Research Article | Open Access
The Existence of Multiple Periodic Solutions of Nonautonomous Delay Differential Equations
We study the multiplicity of periodic solutions of nonautonomous delay differential equations which are asymptotically linear both at zero and at infinity. By making use of a theorem of Benci, some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions.
The existence and multiplicity of periodic solutions of delay differential equations have received a great deal of attention. In 1962, Jones  firstly investigated the existence of periodic solutions to the following scalar equation: By making use of Browder fixed point theorem, the author showed that there exist periodic solutions of (1.1) for each . Since then, various fixed point theorems have been used to study the existence of periodic solutions of delay differential equations(cf. ). As pointed out in , by making change of variable , (1.1) turns into In 1974, Kaplan and Yorke  studied the following more general form of (1.2) They introduced a technique which translates the problem of the existence of periodic solutions of a scalar delay differential equation to that of the existence of critical points of an associated ordinary differential system. Using this method, they proved that (1.3) has a periodic solution with minimal period 4 (resp., 6) when (1.3) has one delay (resp., two delays). In this direction, Fei, Li and He did some excellent work and got some signification results (cf. [5–8]).
Many other approaches, such as coincidence degree theory, the Hopf bifurcation theorem, and the Poincaré-Bendixson theorem, have also been used to study the existence of periodic solutions of delay differential equations (cf. [9, 10]). However, most of those results are concerned with scalar delay equations. In 2005, Guo and Yu  studied vector delay differential system (1.2). They built a variational structure for (1.2) on certain suitable spaces. Then they reduced the existence of periodic solutions of (1.2) to that of critical points of an associated variational functional. By making use of pseudoindex theory, they obtained some sufficient conditions to guarantee the existence of multiple periodic solutions.
In spite of so many papers on periodic solutions of delay differential equations, there are a quite few researches on nonautonomous case (see for example ). The main goal of this paper is to investigate the following nonautonomous system:
We assume that there exists such that is the gradient of with respect to , and as uniformly for , as uniformly for ,
where are symmetric continuous -periodic matrix functions.
Hypothesis is known as asymptotically linear condition at infinity. Hypothesis is an asymptotically linear condition at zero, which implies that 0 is a trivial solution of (1.4). We are interested in nontrivial periodic solutions of (1.4). Similar to , we build a variational structure for (1.4) and convert the existence of periodic solutions to that of critical points of variational functional. Since the asymptotically linear hypothesis at infinity is given by a periodic loop of symmetric matrix, it will be more difficult to deal with more than a constant matrix. However, we can prove the existence of multiple periodic solutions by making use of a multiple critical points theorem of Benci (cf. ).
2. Variational Tools
Denote . The space has been introduced in . The space can be equipped with inner product as follows: where , , , , .
Set Then is a closed subspace of . If , it has Fourier expansion
Let . If for every then is called a weak derivative of , denoted by .
The variational functional defined on , corresponding to (1.4), is Define a linear bounded operator by setting It is easy to prove that is an invariant subspace of with respect to and is self-adjoint if it is restricted to .
Lemma 2.1 (see ). The essential spectrum of the operator restricted to is just .
Define Then can be rewritten as
Similar to the argument as in , we can prove the following two basic lemmas.
Lemma 2.2. Assume that satisfies –. Then is continuous differentiable on and Moreover, is a compact mapping defined as follows:
Lemma 2.3. The existence of -periodic solutions of (1.4) belonging to is equivalent to the existence of critical points of functional restricted to .
Lemma 2.3 implies that we can restrict our discussion on space . At the end of this section, we recall a useful embedding theorem.
Lemma 2.4 (see ). For every , is compactly embedded into the Banach space . In particular, there is an such that
Remark 2.5. Here and hereafter, denotes the real number satisfying (2.11).
3. Main Results
Let be an symmetric continuous -periodic matrix function. We define a bounded self-adjoint linear operator by extending the bilinear forms It is well known that is compact (cf. ).
Lemma 3.1. Suppose that satisfies –. Then
The proof uses the same arguments of .
In order to prove our results, we need an abstract theorem by Benci .
Proposition 3.2. Let satisfy the following:(J1), where is a bounded linear self-adjoint operator and is compact, where denotes the Frechét derivative of ;(J2)every sequence such that and as has a convergent subsequence;(J3), ;(J4)there are two closed subspaces of , , and , and some constant with and such that(a) for ,(b) for .Then the number of pairs of nontrivial critical points of is greater than or equal to . Moreover, the corresponding critical values belong to .
Definition 3.3. Let and be symmetric matrices function in , continuous and -periodic in . A index of and is defined as follows: where are the operators, defined by (3.1), corresponding to and (resp., , denotes the subspace of on which is positive definite (resp., negative definite, null).
Lemma 3.4. If satisfies –, then , defined by (2.8), satisfies (J1), (J3) and (J4).
Proof. Hypothesis , (2.8), and Lemma 2.2 imply both (J1) and (J3). By definition of and Lemma 3.1, we have
Since and are compact operators from to , it follows from Lemma 2.1 and a well-known theorem (cf. ) that the essential spectrum of and is . Thus 0 is either an isolated eigenvalue of finite multiplicity or it belongs to the resolvent. Hence, we decompose as follows:
Setting , , there exists positive constant such that
It follows that, for any , it is
Then there exist constants and such that
Setting , (J4)(b) is satisfied.
By , there exists such that Since is continuous with respect to , denote by . Then is finite. Thus Then, for every , Thus is bounded from below on . Setting with such that , then (J4)(a) is satisfied.
Remark 3.5. Supposing that , any bounded sequence has a convergent subsequence (cf. ).
Theorem 3.6. Suppose that satisfies –, and , then (1.4) has at least pairs of nonconstant periodic solutions if .
Proof. Since , . By Proposition 3.2 and Lemma 3.4, we only need to check (J2). Let be a sequence such that
where , . Suppose to the contrary that we can choose as . Clearly, can be written as . On one hand,
then we have
On the other hand, Since it follows by Lemma 3.1 that Using a similar discussion as (3.8), there exists such that for all . Choosing , we have Thus, which contradicts (3.18). This proves (J2). By Lemma 3.1, (1.4) has at least pairs of nontrivial solutions if . Since the Sobolev space does not contain as its subspace, all nontrivial periodic solutions are nonconstant periodic solutions.
If , then . In this case, we replace by and let and . It is easy to see that (J1)–(J4) are satisfied. Similarly, we can show that (1.4) has at least pairs of nonconstant solutions.
Theorem 3.8. Suppose satisfies – and is bounded, where denotes the derivative of with respect to , as , uniformly for .
Then (1.4) has at least pairs of nonconstant -periodic solutions provided .
Proof. By Proposition 3.2 and Lemma 3.4, it suffices to check condition (J2). Let be a sequence satisfying (3.15). Suppose to the contrary that is unbounded. Clearly, can be written as . Since , for large enough, we get
By , there exists such that . Then the above inequality and (3.8) imply
This gives a uniform bound for . In the same manner, one gets a uniform bound for . Since is convergent, it is bounded and there exist positive constants such that
Therefore, is bounded from above. implies that is bounded. Otherwise, since the kernel of is a finite dimensional space, thus as , which contradicts to (3.26).
If holds, we replace (3.26) by Arguing as above, we can get a contradiction and complete our proof.
Theorem 3.9. Suppose that satisfies – and there exist constants , , , and such that Then (1.4) has at least pairs of nonconstant -periodic solutions provided .
Proof. Let be a sequence satisfying (3.15). We want to show that is a bounded sequence in . Decompose as . Then
Combining the above inequality with (3.15) and Lemma 3.1, we have
Similarly, we have
Then by (3.30) and (3.31), there exists a positive integer such that for It follows that
By , there exist positive constants such that for large Let be such that . Since , the embedding being continuous, the dual space of , contains with continuous embedding. Therefore, by (3.34) Since , taking , it follows that Owing to the fact that ) is a finite dimensional subspace of , there exist two positive constants and such that
Therefore by (3.35), (3.36), and (3.37), Both (3.33) and (3.38) imply that there exists such that which yields a bound for and hence via (3.33). Thus (J2) holds.
Theorem 3.10. Suppose that satisfies – and
there exist positive constants such that Then (1.4) has at least pairs of nonconstant -periodic solutions provided .
Proof. Let be a sequence satisfying (3.15). We want to prove that is bounded in . Suppose, to the contrary, is unbounded in . Decompose as . Clearly, (3.30)–(3.33) still hold.
Assume that holds. Since is a finite dimensional subspace of , we have Combining the above inequality with (3.30), (3.31), we have But this implies the following contradiction: therefore, must be a bounded sequence.
If holds, using a similar argument, we can get a contradiction which completes our proof.
Theorem 3.11. Suppose that satisfies – and there exist constants , , and such that Then (1.4) has at least pairs of nonconstant -periodic solutions provided .
Proof. Let be a sequence satisfying (3.15). Suppose, to the contrary, as . Decompose as . First, we show that for large enough
where and are constants independent of . Since for sufficiently large , therefore, , and we have
This implies that for large enough
By (3.15), (3.22), (3.33) and (3.47), for large enough, we have
Therefore, for sufficiently large ,
This implies that (3.45) holds, where .
By (3.15) and (3.45), for large enough, there exist positive constants such that Now, we claim that there exists such that, for large enough, In fact, for , by (3.45) and the fact that , we have for large enough. This implies (3.51) for .
For , since is a finite dimensional subspace of , we know that for any , where are constants independent of . Now we have Combining (3.52) with (3.54), we get (3.51) for .
On the other hand, by Since that , we get a contradiction from (3.50) and (3.55). Therefore, is bounded.
If holds, using a similar argument as above, we get a contradiction and completes our proof.
Theorem 3.12. Suppose satisfies – and there exist positive constants , , and such that Then (1.4) has at least pairs of nonconstant -periodic solutions provided .
Proof. If holds, for large enough, by (3.45) and (3.51), we have
Since , is bounded, so is . Therefore, satisfies (J2).
In the case that holds, using a similar argument, we can verify (J2). This completes the proof.
Example 3.13. Consider the following nonautonomous delay differential equation
where is a matrix, are constants, .
Case 1. Let , , , and arbitrary. Computing directly, we have . Applying Theorem 3.6, equation (3.58) has at least 10 pairs of -periodic solutions.
Case 2. Let , , , and arbitrary. Then by Theorem 3.8, (3.58) has at least 8 pairs of nonconstant -periodic solutions.
This paper is supported by Xinmiao Program of Guangzhou University and National Natural Science Foundation of China (no. 11026059), SRF of Guangzhou Education Bureau (no. 10A012), and Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. LYM09105).
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Copyright © 2011 Huafeng Xiao and Bo Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.