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Periodic solutions for a class of second order delay differential systems
Boundary Value Problems volume 2016, Article number: 32 (2016)
Abstract
Consider the periodic boundary value problem
where \(\tau>0\) is a given constant, \(r\in\mathbf{R}\) is a parameter. Periodic solutions are obtained by using linking theorems.
1 Introduction
Consider the periodic boundary value problem
where \(\tau>0\) is a given constant, \(r\in\mathbf{R}\) is a parameter. \(F:[0, 2\tau]\times\mathbf{R}^{N}\rightarrow R\) satisfies the following assumptions:
(A) \(F(t,x)\) is measurable with respect to t, for all \(x\in{\mathbf{R}}^{N}\), continuously differentiable in x, for a.e. \(t\in[0, 2\tau]\), and there exist \(a\in C({\mathbf{R}}^{+};{\mathbf{R}}^{+})\) and \(b\in L^{1}([0,2\tau];{\mathbf{R}}^{+})\) such that
for all \(x\in{\mathbf{R}}^{N}\) and a.e. \(t\in[0, 2\tau]\).
Variational methods are very powerful techniques in nonlinear analysis and are extensively used in many disciplines of pure and applied mathematics, including ordinary and partial differential equations, mathematical physics, and geometrical analysis. The existence and multiplicity of solutions for Hamilton systems and Schrödinger equations have been studied extensively via critical point theory; see [1–8].
In the past several years, some results on the existence of periodic solutions for the functional differential equation by the critical point theory have been obtained (see [9–18]). In [16], the authors obtained the multiplicity results for periodic solutions to (1.1) by using critical point theory. Our proof method in this paper is different from the literature [16].
Motivated by the above observation, in this paper, we study the existence of periodic solutions to the system (1.1). The following theorems are the main results of our paper.
- (H1):
-
There exists a constant \(R_{1}>0\) such that
-
(i)
\(F(t,x)\geq0\), \(\forall|x|\leq R_{1}\), \(t\in[0,2\tau]\); or
-
(ii)
\(F(t,x)\leq0\), \(\forall|x|\leq R_{1}\), \(t\in[0,2\tau]\).
-
(i)
- (H2):
-
\(\lim_{|x|\rightarrow\infty}\inf\frac {F(t,x)}{|x|^{2}}>0\) uniformly for \(t\in[0,2\tau]\).
- (H3):
-
\(\lim_{|x|\rightarrow0}\frac{|\nabla F(t,x)|}{|x|}=0\) uniformly for \(t\in[0,2\tau]\).
- (H4):
-
There exist \(\alpha>1 \) and \(a>0\) such that
$$\bigl|\nabla F(t,x)\bigr|\leq a\bigl(|x|^{\alpha}+1\bigr), \quad \forall(t,x)\in[0,2 \tau ]\times{\mathbf{R}}^{N}. $$ - (H5):
-
There exist \(\mu>2\), \(0<\beta<2\), \(R>0\), and a function \(b(t)\in L^{1} ([0,2\tau]; {\mathbf{R}}^{+})\) such that
$$\mu F(t,x)\leq\bigl(\nabla F(t,x), x\bigr)+ b(t)|x|^{\beta},\quad \forall |x|\geq R, t\in[0,2\tau]. $$ - (H5′):
-
There exist \(\beta>\alpha\geq1\) and \(b>0\), \(R>0\) such that
$$\bigl(\nabla F(t,x), x\bigr)-2F(t,x)\geq b|x|^{\beta}, \quad \forall|x| \geq R, t\in[0,2\tau]. $$
Theorem 1.1
Assume that (H1)-(H5) hold, if 0 is an eigenvalue of L (L is defined in Section 2), BVP (1.1) has at least one nontrivial solution.
Theorem 1.2
Assume that (H1)-(H4) and (H5′) hold, if 0 is an eigenvalue of L (L is defined in Section 2), BVP (1.1) has at least one nontrivial solution.
Remark 1.1
The condition (H5) and the condition (H5′) are appeared, respectively, in [7] and [8].
2 Preliminaries
In order to seek 2Ï„-periodic orbits of (1.1), let us transform (1.1) to
by making the change of variable \(t\rightarrow\frac{\pi}{\tau}t=\lambda ^{-1} t \), which implies that a 2Ï€-periodic solution of (2.1) corresponds to a 2Ï„-periodic solution of (1.1). Hence we will only look for the 2Ï€-periodic solutions of (2.1) in the sequel.
Let \(L^{2}(S^{1}, {\mathbf{R}}^{N})\) be the space of square integrable 2Ï€-periodic vector-valued functions with dimension N, and \(C^{\infty}(S^{1}, {\mathbf{R}}^{N})\) be the space of 2Ï€-periodic \(C^{\infty}\) vector-valued functions with dimension N. For any \(u\in C^{\infty}(S^{1}, {\mathbf{R}}^{N})\), it has the following Fourier expansion in the sense that it is convergent in the space \(L^{2}(S^{1}, {\mathbf{R}}^{N})\):
where \(a_{0}, a_{j}, b_{j}\in{\mathbf{R}}^{N}\). Let \(H^{1}(S^{1}, {\mathbf {R}}^{N})\) be the closure of \(C^{\infty}(S^{1}, {\mathbf{R}}^{N})\) with respect to the Hilbert norm
More specifically, \(H^{1}(S^{1}, {\mathbf{R}}^{N})=\{u\in L^{2}(S^{1}, {\mathbf {R}}^{N}): \|u\|<+\infty\}\) with the inner product
for any \(u,v\in H^{1}(S^{1}, {\mathbf{R}}^{N})\), where \((\cdot,\cdot)\) denotes the usual inner product in \({\mathbf{R}}^{N}\). The norm on \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) is defined by
It is well known that \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) is compactly embedded in \(C(S^{1}, {\mathbf{R}}^{N})\) (see Proposition 1.2 of [15]). Define two operators \(L_{0}\) and \(L_{1}\) from \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) into \(H^{*}(S^{1}, {\mathbf{R}}^{N})\) as follows: for any \(u\in H^{1}(S^{1}, {\mathbf{R}}^{N})\), which are given by
for all \(v\in H^{*}(S^{1}, {\mathbf{R}}^{N})\), where \(\dot{u}(t)\) and \(H^{*}(S^{1}, {\mathbf{R}}^{N})\) denote the weak derivative of u and the dual space of \(H^{1}(S^{1}, {\mathbf{R}}^{N})\), respectively. By the Riesz representation theorem, we can identify \(H^{*}(S^{1}, {\mathbf {R}}^{N})\) with \(H^{1}(S^{1}, {\mathbf{R}}^{N})\). Thus, \(L_{i}u\) can also be viewed as an element belonging to \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) such that \(\langle L_{i}u, v\rangle=(L_{i}u)(v)\) for all \(u, v\in H^{1}(S^{1}, {\mathbf{R}}^{N})\), where \(i=0\) or 1. It is easy to check that \(L_{0}\) and \(L_{1}\) are bounded linear operators on \(H^{1}(S^{1}, {\mathbf {R}}^{N})\). Note that \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) is compactly embedded in \(C(S^{1}, {\mathbf{R}}^{N})\), thus \(L_{1}\) is compact on \(H^{1}(S^{1}, {\mathbf{R}}^{N})\).
Lemma 2.1
[16]
The operator \(L:=L_{0}+L_{1}\), that is,
is self-adjoint on \(H^{1}(S^{1}, {\mathbf{R}}^{N})\). The operators \(L_{0}\) and \(L_{1} \) are also self-adjoint on \(H^{1}(S^{1}, {\mathbf{R}}^{N})\).
Lemma 2.2
[16]
The essential spectrum of \(L_{0}\) on \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) is just \(\{-1,1\}\).
In view of the above facts, 0 is not in the essential spectrum of L as it is a compact perturbation of the self-adjoint operator \(L_{0}\). This implies that 0 is at most an eigenvalue of finite multiplicity of L.
Remark 2.1
\(H^{1}(S^{1}, {\mathbf{R}}^{N})\) has an orthogonal decomposition
where \(H^{0}=\operatorname{Ker}L\) is finite dimensional, and \(H^{+}\), \(H^{-}\) are L-invariant subspaces such that for some \(\sigma>0\),
for all \(u\in H^{+}\),
for all \(u\in H^{-}\).
Remark 2.2
By [16], we can choose the proper r to make that the set \(\{j\geq0| \frac{(-1)^{j} j^{2}-r \lambda^{2}}{j^{2}+1}=0\}\neq{\emptyset }\), which means \(H^{0}\neq{\emptyset}\), where \(j\in\mathbf{Z}\).
Lemma 2.3
[5]
There exists \(C>0\) such that
for \(p=1+\alpha, 2\), \(\|u\|_{\infty}=\max_{t\in[0, 2\pi]}|u(t)|\), \(\forall u\in H^{1}(S^{1}, {\mathbf{R}}^{N})\).
Now consider the functional φ defined on \(H^{1}(S^{1}, {\mathbf {R}}^{N})\) given by
Since F satisfies the assumption (A), a standard argument shows the following.
Lemma 2.4
[16]
The functional φ is continuously differentiable on \(H^{1}(S^{1}, {\mathbf{R}}^{N})\) and \(\varphi'\) is defined by
for all \(v\in H^{1}(S^{1}, {\mathbf{R}}^{N})\).
Lemma 2.5
[16]
A critical point \(u\in H^{1}(S^{1}, {\mathbf {R}}^{N})\) of functional φ is equivalent to a 2π-periodic solution of system (2.1).
In this paper, we will use the following local linking Theorem A to prove our theorems. The following concepts appeared in [4, 7, 8].
Let X be a real Banach space with direct decomposition \(X = X^{1}\oplus X^{2}\). Consider two sequences of subspaces:
such that \(X^{j}=\overline{\bigcup_{n\in N}X_{n}^{j}}\), \(j = 1, 2\). For every multi-index \(\alpha=(\alpha_{1}, \alpha_{2})\in{\mathbf{N}}^{2}\), we denote by \(X_{\alpha}\) the space \(X_{\alpha_{1}}\oplus X_{\alpha_{2}}\). We say \(\alpha\leq\beta\) if \(\alpha_{1} \leq\beta_{1}\), \(\alpha_{2} \leq \beta_{2}\).
Definition 2.1
[4]
A sequence \(\{\alpha_{n}\}\subset N^{2}\) is said to be admissible if, for every \(\alpha\in{\mathbf{N}}^{2}\) there is \(m\in N\) such that \(n\geq m\Rightarrow\alpha_{n}>\alpha\).
Definition 2.2
[4]
Let \(f\in C^{1}(X, R)\). Then the function f satisfies the \((PS)^{*}\) condition if every sequence \(\{u_{\alpha_{n}}\} \) such that \(\{\alpha_{n}\}\) is admissible and
contains a subsequence which converges to a critical point of f, where \(f_{\alpha}=f|_{X_{\alpha}}\).
Definition 2.3
[4]
Let X be a Banach space with a direct sum decomposition \(X = X^{1}\oplus X^{2}\). The function \(f\in C^{1}(X, R)\) has a local linking at 0, with respect to \((X^{1}, X^{2})\), if, for some \(r >0\),
Theorem A
[4]
Suppose that \(f\in C^{1}(X, {\mathbf{R}})\) satisfies the following assumptions:
-
(f1)
f has a local linking at 0 and \(X^{1}\neq\{0\}\);
-
(f2)
f satisfies \((PS)^{*}\) condition;
-
(f3)
f maps bounded sets into bounded sets;
-
(f4)
for every \(m\in N\), \(f(x)\rightarrow-\infty\) as \(\|x\| \rightarrow\infty\) on \(X_{m}^{1}\oplus X^{2}\).
Then f has at least one nonzero critical point.
3 Proofs of theorems
In this section, \(c_{i}\) stand for different positive constants for \(i\in \mathbf{Z}^{+}\), \(\mathbf{Z}^{+}\) is the set of all positive integers.
Let
where \(X^{1}=H^{+}\), \(X^{2}=H^{-} \oplus H^{0}\).
Lemma 3.1
Under assumption (H2), (H4)-(H5), the functional φ satisfies the \((PS)^{*}\) condition.
Proof
Let \(\{u_{\alpha_{n}}\}\) be a sequence such that \(\{ \alpha_{n}\}\) is admissible and
For the sake of notational simplicity, set \(u_{n} = u_{\alpha_{n}}\).
Claim 1. \(\{u_{n}\}\) is bounded in X.
If not, passing to a subsequence if necessary, we assume that \(\|u_{n}\|\rightarrow\infty\) as \(n\rightarrow\infty\). Set \(v_{n}=\frac{u_{n}}{\|u_{n}\|}\), then \(\{v_{n}\}\) is bounded in X. Hence, there exists a subsequence, still denoted by \(\{v_{n}\}\). Write \(v_{n}=v_{n}^{+}+v_{n}^{-}+v_{n}^{0}\) and \(v=v^{+}+v^{-}+v^{0}\), then
In view of (H4), we have
For \(|x|\leq R\) and \(t\in[0,2\tau]\), one has
together with (H5), one has
Hence, by (2.6) and (2.7), we have
This implies that
since \(\mu>2\). Together with (3.4), we have
since \(0<\beta<2\). Hence,
That means that
By using (2.3) and (2.4), we discuss two cases:
If \(\|v^{+}\|=0\), we have \(\|v^{-}\|=0\), then we obtain \(\|v\|=\| v^{0}\|=\|v_{n}\|=1\).
If \(\|v^{+}\|\neq0\), we have \(\|v^{-}\|\neq0\), then we obtain \(\|v\|\neq0\).
So we get \(v(t)\not\equiv0\) for \(t\in[0, 2\pi]\). Set
then \(\operatorname{meas}(E_{0})>0\).
From (H2), we get
By the boundedness of \(\varphi(u_{n})\) and (3.4), we have
which together with \(0<\beta<2\) implies that
This contradicts (3.5). Therefore, \(\{u_{n}\}\) is bounded in X.
Claim 2. \(\{u_{n}\}\) possesses a strong convergent subsequence in X.
Write \(u_{n}=u_{n}^{+}+u_{n}^{-}+u_{n}^{0}\) and \(u=u^{+}+u^{-}+u^{0}\), then
In view of (3.2) and \(u_{n}^{-}\rightarrow u^{-}\) in \(C(S^{1}, {\mathbf{R}}^{N})\), it is easy to verify
Note that
Thus,
This yields \(u_{n}^{-}\rightarrow u^{-}\) in X. Similarly, \(u_{n}^{+}\rightarrow u^{+}\) in X. Hence \(u_{n}\rightarrow u\) in X. Hence, \(\{u_{n}\}\) possesses a strong convergent subsequence in X. The proof of Lemma 3.1 is complete. □
Lemma 3.2
Under assumption (H2), (H4), and (H5′), the functional φ satisfies the \((PS)^{*}\) condition.
Proof
Let \(\{u_{\alpha_{n}}\}\) be a sequence such that \(\{ \alpha_{n}\}\) is admissible and
For the sake of notational simplicity, set \(u_{n} = u_{\alpha_{n}}\).
Claim 1. \(\{u_{n}\}\) is bounded in X.
If not, passing to a subsequence if necessary, we assume that \(\|u_{n}\|\rightarrow\infty\) as \(n\rightarrow\infty\). In view of (H5′), there exists \(c_{5}>0\) such that
Hence, we have
This implies that
Let \(u_{n}=u_{n}^{+}+u_{n}^{-}+u_{n}^{0}\in H^{+}\oplus H^{-} \oplus H^{0}\). By (H4), we have
for all n, where we use the Hölder inequality and (2.5). We have
Similarly for \(u_{n}^{-}\), we also get
Again, by (H5′), we have
since \(\beta>\alpha\geq1\). Since dim\(H^{0}<\infty\),
So we get
Hence by (3.8)-(3.10), one has
a contradiction. Therefore, \(\{u_{n}\}\) is bounded in X.
Using similar arguments to the proof of Claim 2 in Lemma 3.1, \(\{u_{n}\}\) possesses a strong convergent subsequence in X. The proof of Lemma 3.2 is complete. □
Lemma 3.3
Under assumption (H1)-(H4), the function φ satisfies the conditions (f1), (f3), and (f4) of Theorem A.
Proof
We only consider the case (i) in (H1). The other case is similar.
(1) We claim that φ has a local linking at 0 with respect to \((X^{1}, X^{2})\).
By (H3), for any \(\varepsilon>0\), there exists \(\delta>0\) such that
By (3.2) and (3.11), there exists \(c_{8}>0\) such that
Hence, for \(u\in X^{1}\), we have
Taking \(\varepsilon=\sigma/(4\lambda^{2} C)\) and noting that \(\alpha>1\) in (H4), we can find a constant \(\rho_{1} > 0\) such that
On the other hand, let \(u=u^{0}+u^{-}\in X^{2}\) satisfying \(\|u\|\leq\rho_{0}=R_{1}/(2C)\) and let
Then we have
for all \(t\in[0,2\pi]\). Consequently,
Hence, from (H1)(i), we obtain
On the one hand, one obtains
It follows from (3.12) that
for all \(t\in\Omega_{2}\), which implies that
Setting \(\varepsilon=\sigma/(16C^{2})\), we have
Consequently,
where \(\rho_{2}<\rho_{0}\) is small enough.
(2) We claim that for every \(m\in{\mathbf{N}}\), \(\varphi (u)\rightarrow-\infty\) as \(\|u\|\rightarrow\infty\) on \(X_{m}^{1}\oplus X^{2}\).
Since \(X_{m}^{1}\) and \(H^{0}\) are finite dimensional, we can choose \(\sigma _{m}>0\), \(C_{m}>0\) such that
By (H2), there exists \(c_{9}>0\) such that
So for \(u\in X_{m}^{1}\oplus X^{2}\),
for \(\|u\|\rightarrow\infty\).
(3) By (3.2), we see that φ maps bounded sets into bounded sets.
The proof of Lemma 3.3 is complete. □
Proof of Theorem 1.1
By Lemmas 3.1 and 3.3, all conditions of Theorem A are satisfied. Thus, problem (2.1) has at least one nonzero critical point. □
Proof of Theorem 1.2
By Lemmas 3.2 and 3.3, all conditions of Theorem A are satisfied. Thus, problem (2.1) has at least one nonzero critical point. □
References
Ding, Y: Variational Methods for Strongly Indefinite Problems. World Scientific, Singapore (2007)
Guo, Y: Nontrivial periodic solutions for asymptotically linear Hamiltonian systems with resonance. J. Differ. Equ. 175, 71-87 (2001)
Su, JB: Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity. J. Differ. Equ. 145, 252-273 (1998)
Li, SJ, Willem, M: Applications of local linking to critical point theory. J. Math. Anal. Appl. 189, 6-32 (1995)
Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)
Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. Am. Math. Soc., Providence (1986)
Zhang, Q, Tang, XH: New existence of periodic solutions for second order non-autonomous Hamiltonian systems. J. Math. Anal. Appl. 369, 357-367 (2010)
He, X, Wu, X: Periodic solutions for a class of nonautonomous second order Hamiltonian systems. J. Math. Anal. Appl. 341, 1354-1364 (2008)
Fei, G: Multiple periodic solutions of differential delay equations via Hamiltonian systems (I). Nonlinear Anal. 65, 25-39 (2006)
Fei, G: Multiple periodic solutions of differential delay equations via Hamiltonian systems (II). Nonlinear Anal. 65, 40-58 (2006)
Guo, Z, Yu, J: Multiplicity results for periodic solutions to delay differential equations via critical point theory. J. Differ. Equ. 218, 15-35 (2005)
Li, J, He, X: Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems. Nonlinear Anal. 31(1/2), 45-54 (1998)
Wu, K, Wu, X: Existence of periodic solutions for a class of first order delay differential equations dealing with vectors. Nonlinear Anal. 72, 4518-4529 (2010)
Wu, K, Wu, X: Multiplicity results of periodic solutions for systems of first order delay differential equation. Appl. Math. Comput. 218, 1765-1773 (2011)
Wu, K: Multiplicity results of periodic solutions for a class of first order delay differential equations. J. Math. Anal. Appl. 390, 427-438 (2012)
Wu, K, Wu, X, Zhou, F: Multiplicity results of periodic solutions for a class of second order delay differential equations. Nonlinear Anal. 75, 5836-5844 (2012)
Zhang, X, Meng, Q: Nontrivial periodic solutions for delay differential systems via Morse theory. Nonlinear Anal. 74, 1960-1968 (2011)
Yu, J, Xiao, H: Multiplicity periodic solutions with minimal period 4 of delay differential equations \(x'(t)=-f(t, x(t-1))\). J. Differ. Equ. 254, 2158-2172 (2013)
Acknowledgements
The authors would like to express their sincere thanks to the referees for their helpful comments. Supported by the Natural Science Foundation of Shanxi Province (No. 2012011004-1) of China and by the National Natural Science Foundation of China (No. 61473180).
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Meng, Q., Ding, J. Periodic solutions for a class of second order delay differential systems. Bound Value Probl 2016, 32 (2016). https://doi.org/10.1186/s13661-016-0542-6
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DOI: https://doi.org/10.1186/s13661-016-0542-6