- Research Article
- Open Access
The Existence of Periodic Solutions for Non-Autonomous Differential Delay Equations via Minimax Methods
Advances in Difference Equations volume 2009, Article number: 137084 (2009)
By using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.
1. Introduction and Main Result
where is odd and is parameter. Since Jone's work in , there has been a great deal of research on problems of existence, multiplicity, stability, bifurcation, uniqueness, density of periodic solutions to (1.1) by applying various approaches. See [2, 4–23]. But most of those results concern scalar equations (1.1) and generally slowly oscillating periodic solutions. A periodic solution of (1.1) is called a "slowly oscillating periodic solution" if there exist numbers and such that for , for and for all .
In a recent paper , Guo and Yu applied variational methods directly to study the following vector equation:
where is odd and is a given constant. By using the pseudo index theory in , they established the existence and multiplicity of periodic solutions of (1.2) with satisfying the following asymptotically linear conditions both at zero and at infinity:
where and are symmetric constant matrices. Before Guo and Yu's work, many authors generally first use the reduction technique introduced by Kaplan and Yorke in  to reduce the search for periodic solutions of (1.2) with and its similar ones to the problem of finding periodic solutions for a related system of ordinary differential equations. Then variational method was applied to study the related systems and the existence of periodic solutions of the equations is obtained.
The previous papers concern mainly autonomous differential delay equations. In this paper, we use minimax methods directly to study the following nonautonomous differential-delay equation:
where is odd with respect to and satisfies the following superlinear conditions both at zero and at infinity
When (1.2) satisfies (1.3), we can apply the twist condition between the zero and at infinity for to establish the existence of periodic solutions of (1.2). Under the superlinear conditions (1.5), there is no twist condition for , which brings difficulty to the study of the existence of periodic solutions of (1.4). But we can use minimax methods to consider the problem without twist condition for .
Throughout this paper, we assume that the following conditions hold.
is odd with respect to and -periodic with respect to .
write . There exist constants and such that(1.6)
with and .
there exist constants , and such that(1.7)
with and .
Then our main result can be read as follows.
Suppose that satisfies (1.5) and the conditions hold. Then (1.4) possesses a nontrivial -periodic solution.
2. Proof of the Main Result
First of all in this section, we introduce a minimax theorem which will be used in our discussion. Let be a Hilbert space with . Let be the projections of onto and , respectively.
where is compact.
Let and be boundary. One calls and link if whenever and for all , then .
A functional satisfies condition, if every sequence that , and being bounded, possesses a convergent subsequence.
Then [25, Theorem ] can be stated as follows.
Theorem 2 A.
Let be a real Hilbert space with , and inner product . Suppose satisfies condition,
, where and is bounded and selfadjoint, ,
is compact, and
there exists a subspace and sets , and constants such that
is bounded and ,
Then possesses a critical value .
Then and , where denotes the gradient of with respect to . We have the following lemma.
Under the conditions of Theorem 1.1, the function satisfies the following.
is 2-periodic with respect to and for all ,
There exist constants , and such that for all with and , , and(2.5)(2.6)
where denotes the inner product in .
The definition of implies (i) directly. We prove case (ii) and case (iii).
Case (ii). Let
Then and or is equivalent to or , respectively.
From (1.5) and L'Hospital rules, we have (2.3) by a direct computation.
Case (iii). By (H2), we have a constant such that for with .
Now we prove for with , that is,
Firstly, it follows from that .
Now we show . Let , . By , , that is, . Then
By reducing method, we have
Thus, the inequality for holds.
Take and . Then (2.5) and (2.6) hold with and .
Below we will construct a variational functional of (1.4) defined on a suitable Hilbert space such that finding -periodic solutions of (1.4) is equivalent to seeking critical points of the functional.
Firstly, we make the change of variable
Then (1.4) can be changed to
where is -periodic with respect to . Therefore we only seek -periodic solution of (2.12) which corresponds to the -periodic solution of (1.4).
We work in the Sobolev space . The simplest way to introduce this space seems as follows. Every function has a Fourier expansion:
where are -vectors. is the set of such functions that
With this norm , is a Hilbert space induced by the inner product defined by
We define a functional by
By Riesz representation theorem, H identifies with its dual space H*. Then we define an operator A:H→H*=H by extending the bilinear form:
It is not difficult to see that is a bounded linear operator on and .
Define a mapping as
Then the functional can be rewritten as
According to a standard argument in , one has for any ,
Moreover according to , is a compact operator defined by
Our aim is to reduce the existence of periodic solutions of (2.12) to the existence of critical points of . For this we introduce a shift operator defined by
It is easy to compute that is bounded and linear. Moreover is isometric, that is, and , where denotes the identity mapping on .
Critical points of over are critical points of on , where is the restriction of over .
Note that any is -periodic and is odd with respect to . It is enough for us to prove for any and being a critical point of in .
For any , we have
This yields , that is, .
Suppose that is a critical point of in . We only need to show that for any . Writing with and noting , one has
The proof is complete.
By Lemma 2.4, we only need to find critical points of over . Therefore in the following will be assumed on .
For , yields that , where is in the Fourier expansion of . Thus . Moreover for any ,
Hence is self-adjoint on .
Let and denote the positive definite and negative definite subspace of in , respectively. Then . Letting , , we see that of Theorem A holds. Since is compact, of Theorem A holds. Now we establish of Theorem A by the following three lemmas.
Under the assumptions of Theorem 1.1, of holds for .
From the assumptions of Theorem 1.1 and Lemma 2.3, one has
By (2.3), for any , there is a such that
Therefore, there is an such that
Since is compactly embedded in for all and by (2.29), we have
Consequently, for ,
Choose and so that . Then for any ,
Thus satisfies of with and .
Under the assumptions of Theorem 1.1, satisfies of .
Set and let
where is free for the moment.
Let . Write
Case (1). If with , one has
Case (2). If , we have
Denote . By appendix, there exists such that ,
Now for , set . By (2.4), for a constant , there is an such that
Choosing , for ,
For , we have
Henceforth, for any and , that is, . Then of holds.
Suppose and for all . Then we claim that for each , there is a such that , that is,
where is a projection. Define
It is easy to see that
According to topological degree theory in , we have
since . Therefore and link.
Now it remains to verify that satisfies -condition.
Under the assumptions of Theorem 1.1, satisfies -condition.
We first show that is bounded. If is not bounded, then by passing to a subsequence if necessary, let as .
By (2.4), there exists a constant such that as . By (2.5), one has
Write . By (2.6), there is a constant such that
This inequality and (2.50) imply that
as , since .
Denote . We have
where is a constant independent of .
By the above inequality, one has
as . This yields
Similarly, we have
Thus it follows from (2.56) and (2.57) that
which is a contradiction. Hence is bounded.
Below we show that has a convergent subsequence. Notice that and is compact. Since is bounded, we may suppose that
Since has continuous inverse in , it follows from
Henceforth has a convergent subsequence.
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1.
It is obviously that Theorem 1.1 holds from Lemmas 2.3, 2.4, 2.6, 2.7, 2.8, and 2.9 and Theorem A.
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This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education for New Teachers and the Science Research Foundation of Nanjing University of Information Science and Technology (20070049).
The purpose of this appendix is to prove the following lemma. The main idea of the proof comes from . Lemma A.1. There exists such that, ,