Multiple periodic solutions to a class of nonautonomous second-order delay differential equation

The existence of the nontrivial periodic solutions for nonautonomous second-order delay differential equation

is investigated, where \lambda >{\pi }^2/{\tau }^2, \tau >0, f\in C^1(\mathbf{R}\times {\mathbf{R}}^2,\mathbf{R}). Multiple periodic solutions are obtained by some recent critical point theorems.

MSC:34K13, 34K18, 58E50.

It is well known that the critical point theory is a powerful tool to deal with the multiplicity of periodic solutions to ordinary differential systems as well as partial differential equations (see [1–6]). In 1998, Li and He [7] first applied the critical point theory to study the multiplicity of periodic solutions for delay differential equations. Especially, in 2005, Guo and Yu [8] established a variational framework for delay differential autonomous systems. In the past several years, some results on the existence of periodic solutions for the functional differential equation have been obtained by the critical point theory (see [7–14]). However, most of these functional differential equations are autonomous, the results on the non-autonomous functional differential equations are relatively few (see [12, 13]).

Motivated by the work of [11, 12], we consider a class of nonautonomous second-order delay differential equation

where \lambda >{\pi }^2/{\tau }^2, \tau >0, f\in C^1(\mathbf{R}\times {\mathbf{R}}^2,\mathbf{R}).

In this paper, we have the following conditions on f.

(f 1) f(t+\tau ,x)=f(t,x), f(t,-x,-y)=-f(t,x,y) and

for all x,y\in \mathbf{R}, t\in [0,\tau ];

(f 2) there exist four τ-periodic and continuous functions a(t), b(t), c(t) and d(t) such that

and

uniformly for t\in [0,\tau ];

(f 3^±) |\mathrm{\nabla }F(t,z)-B(t)z| is bounded for all z=(x,y)\in {\mathbf{R}}^2, t\in [0,\tau ] and

uniformly for t\in [0,\tau ], where

Theorem 1.1 Assume that f satisfies (f 1)-(f 3⁻) with (A(t)z,z)\le 0, (B(t)z,z)>(\lambda -{\pi }^2/{\tau }^2){|z|}^2>0 for z\in {\mathbf{R}}^2\mathrm{\setminus }\{0\}, t\in [0,\tau ]. Then (1.1) possesses at least 2m pairs 2τ-periodic solutions, where

{\mathbf{Z}}^{+} is the set of all positive integers.

Theorem 1.2 Assume that f satisfies (f 1)-(f 3⁺) with (A(t)z,z)\ge 0, for z\in {\mathbf{R}}^2\mathrm{\setminus }\{0\}, m_0\in {\mathbf{Z}}^{+}, t\in [0,\tau ]. Then (1.1) possesses at least 2[m_1-m_0+1] pairs 2τ-periodic solutions, where

In this paper, the main purpose is to study the multiplicity of periodic solutions for systems (1.1) via some recent critical point theorems for strongly indefinite functionals. In order to achieve this, some preliminaries are necessary. Let X and Y be Banach spaces with X being separable and reflexive, and set E=X\oplus Y. Let \mathcal{S}\subset X^{\ast } be a dense subset. For each s\in \mathcal{S}, there is a semi-norm on E defined by

We denote by {\mathcal{T}}_{\mathcal{S}} the topology on E induced by the semi-norm family \{p_s\}, and let ω and {\omega }^{\ast } denote the weak-topology and weak*-topology, respectively. Clearly, the {\mathcal{T}}_{\mathcal{S}} topology contains the product topology on E=X\oplus Y produced by the weak topology on X and the strong topology on Y.

For a functional \mathrm{\Phi }\in C^1(E,\mathbf{R}), we write {\mathrm{\Phi }}_a=\{u\in E:\mathrm{\Phi }(u)\ge a\}. Recall that {\mathrm{\Phi }}^{\prime } is weakly sequentially continuous if u_k\rightharpoonup u in E, one has {lim}_{k\to \mathrm{\infty }}{\mathrm{\Phi }}^{\prime }(u_k)v\to {\mathrm{\Phi }}^{\prime }(u_k)v for each v\in E, i.e., {\mathrm{\Phi }}^{\prime }:(E,\omega )\to (E^{\ast },{\omega }^{\ast }) is sequentially continuous. For c\in \mathbf{R}, we say that Φ satisfies the {(C)}_c condition if any sequence \{u_k\}\subset E, such that \mathrm{\Phi }(u_k)\to c and (1+\parallel u_k\parallel ){\mathrm{\Phi }}^{\prime }(u_k)\to 0 as k\to \mathrm{\infty }, contains a convergent subsequence.

Suppose that

({\mathrm{\Phi }}_0) \mathrm{\Phi }\in C^1(E,\mathbf{R}), {\mathrm{\Phi }}_c is {\mathcal{T}}_{\mathcal{S}}-closed for every c\in \mathbf{R} and {\mathrm{\Phi }}^{\prime }:({\mathrm{\Phi }}_c,{\mathcal{T}}_{\mathcal{S}})\to (E^{\ast },{\omega }^{\ast }) is continuous;

({\mathrm{\Phi }}_1) there exists \rho >0 such that \kappa :=inf\mathrm{\Phi }(B_{\rho }\cap Y)>0=\mathrm{\Phi }(0), where B_{\rho }=\{u\in E:\parallel u\parallel =\rho \};

({\mathrm{\Phi }}_2) there exist a finite dimensional subspace Y_0\subset Y and R>\rho such that and , where E_0:=X\oplus Y_0, and S_0=\{u\in E_0:\parallel u\parallel \le R\}.

The following critical point theorem will be used later (see [6, 14]).

Theorem A Assume that Φ is even and ({\mathrm{\Phi }}_0)-({\mathrm{\Phi }}_2) are satisfied. Then Φ has at least m=dim{Y}_0 pairs of critical points with critical values less than or equal to \overline{c} provided Φ satisfies the {(C)}_c condition for all c\in [\kappa ,\overline{c}].

In this section, we establish a variational structure which enables us to reduce the existence of 2τ periodic solutions of (1.1) to a classic Hamiltonian system. First, we have the following lemma by using similar arguments [11].

Lemma 2.1 Suppose that f\in C^1(\mathbf{R}\times {\mathbf{R}}^2,\mathbf{R}) satisfies (f 1), then the function

satisfies

and H(t,x,y)\in C^2(\mathbf{R}\times {\mathbf{R}}^2,\mathbf{R}).

Proof It is obvious that \frac{\partial H}{\partial x}=f(t,x,y). By (f 1), \frac{\partial f(t,x,y)}{\partial y}=\frac{\partial f(t,y,x)}{\partial x}. Then we have

The proof of Lemma 2.1 is complete. □

Suppose that x is a periodic solution of (1.1) with 2τ, let y(t)=x(t-\tau ), then

We denote z=(x,y). Then

where H(t,z) is defined in (2.1) and \mathrm{\nabla }H(t,z) denotes the gradient of H(t,z) with respect to the z variable. It is easy to obtain the following lemma.

Lemma 2.2 For any 2τ periodic solution of (1.1), let y(t)=x(t-\tau ), then z=(x,y) is a 2τ periodic solution of (2.2). Conversely, for any 2τ periodic solution z=(x,y) of (2.2) satisfying y(t)=x(t-\tau ), x is a 2τ periodic solution of (1.1).

For S^1=\mathbf{R}/(2\tau \mathbf{Z}), let C^{\mathrm{\infty }}(S^1,{\mathbf{R}}^2) denote the space of 2τ periodic C^{\mathrm{\infty }} functions on R with values in {\mathbf{R}}^2. For any z\in C^{\mathrm{\infty }}(S^1,{\mathbf{R}}^2), it has the following Fourier expansion in the sense that it is convergent in the space L^2(S^1,{\mathbf{R}}^2),

where a_0,a_j,b_j\in {\mathbf{R}}^2, j=1,2,\dots .

Let z\in L^2(S^1,{\mathbf{R}}^2). If there exists a function y\in L^2(S^1,{\mathbf{R}}^2) such that, for every x\in C^{\mathrm{\infty }}(S^1,{\mathbf{R}}^2),

then y is called a weak derivative of z denoted by y=z^{\prime }(t).

Let

For any z,y\in H^1(S^1,{\mathbf{R}}^2), 〈\cdot ,\cdot 〉 and \parallel \cdot \parallel can be explicitly expressed by

and

We define an operator L:H^1(S^1,{\mathbf{R}}^2)\to H^1(S^1,{\mathbf{R}}^2) by the Riesz representation theorem

By a direct computation, L is a bounded self-adjoint linear operator on H^1(S^1,{\mathbf{R}}^2).

By (f 2) and (3.6) in Section 3, there exist two continuous functions \alpha (t)>0 and \beta (t)\ge 0 such that

By using similar arguments as in [3], we have that

where

Thus the critical points of \phi (z) in H^1(S^1,{\mathbf{R}}^2) are classical solutions of (2.2).

Denote E=\{z=(x,y)\in H^1(S^1,{\mathbf{R}}^2):y(t)=x(t-\tau )\}. By a direct computation, we have

where a_0,a_j,b_j\in \mathbf{R}, j=1,2,\dots .

Note that for any z=(x,y)\in E, y(t)=x(t-\tau ) and \mathrm{\nabla }H(t,z)=(f(t,x,y),f(t,y,x)). Due to the fact that f(t,x(t-\tau ),y(t-\tau ))=f(t,y(t),x(t)), we know that \mathrm{\nabla }H(t,z)\in E whatever z\in E. Therefore, we have the following lemma.

Lemma 2.3 If z(t) is a critical point of φ in E, then z(t) is a critical point of φ in H^1(S^1,{\mathbf{R}}^2).

Moreover, we also denote by M^{+}(\cdot ), M^{-}(\cdot ) and M^0(\cdot ) the positive definite, negative definite and null subspaces of the self-adjoint linear operator defining it, respectively.

Then E has an orthogonal decomposition

Let

where a_j,b_j\in \mathbf{R}. We have

There exists \sigma >0 such that

where J^{+}=\{j\ge 1:\frac{1}{1+j^2}(\frac{{\pi }^2}{{\tau }^2}{j}^2-\lambda )>0\}, , J^0=\{j\ge 1:\frac{1}{1+j^2}(\frac{{\pi }^2}{{\tau }^2}{j}^2-\lambda )=0\}. Therefore, we get

Remark 2.1 The condition \lambda >{\pi }^2/{\tau }^2 is to ensure J^{-}\ne \mathrm{\varnothing }.

In this section, c_i stand for different positive constants for i\in {\mathbf{Z}}^{+}.

By a direct computation, (f 1) and (f 2) imply that H(t,z) is even and satisfies

uniformly for t\in [0,\tau ].

(f 3^±) implies that

(3.3±)

uniformly for t\in [0,\tau ].

Lemma 3.1 Suppose that f satisfies (f 1)-(f 3^±). Then the function φ satisfies the {(C)}_c condition for any c\in \mathbf{R}.

Proof First we define an operator

for any z,y\in E. By a direct computation, B is a bounded self-adjoint linear operator on E. Thus L+B is also a self-adjoint linear operator on E. Then E has an orthogonal decomposition

Also there exists {\sigma }_1>0 such that

So, set

and

for any z,y\in E, \phi (z) is defined in Section 2.

Let \{z_k\}\subset E be any sequence such that

We first prove that \{z_k\} is bounded. For any \epsilon >0, (3.2) implies that there exists a constant C_{\epsilon }>0 such that

for all z\in {\mathbf{R}}^2 and t\in [0,2\tau ]. Since z_k\in E, we have

where z_k^{+}\in M^{+}(L+B), z_k^{-}\in M^{-}(L+B), z_k^0\in M^0(L+B). By (3.3^±), there exists a constant d>0 such that

Therefore we have

Thus we have that \{\parallel z_k^{+}\parallel \} is bounded. Using similar arguments, we can prove that \{\parallel z_k^{-}\parallel \} is bounded. Consider \{\parallel z_k^0\parallel \}. Arguing indirectly, we suppose \{\parallel z_k^0\parallel \} is unbounded, then we have \parallel z_k^0\parallel \to \mathrm{\infty }. According to the definition of M^0(L+B), this implies that there are constants d_1,d_2>0 such that

By (3.7), we have

Then

By (3.8) and (3.3^±), (3.9) is a contradiction. Hence \{\parallel z_k^0\parallel \} is bounded. Therefore there exists a constant d_3>0 such that

So, we get \{z_k\} is bounded, and going if necessary to a subsequence, we can assume that z_k\rightharpoonup z in E and z_k\to z in C(S^1,{\mathbf{R}}^N). Write z_k=z_k^{+}+z_k^{-}+z_k^0 and z=z^{+}+z^{-}+z^0, then z_k^{\pm }\rightharpoonup z^{\pm } in E, z_k^0\to z^0 in E and z_k^{\pm }\to z^{\pm } in C(S^1,{\mathbf{R}}^N).

In view of (3.6) and z_k^{-}\to z^{-} in C(S^1,{\mathbf{R}}^2), it is easy to verify

and

But then 〈{\mathrm{\Phi }}^{\prime }(z_k)-{\mathrm{\Phi }}^{\prime }(z),z_k^{-}-z^{-}〉\to 0 as k\to \mathrm{\infty }, and

This yields z_k^{-}\to z^{-} in E. Similarly, z_k^{+}\to z^{+} in E and hence z_k\to z in E, that is, Φ satisfies the {(C)}_c condition. Thus φ satisfies the {(C)}_c condition. The proof of Lemma 3.1 is complete. □

Proof of Theorem 1.1 For z\in E, let X=M^{+}(L)\oplus M^0(L), Y=M^{-}(L), E=X\oplus Y, and

where \phi (z) and \psi (z) are from Section 2.

In order to obtain this theorem, we apply Theorem A to the functional \mathrm{\Phi }(z). The proof of this theorem is divided into the following three steps.

Step 1. Φ satisfies ({\mathrm{\Phi }}_0).

We first check that {\mathrm{\Phi }}_c is {\mathcal{T}}_{\mathcal{S}} -closed for any c\in \mathbf{R}. Let \{z_k\} be any sequence {\mathcal{T}}_{\mathcal{S}}-converging to some z\in E. Write z_k=z_k^{+}+z_k^0+z_k^{-} and z=z^{+}+z^0+z^{-}, then z_k^{-}\to u^{-} in E and hence \{z^{-}\} is bounded in the norm topology. Note that (B(t)z,z)>\lambda -{\pi }^2/{\tau }^2>0 for z\in {\mathbf{R}}^2\mathrm{\setminus }\{0\} and for t\in [0,2\tau ], then for any z\in E and small \epsilon >0, we have by (3.10) and (3.6)

which implies that Ψ is bounded from below on E. Consequently, combining z_k\in {\mathrm{\Phi }}_c and \frac{1}{2}〈L{z}_k^{+},z_k^{+}〉=-\frac{1}{2}〈L{z}_k^{-},z_k^{-}〉-\mathrm{\Phi }(z_k)-\mathrm{\Psi }(z_k) shows that \{z_k^{+}\} is bounded in E by (2.3) and hence

Moreover, since {\parallel z_k\parallel }_{L^2}^2={\parallel z_k^{+}\parallel }_{L^2}^2+{\parallel z_k^0\parallel }_{L^2}^2+{\parallel z_k^{-}\parallel }_{L^2}^2, we have

It follows from (3.11) and the above inequality that \{z_k^0\} is also bounded in E since all norms are equivalent in a finite dimensional space. Then {\parallel z_k\parallel }^2 (={\parallel z_k^{+}\parallel }^2+{\parallel z_k^0\parallel }^2+{\parallel z_k^{-}\parallel }^2) is bounded, and hence we can assume that \{z_k\} converges weakly to z=z^{+}+z^0+z^{-} in E. Thus we have \mathrm{\Psi }(z_k)\to \mathrm{\Psi }(z). Note that {〈Ly,y〉}^{1/2} is an equivalent norm on H^{+}. By the lower semi-continuity of the norm, we get

that is, z\in {\mathrm{\Phi }}_c and hence {\mathrm{\Phi }}_c is {\mathcal{T}}_{\mathcal{S}}-closed.

Next, we prove that {\mathrm{\Phi }}^{\prime }:({\mathrm{\Phi }}_c,{\mathcal{T}}_{\mathcal{S}})\to (E^{\ast },{\omega }^{\ast }) is continuous. To achieve this, it is sufficient to demonstrate that {\mathrm{\Psi }}^{\prime } has the same property. Suppose z_k\rightharpoonup u in E, then \{z_k\} converges uniformly to z on [0,2\tau ]. Hence, for every given y\in E, we see that (\mathrm{\nabla }H(t,z_k(t)),y(t)) converges to (\mathrm{\nabla }H(t,z(t)),y(t)) in measure on [0,2\tau ]. Moreover, by (3.6), one has

for all k and t\in [0,2\tau ], where \overline{b}={max}_{t\in [0,\tau ]}\{|B(t)|\} and {\parallel \cdot \parallel }_{\mathrm{\infty }} denotes the natural norm of C(S^1,{\mathbf{R}}^2). Thus, the Vitali theorem is applicable and

for any y\in E. So, Φ satisfies ({\mathrm{\Phi }}_0).

Step 2. Φ satisfies ({\mathrm{\Phi }}_1).

By (3.1), there exist , {\rho }_0>0 such that

By Proposition 1.1 in [3], there is a positive constant ξ such that {\parallel z\parallel }_{\mathrm{\infty }}\le \xi \parallel z\parallel. Set small , then for each z\in Y with \parallel z\parallel \le \rho, one has {\parallel z\parallel }_{\mathrm{\infty }}\le {\rho }_0, and hence by (3.12)

Therefore,

and hence ({\mathrm{\Phi }}_1) holds.

Step 3. Φ satisfies ({\mathrm{\Phi }}_2).

Let

Obviously, Y_0\subset Y and dim{Y}_0=2m. In order to obtain the desired conclusion, it is sufficient to prove that \mathrm{\Phi }(z)\to -\mathrm{\infty } as \parallel z\parallel \to \mathrm{\infty } on E_0:=X\oplus Y_0.

Let \mathrm{\Gamma }=\lambda -\frac{{\pi }^2}{{\tau }^2}. By the definition of m, there exists a constant such that

for t\in [0,\tau ]. Clearly, for any y\in Y_0,

Let \tilde{F}(t,z)=H(t,z)-\frac{1}{2}(B(t)z,z). We claim that

Indeed, for 0\ne z\in E, by (3.6), one has

which implies that (3.15) is true by the arbitrariness of ε. Then, for z=z^{+}+z^0+z^{-}\in E_0, by (2.3), (3.13) and (3.14), one has

Since M^0 and Y_0 are finitely dimensional, (3.15) and the above estimate imply that \mathrm{\Phi }(z)\to -\mathrm{\infty } as \parallel z\parallel \to \mathrm{\infty }, z\in E_0:=X\oplus Y_0. Hence ({\mathrm{\Phi }}_2) holds.

The proof of Theorem 1.1 is complete. □

Proof of Theorem 1.2 For z\in E, let X=M^{-}(L)\oplus M^0(L), Y=M^{+}(L), E=X\oplus Y, and

and

Then the conclusion is obtained by the same argument as in the proof of Theorem 1.1. The proof of Theorem 1.2. is complete. □

Example 3.1 Consider the following equation:

where x(t)\ge 0. Let

Then

Let

Then

By a straightforward computation, we have

and

uniformly for t\in [0,\pi ].

Take \lambda =6, \alpha (t)=-5+sin\pi t, \beta (t)=1-sin\pi t, {\gamma }^2(t)=\frac{1}{6}. By Theorem 1.2, we have a(t)=\frac{1}{{\gamma }^2(t)}+\alpha (t)\ge 0, b(t)=d(t)=1-sin\pi t\ge 0, c(t)=-5+sin\pi t for t\in [0,\pi ], z=(x,y), x\ge 0, y\ge 0,