Optimal Conditions for Maximum and Antimaximum Principles of the Periodic Solution Problem
© Meirong Zhang. 2010
Received: 18 September 2009
Accepted: 11 April 2010
Published: 17 May 2010
Given a periodic, integrable potential , we will study conditions on so that the operator 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, [1–5]. We also refer the reader to Campos et al.  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 [7–9] (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.
respectively. For the precise meaning of these eigenvalues, see Section 2.2.
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 , eigenvalues and rotation numbers 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 of the periodic solution problem (1.2). We will introduce for each potential two matrixes, and , and two functions, and . They are related with the Poincaré matrix and the Green function , 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 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
we have the following classification.
Lemma 2.1 (see ).
2.2. Eigenvalues, Rotation Numbers, and Oscillation of Solutions
Here the equivalence of (2.13) follows from (2.11).
The connection between eigenvalues and oscillation of solutions is as follows.
Given , consider the parameterized Hill's equations (1.3) where . Then
2.3. Continuous Dependence on Potentials
Associated with the Hill's equation (2.1), we have the objects , , , and . All are determined by the potential . It is a classical result that all of these objects are continuously dependent on when the topology is considered. For the fundamental matrix solutions, this can be stated as follows.
In a recent paper , Zhang has proved that these objects have stronger dependence on potentials . Some statements of these facts are as follows.
Lemma 2.5 (Zhang ).
3. Green Functions and Their Variants
3.1. Green Functions
Formula (3.4) can be found from related references. For completeness, let us give the proof.
3.2. Two Matrixes and Two Functions
From equalities in Lemma 3.2, we have the following statements.
Finally, equality (3.29) follows simply from (3.26) and (3.27).
We distinguish two cases.
Since and , we obtain . Hence and . Moreover, . Thus and is stable-parabolic. In conclusion, for unstable-parabolic case, we have . Now it follows from (3.38) that . As proved before, does not change sign. Moreover, it is easy to see from (3.38) that all zeros of 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
We give only the proof for AMP.
From (3.24), we know that
is a solution of (2.1);
is also a solution of (2.1).
Hence (4.11) is also true for this case.
we have the following statements.
Compared with the defining formulas in (4.9), the novelty of formulas in (4.14) is that when is fixed, is a solution of (2.1), while when is fixed, is also a solution of (2.1). A similar observation is used in  as well.
By Lemma 4.2, Theorem 4.1 can be restated as follows.
4.2. Complete Characterizations of MP and AMP Using Eigenvalues
For any , one has and . See . Thus . In the following let us fix any .
Since , we can use the representation (3.35) for where and are nonzero solutions of (2.1). Since , both have at most one zero. See Lemma 2.3. Hence has at most two zeros. However, as is -periodic, does not have any zero. This proves (4.20).
Since has the same nonzero value at the end-points of the interval , it is easy to see from (4.24) and (4.25) that must have another zero which is different from . Consequently, the solution of (2.1) has at least zeros and . This is impossible because . See Lemma 2.3.
See [15, Lemma ] and . 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].
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 and for all . By Lemma 4.5, if , admits MP. By Lemmas 4.6 and 4.8, admits AMP for . By Lemma 4.9, does not admit MP nor AMP for . 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.
Then we have the following complete characterizations for SAMP.
4.3. Explicit Conditions for AMP
Let us recall some known sufficient conditions for AMP.
Lemma 4.12 (Torres and Zhang ).
Lemma 4.13 (Torres [8, Theorem ]).
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 ]).
with the constants being replaced by more general Sobolev constants .
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 of (2.1) are ( , 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.
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).
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