Consider the second-order Hamiltonian systems with impulsive effects
(1.1)
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
for all and a.e. .
When (; ), (1.1) is the Hamiltonian system
(1.2)
In the past years, the existence of solutions for the second-order Hamiltonian systems (1.2) has been studied extensively via modern variational methods by many authors (see [1–13]).
When the gradient is bounded, that is, there exists such that
for all and a.e. , Mawhin-Willem 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 Mawhin-Willem’s results.
For , problem (1.1) gives less results (see [14–16]). 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:
(h1) There exist and such that
for all and a.e. .
(h2) .
(h3) For any ; ,
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:
(h4) .
(h5) For any ; ,
(h6) There exist and such that
Then problem (1.1) has at least one weak solution.
Let
(1.3)
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
(1.4)
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 1.1 Suppose that satisfies assumption (A) and the following conditions hold:
(H1) There exist and such that
(1.5)
for all and a.e. .
(H2) There exists a positive number such that
(1.6)
for all .
(H3)
(1.7)
(H4) For any ; ,
(1.8)
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 1.2 Suppose that satisfies assumption (A) and the condition (H1) of Theorem 1.1. Furthermore, assume that
(H5) There exist , such that
(1.9)
for all .
(H6)
(1.10)
(H7) For any ; ,
(1.11)
(H8) There exist and such that
(1.12)
for every , ; .
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 1.3 Suppose that and satisfy the assumptions (A), (H1), (H2), (H7) and (H8). Furthermore, assume that
(H9)
(1.13)
uniformly for all .
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.