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

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]).

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.

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

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

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

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 $\overline{c}:=sup\mathrm{\Phi }(E_0)<\mathrm{\infty }$ and $sup\mathrm{\Phi }(E_0\mathrm{\setminus }{S}_0)<inf\mathrm{\Phi }(B_{\rho }\cap Y)$, 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].

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

The proof of Lemma 2.1 is complete. □

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).

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

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

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

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

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.

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 $i\in {\mathbf{Z}}^{+}$.

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 $c\in \mathbf{R}$.

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

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 $C(S^1,{\mathbf{R}}^N)$.

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. □

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 (${\mathrm{\Phi }}_0$).

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

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

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

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

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

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 $E_0:=X\oplus Y_0$.