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.
we say that admits the antimaximum principle if
(i) is invertible, and, moreover,
(ii)for any with , one has . Here means that a.e. and on a subset of positive measure.
In an abstract setting, these mean that is a strictly positive operator with respect to the ordering defined by a.e. .
In terminology of differential equations, admits AMP if and only if
has a unique -periodic solution , and, moreover,
(ii)if , one has for all .
We say that admits the maximum principle if for all such that .
Let . Then admits MP iff , and admits AMP iff .
respectively. For the precise meaning of these eigenvalues, see Section 2.2.
Given an arbitrary potential , by introducing the parameterized potentials , , Theorem 1.1 can be stated as follows.
Let . Then admits MP iff , and admits AMP iff .
We will also use Green functions to give complete characterizations on MP and AMP of . 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 , 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
the symplectic group of .
The Floquet multipliers of (2.1) are eigenvalues of . Then , following from (2.6).
We say that (2.1) is elliptic, hyperbolic or parabolic, respectively, if and , , or , respectively. We write the sets of those potentials as , and , respectively.
we have the following classification.
Lemma 2.1 (see ).
Equaqtion (2.1) is elliptic, hyperbolic, or parabolic, iff , , or , respectively. In particular, implies that .
We need to prove the last conclusion. Suppose that . If , we have and . These are impossible.
2.2. Eigenvalues, Rotation Numbers, and Oscillation of Solutions
(i) and as ;
(ii) is an eigenvalue of problem (1.3)–(2.8) (of problem (1.3)–(2.9), resp.) iff or where is even ( is odd, resp.). Here is void;
Here the equivalence of (2.13) follows from (2.11).
where is any nonzero solution of (2.1).
The connection between eigenvalues and oscillation of solutions is as follows.
Given , consider the parameterized Hill's equations (1.3) where . Then
(i)in case , any nonzero solution of (1.3) is nonoscillatory. More precisely, has at most one zero in the whole line
(ii)in case , any nonzero solution of (1.3) is oscillatory. More precisely, has infinitely many zeros.
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.
is continuously Frechét differentiable. Moreover, the Frechét derivatives can be expressed using .
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 ).
are also continuous in .
From this lemma, the set is open in and in .
3. Green Functions and Their Variants
3.1. Green Functions
Here is the -periodic unit Dirac measure located at . The Green function can be expressed using and as follows.
Given , we have the following results.
(i) is given by
(ii) is continuous in and is symmetric
Moreover, can be extended to a continuous -periodic function in both arguments, that is, , .
Formula (3.4) can be found from related references. For completeness, let us give the proof.
- (ii)From formula (3.4), the symmetry (3.5) is obvious. Moreover, . Finally, let us show that can be extended to a continuous function on the torus . By using (2.2), (2.5), and (2.6), one has from (3.4)(3.10)
Thus can be understood as a function on .
In general, is not differentiable at the diagonal .
3.2. Two Matrixes and Two Functions
Some results on and and their connections with the Poincaré matrix are as follows. All of them can be verified directly.
From equalities in Lemma 3.2, we have the following statements.
Given , then
(i) is nonsingular iff , and is nonsingular iff ;
(ii)Equation (2.1) is elliptic, hyperbolic, or parabolic, iff , , or , respectively.
Here is as in (3.12). Note that .
where and 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.
Finally, equality (3.29) follows simply from (3.26) and (3.27).
We remark that, in general, is not true for . Note that (3.29) asserts that is -periodic. Some further properties on are as follows.
Let . Then does not have any zero and therefore does not change sign.
Let . Then has only nondegenerate zeros, if they exist.
- (iii)Let . Then has a constant sign. Moreover,(3.32)
Suppose that is elliptic. We have from Lemma 2.1. By (3.17), . Hence the symmetric matrix is either positive definite or negative definite, according to or . Since for all , we know that on .
- (ii)Suppose that . We have . Thus there exists an orthogonal transformation such that(3.33)
- (iii)Suppose that . We have . Then one eigenvalue of is and another is . In this case,(3.38)
where is a nonzero solution of (2.1). This shows that does not change sign.
We distinguish two cases.
(i) is stable-parabolic, that is, . In this case, one has and .
(ii) is unstable-parabolic, that is, . In this case, we assert that .
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
Using Green functions , we have the following characterizations on MP and AMP.
Let with the Green function . Then admits MP iff and
We give only the proof for AMP.
The sufficiency is as follows. Suppose that satisfies . Then, for any , it is easy to see from (3.1) that for all . We will show that for all and consequently (1.2) admits AMP.
From (3.24), we know that
is a solution of (2.1);
is also a solution of (2.1).
which are evident because and (3.16) shows that .
From the above assertion, we know that (= has only isolated zeros for . As , we have , a contradiction with (4.2).
Hence does not admit AMP.
To this end, let us establish some relation between and .
Hence (4.11) is also true for this case.
we have the following statements.
This is just (4.15) because .
The functionals and are well defined for all potentials . Moreover, by (4.15), and have the same signs with the functionals in (4.9).
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.
Due to the factor which is zero at those , and are in general discontinuous at . However, and are continuous at in the topology or even in the weak topology . See Lemmas 2.4 and 2.5.
By Lemma 4.2, Theorem 4.1 can be restated as follows.
Let . Then admits MP iff , and admits AMP iff .
4.2. Complete Characterizations of MP and AMP Using Eigenvalues
Let be such that . Then and admits MP.
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.
Moreover, it follows from Lemma 2.4 that is continuous in . Thus (4.27) follows simply from (4.32) and (4.33).
Since , . It follows from (4.21), (4.26), and (4.27) that, for all , has the same sign with . Thus . By Corollary 4.4, admits MP.
Suppose that satisfies and . Then and admits AMP.
a contradiction with the characterization of . Thus (4.21) is also true for .
Since , we have from (4.21) and (4.26) that , because we will prove in Lemma 4.7 that for all .
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].
Hence is invariant under transformations (4.42).
Now (4.48) can be obtained as follows. Let . Then . By (4.47), the transformed potential satisfies . By the invariance, we have the desired result (4.48).
Suppose that satisfies . Then and admits AMP.
In conclusion we have .
Suppose that satisfies . Then does not admit neither MP nor AMP.
We need not to consider the case because is not invertible.
In the following let us assume that satisfies . Then . The following is a modification of the last part of the proof of Lemma 4.8.
Hence 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 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.
Let . Then admits SAMP iff iff .
4.3. Explicit Conditions for AMP
Let us recall some known sufficient conditions for AMP.
Lemma 4.12 (Torres and Zhang ).
Then admits AMP.
In order to overcome the technical assumption (4.60) on positiveness of , one observation is as follows.
Lemma 4.13 (Torres [8, Theorem ]).
Let . Suppose that all gaps of consecutive zeros of all nonzero solutions of (2.1) are strictly greater than the period 1. Then the Green function 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 ]).
Then admits AMP.
with the constants being replaced by more general Sobolev constants .
- (i)Recall the following trivial upper bound:(4.69)
- (ii)It is also possible to construct many potentials for which admits AMP, while (4.70) is violated. For example, let and be defined by(4.71)
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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