Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem
A class of second order impulsive Hamiltonian systems are considered. By applying a local linking theorem, we establish the new criterion to guarantee that this impulsive Hamiltonian system has at least one nontrivial T-periodic solution under local superquadratic condition. This result generalizes and improves some existing results in the known literature.
1. Introduction and Main Results
Consider the second order Hamiltonian systems with impulsive effects where , , , , where and denote the right and left limits of at , respectively, are continuous, and , . is a symmetric constant matrix.
Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. The theory of impulsive differential systems has been developed by numerous mathematicians (see [1–4]). These kinds of processes naturally occur in control theory, biology, optimization theory, medicine, and so on (see [5–9]).
In recent years, many existence results are obtained for impulsive differential systems by critical point theory, such as [10–23] and their references. In most superquadratic cases, there is so-called Ambrosetti-Rabinowitz condition (see [18–23]): where is a constant, which implies that is of superquadratic growth as ; that is,
Moreover, Wu and Zhang  study the homoclinic solutions without any periodicity assumption under the local Ambrosetti-Rabinowitz type condition. Two key conditions of the main results of  are listed as follows.(A1)There exist and such that (A2)There exists such that , uniformly in .
In recent paper , Zhang and Tang had obtained some results of the nontrivial T-periodic solutions under much weaker assumptions instead of (A1) and (A2).(B1)There exist constants , , and and a function such that (B2)There exists a subset of with such that
Remark 1. Condition (B2) is weaker than (A2) because condition (A2) implies
Recently, applying the local linking theorem (see ), the works in [27–30] obtained the existence of periodic solutions or homoclinic solutions with (3) superquadratic condition under different systems. As shown in , condition (B2) is a local superquadratic condition; this situation has been considered only by a few authors.
Motivated by papers [24, 25, 31], in this paper, we aim to consider problem (1) under local superquadratic condition via a version of the local linking theorem (see ). In particular, the impulsive function satisfies a kind of new superquadratic condition which is different from that in the known literature. Our main results are the following theorems.
Theorem 2. Suppose that and , , satisfies (B2) and consider the following.(H1)There exists a positive constant such that (H2) uniformly for .(H3)There exist constants , and such that, for every and with , (H4)There exist constants , and such that, for every and with , (I1)There exist constants and such that (I2)There are two constants and such that (I3) satisfies , for all .Then problem (1) has at least one nontrivial T-periodic solution.
Remark 3. Noting (3), obviously, conditions (B2) and (H4) are weaker than those of (2). From (B2), we only need to hold in a subset of . What is more, in (2) is asked to be positive globally. Here need not be nonnegative globally; we also generalized Theorems 1.3 and 1.4 in . For example, let where Let ; then satisfies our Theorem 2 but does not satisfy (2) and (3) and does not satisfy the corresponding conditions in .
Theorem 4. Suppose that and , , satisfies (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.(H5)There exist constants and such that where Then problem (1) has at least one nontrivial T-periodic solution.
Theorem 5. Suppose that and , , satisfies (B2), (H1), (H2), (H3), (I1), (I2), and (I3) and the following condition holds.(H6)There exist constants , , and and a function and such that, for every and with , Then problem (1) has at least one nontrivial T-periodic solution.
Let be a real Banach space with direct sum decomposition . Consider two sequences of subspaces , such that . For every multi-index , let ; we define . A sequence is admissible if for every there is such that . For every , we define by the function restricted to .
Definition 6 (see ). Let . The function satisfies the condition if every sequence , such that is admissible and possesses a subsequence which converges to a critical point of .
Definition 7 (see ). Let be a Banach space with direct sum decomposition . The function has local linking at with respect to , if there exists such that
Theorem 8 (see ). Suppose that satisfies the following assumptions:(A1) has local linking at 0 and ,(A2) satisfies condition,(A3) maps bounded sets into bounded sets,(A4)for every and , as .Then has at least three critical points.
Let us recall some basic notation. In the Sobolev space , consider the inner product for any . The corresponding norm is defined by for any . Moreover, it is well known that is compactly embedded in , which implies that for some constant , where .
Define the corresponding functional on by
By the conditions of and , , we get that functional is a continuously Gáteaux differential functional whose Gáteaux derivative is the functional , given by
If , then is absolutely continuous and . In this case, may not hold for some ; this leads to impulsive effects.
To prove our main results, we have the following facts (see ).
Letting we see that where is the liner self-adjoint operator defined and is the identity matrix. By the Riesz representation theorem, we have
The compact imbedding of into implies that is compact. Summing up the above discussion, can be rewritten as
By classical spectral theory, we can decompose into the orthogonal sum of invariant subspace for where and and are such that, for some , Notice that is finite dimensional.
In this paper, we set .
3. Proof of Main Results
3.1. The Proof of Theorem 2
Let and ; then . Suppose is an orthogonal basis of . Correspondingly, let then , . We divide our proof into four steps.
Step 1. has local linking at .
In view of (I1), we obtain Combining this inequality, we have Since , this implies Applying (H2) and (33), for any , there exists such that which implies that
On one hand, by (34) and (35), for all with . Choose ; then one has
On the other hand, since is finite, there exists a constant such that For all with . Choose ; by (34), (35), and (37), we obtain Let ; one has
Step 2. maps bounded sets into bounded sets.
By (H3) and , there exists such that which implies that Note that It implies that maps bounded sets into bounded sets.
Step 3. satisfies the condition.
Consider a sequence such that is admissible. Then there exists a constant such that By (41), for , one has together with (H4), one has for all .
It follows from (21), (43), (45), (H1), and (I2) that
Since , , and , (46) shows that is bounded in . By a method similar to that of , we can prove that has a convergent subsequence. Thus satisfies condition.
Step 4. For every and , as .
Since is finite, there exists such that In fact, for , by (B2), there exists such that Hence, it follows from (47), (48), and (I3) that for and a.e. , which implies that Therefore, all the assumptions of Theorem 8 are verified. Then, the proof of Theorem 2 is completed.
3.2. The Proof of Theorem 4
To prove (A4), set , ; then by (H5), we have When , it follows from (51) that this implies Set ; then by (53), we have ; since is finite, there exists such that By (54), (55), and (I3), we have Since and , (56) implies Consequently, the conclusion follows from Theorem 8. This completes the proof.
3.3. The Proof of Theorem 5
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (no. 11271371).
B. Dai, Y. Li, and Z. Luo, “Multiple periodic solutions for impulsive Gause-type ratio-dependent predator-prey systems with non-monotonic numerical responses,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7478–7487, 2011.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Sun, H. Chen, and L. Yang, “The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 2, pp. 440–449, 2010.View at: Publisher Site | Google Scholar | MathSciNet