Existence of periodic solutions for first-order delay differential equations via critical point theory

By the weak linking theorem and the linking theorem, we study the existence of periodic solutions for the following system of delay differential equations:

where f\in C(R^n,R^n), and r>0 is a given constant. Two existence theorems of 4r-periodic solutions of (1) are obtained.

Consider the following system of delay differential equations:

where f\in C(R^n,R^n), and r>0 is a given constant.

As n\equiv 1, the existence of the periodic solutions for (1.1) has been extensively studied in the past years (for example, see [1–5]). However, their methods are not variational. Few results of the existence of periodic solutions for delay differential equations have been obtained by the variational method. In 2005, Guo and Yu [6] took the lead in using the variational approaches to study the existence of multiple periodic solutions for (1.1), and a multiplicity result was given. Recently, using the variational approaches, the multiplicity of the periodic solutions for the following system:

was studied by Wu and Wu in [7]. In the present paper, our main purpose is to study the existence of the periodic orbits for system (1.1) via the linking and weak linking theory.

Throughout this paper, we always assume that

(f_1) f is odd, i.e., for any x\in R^n, f(-x)=-f(x);

(f_2) there exists a continuously differentiable function F such that \mathrm{\nabla }F(x)=f(x) for all x\in R^n, and F(0)=0;

(f_3) f(x)=Ax+o(|x|) as |x|\to \mathrm{\infty }, where A={(a_{ij})}_{n\times n} is an n\times n matrix with

and {\lambda }^{-1}{(-1)}^{j+1}(2j-1)\notin \sigma (A) (the set of all eigenvalues of A) for any j\in Z^{+}, where \lambda =\frac{2r}{\pi } and Z^{+} is the set of all positive integers.

In the following, we give some preliminaries.

Definition 1.1 ([8])

Let E be a Hilbert space and I\in C^1(E,R). The function I^{\prime } is called weak-to-weak continuous if

Definition 1.2 ([8])

A subset A of a Banach space E links a subset B of E weakly if for every I\in C^1(E,R) satisfying (1.2) and

there are a sequence \{u_k\}\subset E and a constant c such that

and

The following lemma is Example 1 in [8].

Lemma 1.1 Let E be a separable Hilbert space, and let M, N be a closed subspace such that E=M\oplus N. Let

and take A=\partial B_R\cap N, B=M. Then A links B weakly.

One can easily find that (1.1) can be changed to the equation

by making the change of variable t\mapsto \frac{\pi }{2r}t={\lambda }^{-1}t. Thus, a 4r-periodic solution of (1.1) corresponds to a 2π-periodic solution of (1.7).

Similar to the treatment in Guo and Yu [6], we introduce the following spaces. Let L^2(S^1,R^n) denote the set of n-tuples of 2π periodic functions which are square integrable. Let C^{\mathrm{\infty }}(S^1,R^n) be the space of 2π-periodic C^{\mathrm{\infty }} vector-valued functions with dimension n. For any u\in C^{\mathrm{\infty }}(S^1,R^n), it has the following Fourier expansion in the sense that it is convergent in the space L^2(S^1,R^n):

where a_0^u,a_j^u,b_j^u\in R^n. Set H^{\frac{1}{2}}(S^1,R^n) is the closure of C^{\mathrm{\infty }}(S^1,R^n) with respect to the Hilbert norm

More specifically, with the inner product

for any u,v\in H^{\frac{1}{2}}(S^1,R^n), where (\cdot ,\cdot ) denotes the usual inner product in R^n. In the sequel, we denote by H the Hilbert space H^{\frac{1}{2}}(S^1,R^n). The norm on H is defined by

Now consider a functional I defined on H

where \dot{u}(t) denotes the weak derivative of u.

We define an operator L:H\to H^{\ast } as follows: for any u\in H, Lu is defined by

where H^{\ast } denotes the dual space of H. By the Riesz representation theorem, we can identify H^{\ast } with H. Thus, Lu can also be viewed as a function belonging to H such that 〈Lu,v〉=Lu(v) for any u,v\in H. Define

Then I(u) can be rewritten as

Define the bounded linear operator \zeta :H\to H as follows: for any u\in H, \zeta u(\cdot )=u(\cdot +\frac{\pi }{2}). Next, we set E=\{u\in H:{\zeta }^{2}u=-u\}. Then E is a closed subspace of H and is invariant with respect to L. It is easy to check that L is a bounded linear operator on H, L{|}_E is self-adjoint, and E is also invariant with respect to {\mathrm{\Phi }}^{\prime } under condition (f_1) (see Guo and Yu [6]). By Proposition B.37 in [9] and Lemma 2.2 in [6], we have the following two lemmas.

Lemma 1.2 Assume that f satisfies (f_2) and the following condition:

(f_0) there are constants a_1,a_2>0 and \alpha \ge 1 such that

for all x\in R^n.

Then the functional I is continuously differentiable on H and I^{\prime }(u) is defined by

Moreover, {\mathrm{\Phi }}^{\prime }:H\to H^{\ast } is a compact mapping defined as follows:

By the Riesz theorem, we can view {\mathrm{\Phi }}^{\prime }(u) as an element of H for any u\in H. As usual, we identify u\in H and its continuous representative.

We have the following fact.

Lemma 1.3 Assume that f satisfies (f_0), (f_1), (f_2). Then critical points of functional I restricted to E are 2π-periodic solutions of system (1.7).

Remark 1.1 It is pointed in [6] that a critical point u of I in H will be a weak solution of (1.7). However, a simple regularity argument shows that u\in C^1(S^1,R^n).

Remark 1.2 As usual, we should deal with (1.10) in the space H. But, according to Lemma 1.3, we only need to treat the functional I in the subspace E of H.

Lemma 1.4 ([9])

For each s\in [1,\mathrm{\infty }), H^{\frac{1}{2}}(S^1,R^n) is compactly embedded in L^s(S^1,R^n). In particular there is {\alpha }_s>0 such that

for all u\in H^{\frac{1}{2}}(S^1,R^n).

Theorem 2.1 Assume that f satisfies (f_1), (f_2) and (f_3). Then (1.1) possesses at least one 4r-periodic solution.

Proof Let e_1,e_2,\dots ,e_n denote the usual normal orthogonal bases in R^n and set

where Z^{+} is the set of all positive integers. Then E=M\oplus N. For any u\in E, it has a Fourier expansion as follows:

where all a_{2j-1}^u,a_{4j-1}^u,a_{4j-3}^u,b_{2j-1}^u,b_{4j-1}^u,b_{4j-3}^u\in R^n,

Consequently, we have

Let

Then (2.2) and (2.3) show that \parallel \cdot \parallel and {\parallel \cdot \parallel }_H are two equivalent norms on E. Henceforth we use the norm \parallel \cdot \parallel as the norm for E. And the spaces M, N are mutually orthogonal with respect to the associated inner product.

First, we prove that I(u) satisfies (1.2) in E.

Let \{u_k\} be any sequence which converges to some u weakly in E. By the compactness of the embedding E\hookrightarrow L^2(S^1,R^n), we assume that

Thus, (f(u_k),v)\to (f(u),v) a.e. for all v\in E. Since f(x)\in C(R^n,R^n) satisfies (f_3), there exist positive constants M_1 and M_2 such that

Note that the right-hand side of (2.4) converges to M_1|v|+M_2|u||v| in L^1(S^1,R^n). Hence \{(f(u_k),v)\}\subset L^1(S^1,R) is uniformly absolutely continuous. Hence, by Vitali’s theorem,

Moreover, since L is a bounded self-adjoint linear operator on E,

According to (2.5) and (2.6), we get that I^{\prime } is weak-to-weak continuous.

Next, we prove

and

Indeed, by (f_3), we know that there exists a positive constant c such that

Thus, for u\in M, by (2.9), we have

where c_1>0 is a given constant. Since \frac{3-3\lambda \parallel A\parallel }{8}>0, (2.10) implies (2.7). The proof of (2.8) is similar. In fact, when v\in N, by (2.9), we get that

This implies (2.8).

Note that (2.10) implies

The combination of (2.8) and (2.11) implies that there is R>0 such that (1.3) holds with A=\partial B_R\cap N, B=M. By Lemma 1.1 we know that there are a sequence \{u_k\}\subset E and a constant c such that

Finally, we show that the sequence \{u_k\} is bounded in E. To do this, assume that {\rho }_k=\parallel u_k\parallel \to \mathrm{\infty }, and write {\tilde{u}}_k=\frac{1}{{\rho }_k}{u}_k. Then \parallel {\tilde{u}}_k\parallel =1. From Lemma 1.4, there is a renamed subsequence such that

By (f_3), for any ϵ>0, there exists a constant \overline{r}>0 such that

Moreover, by the continuity of f, there is a constant c_2>0 such that

By (2.13) and (2.14), for any v\in E, we have

as ϵ\to 0 and k\to \mathrm{\infty }. This shows

Hence

By (2.12) and (2.17), we see that

for all v\in E, i.e.,

for all v\in E.

Set

Then, by (2.18), one can obtain

where I is the n\times n unit matrix. For any j, take v(t)=\frac{1}{\sqrt{\pi }}{e}_{i}cos(2j-1)t and v(t)=\frac{1}{\sqrt{\pi }}{e}_{i}sin(2j-1)t, where i=1,2,\dots ,n. An easy computation shows that

and

Hence, by {\lambda }^{-1}{(-1)}^{j+1}(2j-1)\notin \sigma (A), we get that \tilde{u}\equiv 0.

Let {\tilde{u}}_k={\tilde{x}}_k+{\tilde{y}}_k, where {\tilde{x}}_k\in M, {\tilde{y}}_k\in N. A proof similar to (2.16) shows that

and

where \tilde{u}=\tilde{x}+\tilde{y}, \tilde{x}\in M, \tilde{y}\in N. Thus

On the other hand, (2.12) implies

which contradicts (2.23). Thus {\rho }_k must be bounded. Consequently, there is a renamed subsequence of \{u_k\} such that u_k\rightharpoonup u in E. Hence, by the weak-to-weak continuity of I^{\prime }, we have

Now, the combination of (2.12) and (2.24) implies that I^{\prime }(u)=0. This completes the proof. □

Remark 2.1 Let n=1, r=\frac{\pi }{3} and f(x)=x. Then f(x) satisfies all the conditions of Theorem 2.1.

In order to give our another result, we still need the following preliminaries.

Let P_M, P_N be the projectors of E onto M, N associated with the given splitting of E. Set

Recall that K is continuous and maps bounded sets to relatively compact sets since K is compact. Let S,Q\subset E with Q\subset \tilde{E}, a given subspace of E. Then ∂Q will refer to the boundary of Q in \tilde{E}.

Definition 2.1 We say S and ∂Q link if whenever \mathrm{\Psi }\in \mathcal{H} and \mathrm{\Psi }(t,\partial Q)\cap S=\mathrm{\varnothing } for all t\in [0,1], then \mathrm{\Psi }(t,Q)\cap S\ne \mathrm{\varnothing } for all t\in [0,1].

Lemma 2.1 ([9])

Let \rho >0, S\equiv \partial B_{\rho }\cap M, e\in M\cap \partial B_1, r_1>\rho, r_2>0, Q\equiv \{re:r\in (0,r_1)\}\oplus (B_{r_2}\cap N), and \tilde{E}\equiv span\{e\}\oplus N. Then S and ∂Q link.

Lemma 2.2 ([9])

Suppose I\in C^1(E,R) satisfies the (PS) condition and

(I_1) I(u)=\frac{1}{2}〈Lu,u〉+\mathrm{\Phi }(u), where Lu=L_1{P}_{M}u+L_2{P}_{N}u and L_1:M\to M, L_2:N\to N are bounded self-adjoint,

(I_2) {\mathrm{\Phi }}^{\prime } is compact,

(I_3) there exist a subspace \tilde{E}\subset E and sets S\subset E, Q\subset \tilde{E} and constants \alpha >\beta such that

1. (i)

   S\subset M and I{|}_S\ge \alpha,

2. (ii)

   Q is bounded and I{|}_{\partial Q}\le \beta,

3. (iii)

   S and ∂Q link.

Then I possesses a critical value c\ge \alpha.

The following is our another main result.

Theorem 2.2 Assume that f satisfies (f_1), (f_2) and the following conditions:

(f_4) F(x)\le 0 for all x\in R^n,

(f_5) f(x)=o(|x|) as |x|\to 0,

(f_6) there exist constants c>0, p>2 and \tilde{r}>0 such that

and

Then (1.1) possesses at least one nonconstant 4r-periodic solution.

Proof We will show that I satisfies the hypotheses of Lemma 2.2. This will lead to a nonconstant 4r-periodic solution of (1.1). We divide the proof of Theorem 2.2 into the following three parts.

First, we prove that I satisfies (I_1) and (I_2) of Lemma 2.2.

Note that L(M)\subset M, L(N)\subset N and L is bounded self-adjoint on E. We see that I satisfies (I_1) of Lemma 2.2 with L_1=L{|}_M, L_2=L{|}_N and

By Proposition B.37 in [9], (f_6) implies that {\mathrm{\Phi }}^{\prime } is compact. Hence (I_2) holds.

Next, we show that I satisfies (I_3) of Lemma 2.2.

By (f_5), for any ϵ>0, there is \delta >0 such that

whenever |x|\le \delta. By (f_6), there is a constant c=c(ϵ) such that

Hence

By (2.25) and Lemma 1.4, for any u\in M, we have

Choose ϵ={(4\lambda {\alpha }_2^2)}^{-1}. Since p>2, there is small \rho >0 such that \frac{1}{8}{\rho }^2\ge \lambda c{\alpha }_p^p{\rho }^p. Then, for u\in \partial B_{\rho }\cap M, (2.26) implies that I(u)\ge \frac{1}{8}{\rho }^2:=\alpha >0. Consequently, I satisfies (I_3)(i) with S=\partial B_{\rho }\cap M.

Set e\in \partial B_1\cap M and

where r_1>\rho and r_2>0 are free constants for the moment. Define \tilde{E}=span\{e\}\oplus N. Then Q\subset \tilde{E} and S and ∂Q link by Lemma 2.1.

By (f_6), there are constants c_1,c_2>0 such that

for all x\in R^n. Thus, for v\in B_{r_2}\cap N, by the Hölder inequality (note that p>2) and orthogonality, we get that

where c_3,c_4>0 are constants. Now, choose large r_1>\rho and r_2>r_1 such that

and

By (2.27), (2.29), (2.30), (2.31) and (f_4), one can easily check that I{|}_{\partial Q}\le 0:=\beta. Hence I satisfies (I_3)(ii).

To sum up, I satisfies (I_3) of Lemma 2.2.

Finally, we check that I satisfies the (PS) condition. Let \{u_k\}\subset E be a sequence such that |I(u_k)|\le c_0 and I^{\prime }(u_k)\to 0 as k\to \mathrm{\infty }. Then, for large k, by (f_6) and (2.28), we have

where c_5,c_6,c_7>0 are constants.

Let u_k=x_k+y_k, where x_k\in M, y_k\in N. Then, for large k, by (f_6), Lemma 1.4 and the Hölder inequality, we get that

This implies that

Similarly, one can easily see that

By (2.32), (2.34) and (2.35), there is a constant c_8>0 such that

which implies that \{u_k\} is bounded in E.

By the compactness of {\mathrm{\Phi }}^{\prime }, going if necessary to a subsequence, we can assume that

and

Let u=x+y, where x\in M and y\in N. Then

as k\to \mathrm{\infty }. Similarly, we have {\parallel y_k-y\parallel }^2\to 0 as k\to \mathrm{\infty }. Hence u_k\to u in E. Hence I satisfies the (PS) condition.

Therefore, Theorem 2.2 follows from Lemma 2.2. □

Remark 2.2 Let n=1 and f(x)=-x^{\frac{5}{3}}. Then f(x) satisfies all the conditions of Theorem 2.2 with p=\frac{8}{3} and \tilde{r}>0.

Many thanks to the reviewers for carefully reading the manuscript and valuable comments on improving the paper. This work is supported partly by the Professor Program Foundation of Zhaotong University.

Affiliations

1. Department of Mathematics and Statistics, Zhaotong University, Zhaotong, Yunnan, 657000, P.R. China

   • Shao-Kang Zhang
   •  & Lichun Chen

Corresponding author

Correspondence to Shao-Kang Zhang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in the paper. They read and approved the final manuscript.

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