Functional Differential and Difference Equations with Applications 2013View this Special Issue
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Infinitely Many Periodic Solutions to Delay Differential Equations via Critical Point Theory
By the critical point theory, infinitely many 4σ-periodic solutions are obtained for the system of delay differential equations , where and . It is shown that all the periodic solutions derived here are brought about by the time delay.
This paper is concerned with the existence of periodic solutions to the system of delay differential equations where and .
Delay differential equations have widely been applied to describe the dynamics phenomena in both natural and manmade processes such as chemistry, physics, engineering, and economics. The existence of the periodic solutions for delay differential equations has been extensively investigated by using various methods, including fixed point theorems [1–5], Hopf bifurcation theorems [6–8], variational methods [9–14], the methods of differential inequalities [15–21], and other effective approaches (e.g., see [22–24]). In [25–31], the minimal periods of the periodic solutions to Lipschitzian differential equations are estimated through the Lipschitz constants (see Remark 4).
The use of variational methods in the study of -periodic solutions of system (1) having a variational structure was introduced in 2005 by Guo and Yu . Assume that is odd in ; that is, , for all ;there exists such that , for all , where denotes the gradient of .
In , the authors obtained the multiplicity results for periodic solutions to (1) in the case that is asymptotically linear. Later, the existence of the periodic solution of (1) was investigated by using Morse theory and Galerkin methods . For the other relative investigations, we refer the reader to [11–14].
Many practical problems, such as nonlinear population growth models and control systems working with potentially explosive chemical reactions, can be transformed into the form of (1). For example, by the change of variables , the following generalized food-limited population model is transformed equivalently into (1) with , , and , where and are positive numbers. When , . It is known from  that, with the slope increasing and tending to infinity, the number of the periodic solutions of (2) increases and tends to infinity. Naturally, one would conjecture that when , (2) possesses infinitely many periodic solutions, since in this case .
Motivated by the above observation, in this paper, we study the existence of infinitely many periodic solutions to the system (1) under the assumptions , , andthere are , and such that(i), for all ;(ii), for all , where , .
Here and subsequently, , denote the inner product and the standard norm in , respectively, and the bold face represents the coordinate origin of . The main result of this paper is stated as follows.
Theorem 1. Assume that hold. Then (1) possesses a sequence of nonconstant -periodic solutions satisfying as .
Example 2. When , it is easy to check that satisfies with ; then (2) has a sequence of nonconstant -periodic solutions satisfying as .
Remark 3. Let us compare the result here with that in the case of ordinary differential equations (ODE). Without the time delay, (1) reduces to the following system of ODE Let be the solution of (3) satisfying the initial condition . Then the derivative of the Lyapunov function along reads From ()-(i), we see that for , which implies that there is no any periodic orbit of (3) across ; that is, the trivial solution is an isolated periodic solution. However, by the above theorem, with the time delay, the system (1) possesses infinitely many periodic solutions in any neighborhood of the origin.
Remark 4. Consider the following system of th order functional differential equations: where satisfies the Lipschitz condition and is a measurable function. The lower bounds for the periods of the periodic solutions to (5) and their special forms are estimated in [25–31]. From this perspective, Theorem 1 complements the information in the case of non-Lipschitzian differential equations. For the unique solvability of the periodic problems on functional differential equations, we refer the reader to [1, 15–21].
The remainder of this paper is divided into two parts. In the next section, we state the preliminaries on the variational structure for (1). In the final section, the proof of Theorem 1 will be given via the -genus theory, together with an approximating argument.
Let denote the set of -tuples of -periodic functions which are square integrable. If , it has a Fourier expansion where and the series converges in the space . For with its expansion (6), set , where Then , equipped with the norm , is a Sobolev space.
On the other hand, for with its expansion (6), set Then possesses another norm which is equivalent to . In the following, we always employ as the norm of . The associated inner product with is denoted by .
Now set Then is a closed subspace of and the Fourier expansion of reduces to Thus with being expanded as we have
For , we call a weak derivative of and denote it by if Further, for with its expansion (10), define Then it is easy to check that for . Therefore extends to all of as a continuous quadratic form. This extension will still be denoted by .
Let and satisfy for some . Define and , . The following lemma is derived from [9, Lemma 2.2].
Let be the orthonormal basis of . For , set For , define where the closure is of sense. Set ; then . In the rest of this paper, this decomposition will always be referred to when a point is written as , where .
Lemma 7 (see ). For each there is such that for all with , the orthogonal complement in , where (and below) denotes the usual -norm.
3. Proof of Theorem 1
Without loss of generality we assume that since, under the change of variables , (1) can be transformed into the system where still satisfies with being replaced by .
Let be such that for , for , and for . Define by where .
Let be such that . By we get where (and below) 's stand for positive constants.
Lemma 8. Let be defined by (25); then , , and
Proof. From (25), it is easy to see that . Now we start to prove (28). Let be such a constant that for . For , , set ; then . Define , . Then, by (26),
which implies that ; that is,
It follows that for , which, combining with (25), leads to the inequality on the left hand of (28) with being chosen adequately.
Again, for , , set and define , . Then by the first inequality in (27), Thus , which leads to , where . In the same way, from the second inequality in (27), we can arrive at for , where the constant only depends on and . With , the inequalities on the right hand of (28) hold. Thus we get (28), which implies that for and that . The proof is complete.
Now we consider the functional
Lemma 9. satisfies condition; that is, every sequence such that is bounded and as has a convergent subsequence.
Proof. By Lemma 5, for , is defined by
To verify that satisfies condition, we suppose and as . Note that, for large , . Thus for large and , from (32) and (33),
Noticing that , we see from (25) that, for all ,
which, combining with (26) and (34), implies
Next for large , taking and in and using (23), (27), and (28) and the Hölder inequality (, ), we get where the last inequality holds since is compactly embedded in for . It follows from (36) that Similarly, (40) works with being replaced by . Combining these inequalities shows which implies that is bounded in .
Let be defined by (16). By [33, Proposition B.37], is precompact in . Moreover, from (23) and (33), It follows that has a convergent subsequence. The proof is complete.
Lemma 10. For each , there are , , and such that (a), for all and , where ;(b).
Proof. Noticing that and that , we have, for ,
where is a positive constant depending on . It follows by (28) that, for ,
which implies (a) by setting and .
Let . By (28) and Lemma 7, where . Noticing , one can see that as and (b) follows. The proof is complete.
In the following, let denote the family of closed (in ) subsets of symmetric with respect to the origin, and the -genus map (see ). For , set and define
Lemma 11. For all , is a critical value of and
Proof. We first prove that (49) holds. For each , let be chosen as that in Lemma 10; then it follows by Lemma 10(a) that . Denote ; then and . Since , we have
On the other hand, for every , by the property of genus, , which, from Lemma 10(b), leads to for every . Thus and (49) holds.
By and (25), is even with respect to , which implies that is even. We claim that is a critical point of . Otherwise, there exists , such that there is no any critical point in the interval . By the definition of , there exists , such that For , denote . Use a positive rather than a negative gradient flow [33, Remark A.17], we get such that is odd and Since , we have ; that is,
On the other hand, by the property of genus, we know that , which, by the definition of , leads to This contradiction implies that is a critical value of . The proof is complete.
Now we are in a position to give the following proof.
Proof of Theorem 1. In view of Lemma 11, let be such that
Then by condition, along a subsequence as , such that
which implies that is nonzero. Moreover, by Lemma 5,
We claim that, for sufficiently large , solves (1). In fact, from (26) and (56) By (27), (58), and Hölder inequality Similarly, the above inequality works with replaced by . These inequalities yield Since as , it follows that
Furthermore, from (27) and (57), we have It follows from (58) that as . Recalling (61), we get which implies that as . Thus for sufficiently large, and therefor . It follow from (57) that, for sufficiently large, solves (1). In addition, by (1) and (i), the only constant solution of (1) is the trivial solution. Then (56) yields that is nonconstant and the proof of Theorem 1 is complete.
The authors are grateful to the referees’ careful reviewing and helpful comments. The work was supported by BJJW (KZ201310028031 and KM2014).
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Copyright © 2013 Xiaosheng Zhang and Duo Wang. 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.