Abstract
In this paper, a class of secondorder Hamiltonian systems with impulsive effects are considered. By using critical point theory, we obtain some existence theorems of solutions for the nonlinear impulsive problem. We extend and improve some recent results.
MSC: 334B18, 34B37, 58E05.
Keywords:
Hamiltonian system; impulsive; critical point theory1 Introduction and main results
Consider the secondorder Hamiltonian systems with impulsive effects
where , , (; ) are continuous and satisfies the following assumption:
(A) is measurable in t for every and continuously differentiable in x for a.e. and there exist , such that
When (; ), (1.1) is the Hamiltonian system
In the past years, the existence of solutions for the secondorder Hamiltonian systems (1.2) has been studied extensively via modern variational methods by many authors (see [113]).
When the gradient is bounded, that is, there exists such that
for all and a.e. , MawhinWillem in [1] proved the existence of solutions for problem (1.2) under the condition
or
When the gradient is bounded sublinearly, that is, there exist and such that
for all and a.e. , Tang [2] proved the existence of solutions for problem (1.2) under the condition
or
which generalizes MawhinWillem’s results.
For , problem (1.1) gives less results (see [1416]). In [14], Zhou and Li extended the results of [2] to impulsive problem (1.1); they proved the following theorems.
Theorem A[14]
Assume that (A) and the following conditions are satisfied:
Then problem (1.1) has at least one weak solution.
Theorem B[14]
Suppose that (A) and the condition (h1) of Theorem A hold. Assume that:
Then problem (1.1) has at least one weak solution.
Let
where is convex in (e.g., ), , satisfying (e.g., ), , and . It is easy to see that satisfies the condition (h2) but does not satisfy the condition (h1). The above example shows that it is valuable to further improve Theorem A.
Let
where satisfies that the gradient is Lipschitz continuous and monotone in (e.g., ), , satisfies (e.g., ), , and . It is easy to see that satisfies the condition (h4) but does not satisfy the condition (h1). The above example shows that it is valuable to further improve Theorem B.
In this paper, we further study the existence of solutions for impulsive problem (1.1). Our main results are the following theorems.
Theorem 1.1Suppose thatsatisfies assumption (A) and the following conditions hold:
(H2) There exists a positive numbersuch that
(H3)
Then impulsive problem (1.1) has at least one weak solution.
Remark 1.1 Theorem 1.1 generalizes Theorem A, which is a special case of our Theorem 1.1 corresponding to .
Example 1.1 Let , . Consider the following impulsive problem:
where
Take
which is bounded and
, , . Then all the conditions of Theorem 1.1 are satisfied. According to Theorem 1.1, the above problem has at least one weak solution. However, F does not satisfy the condition (h1) in Theorem A. Therefore, our result improves and generalizes the Theorem A.
Theorem 1.2Suppose thatsatisfies assumption (A) and the condition (H1) of Theorem 1.1. Furthermore, assume that
(H6)
Then impulsive problem (1.1) has at least one weak solution.
Remark 1.2 Theorem 1.2 generalizes Theorem B, which is a special case of our Theorem 1.2 corresponding to .
Example 1.2 Let , . Consider the following impulsive problem:
where
Take
which is bounded from above, and
, , , , (). Then all the conditions of Theorem 1.2 are satisfied. According to Theorem 1.2, the above problem has at least one weak solution. However, F does not satisfy the condition (h4) in Theorem B. Therefore, our result improves and generalizes Theorem B.
Theorem 1.3Suppose thatandsatisfy the assumptions (A), (H1), (H2), (H7) and (H8). Furthermore, assume that
(H9)
Then impulsive problem (1.1) has at least one weak solution.
Example 1.3 Let , . Consider the following impulsive problem:
where
Take
Then all the conditions of Theorem 1.3 are satisfied. According to Theorem 1.3, the above problem has at least one weak solution. However, is neither superquadratic in X nor subquadratic in X.
2 Preliminaries
inducing the norm
The corresponding functional ϕ on given by
is continuously differentiable and weakly lower semicontinuous on . For the sake of convenience, we denote
where
and
Definition 2.1 We say that a function is a weak solution of (1.1) if the identity
It is well known that the solutions of impulsive problem (1.1) correspond to the critical point of ϕ.
Definition 2.2[1]
Let X be a Banach space, and .
(1) ϕ is said to satisfy the condition on X if the existence of a sequence such that and as implies that c is a critical value of ϕ.
(2) ϕ is said to satisfy the condition on X if any sequence for which is bounded and as possesses a convergent subsequence in X.
Remark 2.1 It is clear that the condition implies the condition for each .
Lemma 2.1[1]
Ifϕis weakly lower semicontinuous on a reflexive Banach spaceX (i.e., if, then) and has a bounded minimizing sequence, thenϕhas a minimum onX.
Remark 2.2 The existence of a bounded minimizing sequence will be in particular ensured when ϕ is coercive, i.e., such that
Lemma 2.2[1]
LetXbe a Banach space and let. Assume thatXsplits into a direct sum of closed subspaceswithand
where
Let
and
Then ifϕsatisfies thecondition, cis a critical value ofϕ.
Lemma 2.3[1]
If the sequenceconverges weakly touin, thenconverges uniformly touon.
Lemma 2.4[1]
and
Lemma 2.5There existssuch that if, then
3 Proof of main results
Proof of Theorem 1.1 It follows from (H1) and Sobolev’s inequality that
for all and some positive constants , and . By (H2) and Wirtinger’s inequality, we have
From (H4), we obtain
for all . As if and only if , (3.3) and (H3) imply that
By Lemma 2.1 and Remark 2.1, ϕ has a minimum point on , which is a critical point of ϕ. Therefore, we complete the proof of Theorem 1.1. □
Lemma 3.1Assume that the conditions of Theorem 1.2 hold. Thenϕsatisfies thecondition.
Proof Let be bounded and as . From (H1) and Lemma 2.4, we have
for all large n and some positive constants , and . It follows from (H5) and Lemma 2.4 that
By (3.4), (3.5), (H8) and Young’s inequality, we have
where , . From Wirtinger’s inequality, we obtain
The inequalities (3.6) and (3.7) imply that
for all large n and some positive constants and . It follows from (H5), CauchySchwarz’s inequality and Wirtinger’s inequality that
Like in the proof of Theorem 1.1, we get
From (H7), we have
for all . Since is bounded, from (3.9) and (3.10), there exists a constant C such that
for all large n and some constant . By the above inequality and (H6), we know that is bounded. In fact, if not, without loss of generality, we may assume that as . Then, from (3.8) and the above inequality, we have
which contradicts (H6). Hence is bounded. Furthermore, is bounded from (3.7) and (3.8). Hence, there exists a subsequence of defined by such that
By Lemma 2.3, we have
On the other hand, we get
It follows from the above equality, (A) and the continuity of that
Thus, we conclude that ϕ satisfies the condition. □
Now, we give the proof of our Theorem 1.2.
Proof of Theorem 1.2 Let W be the subspace of given by
In fact, for , then , by the proof of Theorem 1.1, we have
Like in the proof of Lemma 3.1, we obtain
By (H8) and Lemma 2.5, we find
for all . By (3.14), (3.15) and (3.16), we have
By Lemma 2.4, we have on W. Hence (3.13) follows from (3.17).
On the other hand, by (H7), we get
for all . Therefore, from (3.18) and (H6), we obtain
as in . It follows from Lemma 2.2 and Lemma 3.1 that problem (1.1) has at least one weak solution. □
Proof of Theorem 1.3 First we prove that ϕ satisfies the condition. Suppose that is a sequence for ϕ, that is, as and is bounded. In a way similar to the proof of Theorem 1.1, we have
and
From (3.7) and the above inequalities, we obtain
for some positive constants and .
By (H9) there exists such that
for all and , which implies that
for all and . It follows from assumption (A) that
By the boundedness of , (H7) and (3.19), there exists a constant such that
which implies that is bounded. Like in the proof of Lemma 3.1, we know that ϕ satisfies the condition.
Furthermore, we can prove Theorem 1.3 using the same way as in the proof of Theorem 1.2. Here, we omit it. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
We express our gratitude to the referees for their valuable criticism of the manuscript and for helpful suggestions. Supported by the National Natural Science Foundation of China (11271371, 10971229).
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