperiodicity of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq487_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq488_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq489_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ116_HTML.gif
(4.54)

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

Lemma 4.9.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq491_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq492_HTML.gif . Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq494_HTML.gif because https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq496_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq497_HTML.gif . Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq499_HTML.gif associated with https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq501_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq502_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq503_HTML.gif . Denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ117_HTML.gif
(4.55)
Then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq506_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq507_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq508_HTML.gif must have some zero https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq509_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq510_HTML.gif , the interior of the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq511_HTML.gif because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq512_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq513_HTML.gif are consecutive zeros of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq514_HTML.gif . As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq515_HTML.gif , one must have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ118_HTML.gif
(4.56)
Thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq516_HTML.gif changes sign near https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq517_HTML.gif . Consequently,
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ120_HTML.gif
(4.58)

Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq520_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq521_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq522_HTML.gif . By Lemma 4.5, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq523_HTML.gif , https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq525_HTML.gif admits AMP for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq526_HTML.gif . By Lemma 4.9, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq529_HTML.gif , we say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq530_HTML.gif admits strong antimaximum principle (SAMP) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq531_HTML.gif admits AMP and, moreover, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq532_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq533_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq534_HTML.gif admits SAMP iff https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq535_HTML.gif iff https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq537_HTML.gif satisfies the following two conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ122_HTML.gif
(4.60)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ123_HTML.gif
(4.61)

Then https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ124_HTML.gif
(4.62)
Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq540_HTML.gif . These constants https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq542_HTML.gif is established in [22]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ125_HTML.gif
(4.63)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ126_HTML.gif
(4.64)
Now such an https://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 https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq547_HTML.gif ]).

Let https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq551_HTML.gif ]).

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

Then https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq555_HTML.gif -Laplacian problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ130_HTML.gif
(4.68)

with the constants https://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:
    https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq557_HTML.gif admits SAMP. For AMP of https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq560_HTML.gif for which https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq562_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq563_HTML.gif be defined by
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ133_HTML.gif
    (4.71)
     
Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq565_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq566_HTML.gif , where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ134_HTML.gif
(4.72)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ135_HTML.gif
(4.73)
we conclude that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq568_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq569_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq570_HTML.gif admits AMP. However, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq571_HTML.gif is large and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq572_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_Equ136_HTML.gif
(4.74)
is also large. Hence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq574_HTML.gif of (2.1) are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F410986/MediaObjects/13661_2009_Article_923_IEq575_HTML.gif ( https://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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