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Multiple periodic solutions for second order Josephson-type differential systems
Boundary Value Problems volume 2017, Article number: 99 (2017)
Abstract
In this paper, some new existence theorems are obtained for multiple periodic solutions of second order Josephson-type differential systems with partially periodic potential by using the minimax methods in critical point theory, which generalize and improve some known results in the literature.
1 Introduction
In this paper, we study the second order Josephson-type differential systems
where A is an \((N\times N)\)-symmetric matrix, \(h(t)\in L^{1}([0,T]; \mathbb{R}^{N})\), \(T>0\), \(\nabla F(t,x)\) denotes its gradient with respect to the second variable, that is,
and \(F:[0,T]\times \mathbb{R}^{N}\rightarrow \mathbb{R}\) satisfies the following assumptions:
- (H1):
-
\(F(t,x)\) is measurable in t for each \(x\in \mathbb{R}^{N}\) and continuously differentiable in x for a.e. \(t\in [0,T]\), and there exist \(a\in C(\mathbb{R}^{+},\mathbb{R}^{+})\), \(b\in L^{1}([0,T]; \mathbb{R}^{+})\) such that
$$ \bigl\vert F(t,x) \bigr\vert \leq a \bigl(\vert x \vert \bigr)b(t), \qquad \bigl\vert \nabla F(t,x) \bigr\vert \leq a \bigl(\vert x \vert \bigr)b(t) $$for all \(x\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\).
- (H2):
-
\(\operatorname{dim} N(A)=m\geq 1\) and matrix A has no eigenvalue of the form \(k^{2}\upsilon^{2}\) (\(k\in \mathbb{N}/\{0\}\)), where \(\upsilon =2 \pi /T\).
- (H3):
-
There exist linearly independent vectors \(e_{j}\in \mathbb{R}^{N}\) (\(1 \leq j\leq m\)) such that
$$ N(A)=\operatorname{span} \{e_{1},e_{2},\ldots ,e_{m}\} $$and
$$ \int_{0}^{T} \bigl(h(t),e_{j} \bigr) \,dt=0. $$
This problem (1.1) occurs in various branches of mathematical physics, for example, when \(A=N^{2}D^{2}\) and \(-\nabla F(t,u(t))=f(u(t))=(a _{1}\sin u_{1},\ldots ,a_{N}\sin u_{N})\), problem (1.1) reduces to the nonlinear systems of the form
where D is an \((N\times N)\)-symmetric matrix. This type of problem can be applied to describe the motion of forced linearly coupled pendulums.
During the past two decades, the existence of periodic solutions for second order differential systems have been studied extensively, and many solvability conditions have been obtained via variational methods and critical point theory. In this direction we mention the papers [1–15], and we refer the reader to [16–19] for a broad introduction to variational methods and critical point theory. It might be also interesting to study the above mentioned abstract equations with more general potentials, see the paper [20].
In the classical monograph [6], Mawhin and Willem proved that problem (1.1) has at least one solution by using the saddle point theorem under the following bounded condition: there exists \(g\in L^{1}([0,T]; \mathbb{R}^{+})\) such that
for all \(x\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\). They obtained the following result.
Theorem A
[6]
Suppose that F satisfies (H1)-(H3), (1.3) and
- (H4):
-
there exists \(T_{j}>0\) such that
$$ F ( t,x+T_{j}e_{j} ) =F(t,x), \quad 1\leq j\leq m, $$for all \(x\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\). Then Eq. (1.1) has at least one solution in \(H_{T}^{1}\), where the Sobolev space \(H_{T}^{1}\) is defined by
$$\begin{aligned} H_{T}^{1}&= \bigl\{ u:[0,T]\rightarrow \mathbb{R}^{N} \mid \textit{u is absolutely continuous}, \\ &\quad u(0)=u(T) \textit{ and } \dot{u}\in L^{2} \bigl([0,T]; \mathbb{R}^{N} \bigr) \bigr\} \end{aligned}$$and \(H_{T}^{1}\) is a Hilbert space with the norm
$$ \Vert u \Vert = \biggl( \int_{0}^{T} \bigl\vert \dot{u}(t) \bigr\vert ^{2}\,dt+ \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}}, \quad u\in H_{T}^{1}. $$
When the nonlinearity \(\nabla F(t,x)\) is sublinear, that is, there exist \(f,g\in L^{1}([0,T];\mathbb{R}^{+})\) and \(\alpha \in [0,1)\) such that
for all \(x\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\), Tang [12, 13] researched the existence of periodic solutions for system (1.1) in the case \(A\equiv 0\).
Subsequently, Feng and Han [3] generalized Theorem A to the sublinear case, and they assume that the following assumptions hold:
- (H5):
-
There exist \(T_{j}>0\), \(1\leq r\leq m\) such that
$$ F ( t,x+T_{j}e_{j} ) =F(t,x), \quad 1\leq j\leq r, $$for all \(x\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\).
- (H6):
-
\(\lim_{\Vert x \Vert \rightarrow \infty }\frac{\int_{0}^{T}F(t,x)\,dt}{ \Vert x \Vert ^{2\alpha }}=-\infty , \mbox{as } x\in N(A)\ominus \operatorname{span}\{e_{1},e_{2},\ldots ,e_{r}\}\).
Theorem B
[3]
Suppose that F satisfies (H1)-(H3), (H5)-(H6) and (1.4). Then Eq. (1.1) has at least \(r+1\) distinct solutions in \(H_{T}^{1}\).
In [15], the author obtained the following result.
Theorem C
[15]
Suppose that F satisfies (H1)-(H3), (H5), (1.4) and the following generalized Ahmad-Lazer-Paul type coercive conditions:
- (H7):
-
\(\lim_{\Vert x \Vert \rightarrow \infty }\frac{\int_{0}^{T}F(t,x)\,dt}{\Vert x \Vert ^{2\alpha }}<-L\), as \(x\in N(A)\ominus \operatorname{span}\{e_{1},e_{2},\ldots ,e_{r}\}\), where L is a positive constant. Then Eq. (1.1) has at least \(r+1\) distinct solutions in \(H_{T}^{1}\).
In this paper, we use a more general control function instead of \(\vert x \vert ^{\alpha }\) in (1.4). By using the generalized saddle point theorem due to Liu [5], we can prove the existence of multiple periodic solutions for the second order Josephson-type differential systems for a new and large range of the nonlinear term.
2 Preliminaries
In [6], Mawhin and Willem established a variational structure which enables us to reduce the existence of solutions for problem (1.1) to the existence of critical points of the following energy functional. Define the energy functional associated with problem (1.1) on \(H_{T}^{1}\)
It follows from assumption (H1) that the functional φ is continuously differentiable. Moreover, one has
Then the solutions of problem (1.1) correspond to the critical points of φ (see [6]).
Let
Therefore, we can see that
where I denotes the identity operator on \(H_{T}^{1}\) and \(K:H_{T} ^{1}\rightarrow H_{T}^{1}\) is the linear self-adjoint operator defined, using Riesz representation theorem, by
It is easy to see that K is compact. By classical spectral theory, we can decompose \(H_{T}^{1}\) into the orthogonal sum of invariant subspaces for \(I-K\)
where \(H^{0}=\mathrm{Ker}(I-K)=N(A)\) and \(\operatorname{dim} H^{-}<+\infty \), for some \(\delta >0\), we have
Lemma 2.1
[6]
There is a continuous embedding \(H_{T}^{1}\hookrightarrow C([0,T],\mathbb{R}^{N})\), and the embedding is compact. Then there exists \(C_{0}>0\) such that
Define
then
where \(u^{-}\in H^{-}, u^{+}\in H^{+}\), \(Pu^{0}\in Y_{1}\) and \(Qu^{0}=\sum_{j=1}^{r}c_{j}e_{j}\). Let
be a discrete subgroup of \(H_{T}^{1}\), where \(\mathbb{Z}\) is the set of all integers, and let \(\pi :H_{T}^{1}\rightarrow H_{T}^{1}/G\) be the canonical surjection. Let
where \(W=H^{+}\), \(Z=H^{-}\oplus Y_{1}\), \(V=Y_{0}/G\), then \(\dim Z<+ \infty \), \(\dim V<+\infty \), and V is isomorphic to the torus \(T^{r}\). The element in V can be represented as
where \(\hat{c}_{j}=c_{j}-k_{j}T_{j}\), \(0\leq \hat{c}_{j}< T_{j}\). Let
By (H3) and (H5), we have
and
Thus, \(\varphi (u)=\varphi (\hat{u})\), \(\varphi '(u)=\varphi '( \hat{u})\). Define \(\psi : X\times V\mapsto \mathbb{R}\): \(\psi (\pi (u))= \varphi (u)\), then ψ is well defined. Moreover, ψ is continuously differentiable and
Definition 2.1
[6]
Suppose that ψ satisfies the (PS) condition, that is, every sequence \(\{x_{n}\}\) of \(X\times V\) such that \(\psi \{x_{n}\}\) is bounded and \(\psi '\{x_{n}\}\rightarrow 0\) as \(n\rightarrow \infty \) possesses a convergent subsequence.
Lemma 2.2
The generalized saddle point theorem [5]
Let X be a Banach space with a decomposition \(X=Z+W\), where Z and W are two subspaces of X with \(\dim Z<+\infty \). Let V be a finite-dimensional, compact \(C^{2}\)-manifold without boundary. Let \(\psi :X\times V\rightarrow \mathbb{R}\) be a \(C^{1}\)-function satisfying the (PS) condition. Suppose that there exist constants \(\rho >0\) and \(\gamma <\beta \) such that
where \(S=\partial D\), \(D=\{z\in Z\mid \vert z \vert \leq \rho \}\). Then the functional ψ has at least \(\operatorname{cuplength}(V)+1\) critical points.
3 Main results
Here are our main results.
Theorem 3.1
Suppose that assumptions (H1)-(H3), (H5) hold and there exist constants \(M_{i}>0\), \(i=0,1,2\), and a nonnegative function \(\omega \in C([0,\infty ),[0,\infty ))\) with the properties:
- (ω1):
-
\(\omega (s)\leq \omega (t)\), \(\forall s\leq t, s,t \in [0,\infty )\);
- (ω2):
-
\(\omega (s+t)\leq M_{0}(\omega (s)+\omega (t))\), \(\forall s,t\in [0,\infty )\);
- (ω3):
-
\(0\leq \omega (s)\leq M_{1}s+M_{2}\), \(\forall s,t \in [ 0,\infty )\);
- (ω4):
-
\(\omega (s)\rightarrow +\infty \) as \(s\rightarrow +\infty \).
Moreover, there exist constant \(a>3\) and \(f,g\in L ^{1}([0,T];\mathbb{R}^{+})\) with
such that
for all \(x\in \mathbb{R}^{N}\) and a.e. \(t\in [0,T]\), and
as \(x\in N(A)\ominus \operatorname{span}\{e_{1},e_{2},\ldots ,e_{r}\}\). Then Eq. (1.1) has at least \(r+1\) distinct solutions in \(H_{T}^{1}\).
Theorem 3.2
Suppose that assumptions (H1)-(H3), (H5), (3.1), (3.2) hold and
as \(x\in N(A)\ominus \operatorname{span}\{e_{1},e_{2},\ldots ,e_{r}\}\). Then Eq. (1.1) has at least \(r+1\) distinct solutions in \(H_{T}^{1}\).
By Theorems 3.1 and 3.2, it is easy to obtain the following corollary.
Corollary 3.1
Suppose that assumptions (H1)-(H3), (H5), (3.1), (3.2) hold and
as \(x\in N(A)\ominus \operatorname{span}\{e_{1},e_{2},\ldots ,e_{r}\}\). Then Eq. (1.1) has at least \(r+1\) distinct solutions in \(H_{T}^{1}\).
Remark 3.1
(i) When \(A\equiv 0\), assumptions (ω1)-(ω4) and condition (3.2) were introduced in [10]. Comparing with the results in [10], the periodicity and coercivity conditions in our Theorem 3.1 are only in a part of variables of potentials, and we obtained multiplicity of periodic solutions for problem (1.1).
(ii) To show that our Theorem 3.1 is new, we give an example to illustrate our result. For example, let \(1\leq r\leq m\), \(x=(x_{1},x _{2},\ldots ,x_{N})^{T}\in \mathbb{R}^{N}\), and
let \(\omega (\vert x \vert )=\ln [1+(r+1+\vert x \vert ^{2})]\). Then F satisfies all the conditions of Theorem 3.1, but not covered by the results of [1–15].
For example, let \(x=(x_{1},x_{2},\ldots ,x_{N})^{T}\in \mathbb{R}^{N}\), \(\omega (\vert x \vert )=\vert x \vert \) and
where \(1\leq r\leq m\). Then F satisfies all the conditions of Theorem 3.2, but not covered by the results of [1–15].
For the sake of convenience, we denote by \(C_{i}\) (\(i=1,2,3,\ldots ,33\)) various positive constants.
Proof of Theorem 3.1
First, we prove that ψ satisfies the (PS) condition. Let \(\pi :W_{T}^{1,p(t)}\rightarrow W_{T}^{1,p(t)}/G\) be the canonical surjection. Define \(\psi : X\times V\mapsto \mathbb{R}\) by \(\psi ( \pi (u))=\varphi (u)\). Assume that \((\pi (u_{n}))\) is a (PS) sequence for ψ, that is, \(\psi (\pi (u_{n}))\) is bounded and \(\psi '( \pi (u_{n}))\rightarrow 0\). Then \(\varphi (u_{n})\) is bounded and \(\varphi '(u_{n})\rightarrow 0\).
We can get from (ω1), (ω2), and (ω3) that
By (2.3) and the boundedness of \(\vert Q\hat{u}^{0} \vert \), we have
From (H3) and (2.3), we obtain that
It follows from (2.2), (3.5), and (3.6) that
for large n. So we have
where \(C_{5}=\min_{s\in [0,+\infty )} \{ [ \delta -(2+a)M _{0}M_{1}C_{0}^{2}\int_{0}^{T}f(t)\,dt ] s^{2}-C_{4}s \} \).
From (3.1), one has that \((2+a)M_{0}M_{1}C_{0}^{2}\int_{0}^{T}f(t)\,dt<\delta \), then \(C_{5}<0\). Hence
In a similar way, we have
Combining the above two inequalities, one has that
Consequently,
Using similar arguments, we can prove that
By (3.5), (3.6), (3.7), (3.8), and (3.9), we have
In a similar way, we can obtain
By (3.2) and (ω1), one has
From (3.4), (2.3), and the boundedness of \(\vert Q\hat{u}^{0} \vert \), we have
Hence, we have
By (3.8), (3.9), and (3.12), we have
From (H3), (2.3), (3.8), and (3.9), one has
It follows from (3.10), (3.11), (3.13), and (3.14) that
which implies \(\vert Pu^{0}_{n} \vert \) is bounded. Otherwise, we assume \(\vert Pu^{0}_{n} \vert \rightarrow \infty \) as \(n\rightarrow \infty \). From (ω4), we obtain that
By (3.3), we conclude that
this contradicts the boundedness of \(\{\varphi (u_{n})\}\), so \(\vert Pu^{0}_{n} \vert \) is bounded. Combining (3.8) and (3.9), we obtain that \(\Vert u^{+}_{n} \Vert \) and \(\Vert u^{-}_{n} \Vert \) are bounded. Furthermore, \(\vert Q\hat{u}^{0} \vert \) is bounded, so \(\{\hat{u}_{n}\}\) is bounded in \(H^{1}_{T}\). Arguing then as in Proposition 4.1 in [6], \(\{\hat{u} _{n}\}\) has a convergent subsequence. By \(\pi (\hat{u}_{n})=\pi (u _{n})\), we conclude that ψ satisfies the (PS) condition.
Next, we only need to verify the linking conditions of the generalized saddle point theorem:
(a) For \(\pi (u)\in W\times V\), \(u(t)=u^{+}(t)+Qu^{0}\). By the proof of (3.12), we have
Hence
Note the boundedness of \(\vert Qu^{0} \vert \) and (3.1), we obtain that \(\psi (\pi (u))\rightarrow +\infty \) as \(\Vert u \Vert \rightarrow -\infty \) for all \(\pi (u)\in W\times V\), which implies that there exists \(\beta \in \mathbb{R}\) such that \(\psi (\pi (u))\geq \beta \) on \(W\times V\).
(b) In a way similar to the proof of (3.12), we have
Consequently,
From \(a>3\) and (3.1), one has that \({-}\delta +5M_{0}M_{1}C_{0}^{2} \int_{0}^{T}f(t) \, dt<0\).
By (3.3), we deduce that
so we obtain that \(\psi (\pi (u))\rightarrow +\infty \) as \(\Vert u \Vert \rightarrow -\infty \) for all \(\pi (u)\in Z\times V\), which implies that there exists \(\gamma <\beta \) such that \(\psi (\pi (u))\leq \gamma \) on \(Z\times V\).
The functional ψ satisfies all the assumptions of Lemma 2.2, so it has at least \(\operatorname{cuplength}(V)+1\) critical points, and since V is the torus \(T^{r}\), then \(\operatorname{cuplength}(V)=r\). Hence φ has at least \(r+1\) critical points. Therefore, problem (1.1) has at least \(r+1\) distinct solutions in \(H_{T}^{1}\). The proof of Theorem 3.1 is completed. □
Proof of Theorem 3.2
The proof of Theorem 3.2 is similar to the proof of Theorem 3.1, so we omit the discussions here. □
Remark 3.2
Using the parallel arguments with little change as in the proofs of Theorems 3.1 and 3.2, the conclusions of Theorems 3.1 and 3.2 hold if we replace (ω3) with
- (ω3)′:
-
\(0\leq \omega (s)\leq M_{1}s^{\alpha }+M_{2}\), \(\forall s,t\in [ 0, \infty )\), where \(0\leq \alpha <1\).
References
Chang, KQ: On the periodic nonlinearity and the multiplicity of solutions. Nonlinear Anal., Theory Methods Appl. 13, 527-537 (1989)
Faraci, F: Multiple periodic solutions for second order systems with changing sign potential. J. Math. Anal. Appl. 319, 567-578 (2006)
Feng, JX, Han, ZQ: Periodic solutions to differential systems with unbounded or periodic nonlinearities. J. Math. Anal. Appl. 323, 1264-1278 (2006)
Han, ZQ, Wang, SQ: Multiple solutions for nonlinear systems with gyroscopic terms. Nonlinear Anal., Theory Methods Appl. 75, 5756-5764 (2012)
Liu, JQ: A generalized saddle point theorem. J. Differ. Equ. 82, 372-385 (1989)
Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)
Ning, Y, An, TQ: Periodic solutions of a class of nonautonomous second-order Hamiltonian systems with nonsmooth potentials. Bound. Value Probl. 2015, 34 (2015)
Pipan, J, Schechter, M: Non-autonomous second order Hamiltonian systems. J. Differ. Equ. 257, 351-373 (2014)
Schechter, M: Periodic second order superlinear Hamiltonian systems. J. Math. Anal. Appl. 426, 546-562 (2015)
Wang, ZY, Zhang, JH: Periodic solutions of a class of second order non-autonomous Hamiltonian systems. Nonlinear Anal., Theory Methods Appl. 72, 4480-4487 (2010)
Xiao, L: Existence of periodic solutions for second order Hamiltonian system. Bull. Malays. Math. Soc. 35, 785-801 (2012)
Tang, CL: Periodic solutions of second order nonautonomous systems with sublinear nonlinearity. Proc. Am. Math. Soc. 126, 3263-3270 (1998)
Tang, CL: A note on periodic solutions of second order systems. Proc. Am. Math. Soc. 132, 1295-1393 (2003)
Zhang, XY, Tang, XH: Periodic solutions for an ordinary p-Laplacian system. Taiwan. J. Math. 12, 1369-1396 (2011)
Zhang, SG: Multiple periodic solutions for a class of sublinear nonautonomous second order system. Acta Anal. Funct. Appl. 15, 12-20 (2013)
Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 65. Am. Math. Soc., Providence (1986)
Willem, M: Minimax Theorems. Birkhäuser, Boston (1996)
Schechter, M: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999)
Bartsch, T: Critical point theory on partially ordered Hilbert spaces. J. Funct. Anal. 186, 117-152 (2001)
Shahmurov, R: Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annulus. J. Differ. Equ. 249, 526-550 (2010)
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This work is supported by the National Natural Science Foundation of China (No. 31260098).
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Zhang, S. Multiple periodic solutions for second order Josephson-type differential systems. Bound Value Probl 2017, 99 (2017). https://doi.org/10.1186/s13661-017-0829-2
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DOI: https://doi.org/10.1186/s13661-017-0829-2