Periodic Solutions for Second Order Hamiltonian Systems with Impulses via the Local Linking Theorem

and Applied Analysis 3 2. Preliminaries LetX be a real Banach space with direct sum decomposition X = X 1 ⊕ X . Consider two sequences of subspaces X 0 ⊂ X 1 1 ⊂ ⋅ ⋅ ⋅ ⊂ X , X 0 ⊂ X 2 1 ⊂ ⋅ ⋅ ⋅ ⊂ X 2 such that X = ⋃ n∈N X j n, j = 1, 2. For every multi-index α = (α1, α2) ∈ N , let X α = X α1 ⊕ X α2 ; we define α ≤ β ⇔ α 1 ≤ β 1 , α 2 ≤ β 2 . A sequence {α n } ⊂ N 2 is admissible if for every α ∈ N there is m ∈ N such that n ≥ m ⇒ α n ≥ α. For every φ : X → R, we define by φ α the function φ restricted toX α . Definition 6 (see [26]). Let φ ∈ C(X,R). The function φ satisfies the (PS) condition if every sequence {u αn }, such that {α n } is admissible and u αn ∈ X αn , sup n φ (u αn ) < ∞, φ 󸀠 αn (u αn ) 󳨀→ 0, as n 󳨀→ ∞, (17) possesses a subsequence which converges to a critical point of φ. Definition 7 (see [26]). Let X be a Banach space with direct sum decompositionX = X⊕X.The function φ ∈ C(X,R) has local linking at 0 with respect to (X, X), if there exists r > 0 such that φ (u) ≥ 0, ∀u ∈ X 1 with ‖u‖ ≤ r, φ (u) ≤ 0, ∀u ∈ X 2 with ‖u‖ ≥ r. (18) Theorem 8 (see [26]). Suppose that φ ∈ C(X,R) satisfies the following assumptions: (A1) φ has local linking at 0 and X ̸ = {0}, (A2) φ satisfies (PS) condition, (A3) φmaps bounded sets into bounded sets, (A4) for every m ∈ N and u ∈ X m ⊕ X , φ(u) → −∞ as ‖u‖ → ∞. Then φ has at least three critical points. Let us recall some basic notation. In the Sobolev space X := H 1 0 (0, T), consider the inner product (u, V) = ∫ T 0 (u (t) , V (t)) dt + ∫ T 0 (?̇? (t) , V̇ (t)) dt, (19) for any u, V ∈ X. The corresponding norm is defined by ‖u‖ = (∫ T 0 |u (t)| 2 dt + ∫ T 0 |?̇? (t)| 2 dt) 1/2 , (20) for any u ∈ X. Moreover, it is well known thatX is compactly embedded in C[0, T], which implies that ‖u‖∞ ≤ C ‖u‖ , (21) for some constant C > 0, where ‖u‖ ∞ = max t∈[0,T] |u(t)|. Define the corresponding functional φ onX by

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][2][3][4]).These kinds of processes naturally occur in control theory, biology, optimization theory, medicine, and so on (see [5][6][7][8][9]).
Moreover, Wu and Zhang [24] study the homoclinic solutions without any periodicity assumption under the local Ambrosetti-Rabinowitz type condition.Two key conditions of the main results of [24] are listed as follows.
In recent paper [25], Zhang and Tang had obtained some results of the nontrivial T-periodic solutions under much weaker assumptions instead of (A1) and (A2).
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 [26]).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.
(H1) There exists a positive constant  such that 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 [25].For example, let where Let  0 = [/8, /4]; then  satisfies our Theorem 2 but does not satisfy (2) and (3) and does not satisfy the corresponding conditions in [25].
The remaining of this paper is organized as follows.In Section 2, some fundamental facts are given.In Section 3, the main results of this paper are presented.

Preliminaries
Let  be a real Banach space with direct sum decomposition  =  1 ⊕  2 .Consider two sequences of subspaces For every  :  → R, we define by   the function  restricted to   .Definition 6 (see [26]).Let  ∈  1 (, R).The function  satisfies the () * condition if every sequence {   }, such that {  } is admissible and possesses a subsequence which converges to a critical point of .
Following the ideas of [11,12], we can prove that the critical points of  are the weak solutions of problem (1).
To prove our main results, we have the following facts (see [32]). Letting we see that where  :  1  →  1  is the liner self-adjoint operator defined and  is the  ×  identity matrix.By the Riesz representation theorem, we have The compact imbedding of  1  into ([0, ],   ) 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  0 = ker( − ) and  − and  + are such that, for some  > 0, Notice that  − is finite dimensional.
Then, the proof of Theorem 2 is completed.

3. 3 .
The Proof of Theorem 5. Similar to the proof of Theorem 2,  satisfies all conditions of Theorem 8. Thus, problem (1) has at least one nontrivial T-periodic solution.