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

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

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 $F(0)=0$;

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])

Definition 1.2 ([8])

The following lemma is Example 1 in [8].

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

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).

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

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:

for all $x\in R^n$.

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 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.

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.

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

This implies (2.8).

for all $v\in E$.

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

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.

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,

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 $|x|\to 0$,

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.

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.

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 $S=\partial B_{\rho }\cap M$.

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 (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.

where $c_5,c_6,c_7>0$ are constants.