INFINITELY MANY SOLUTIONS FOR NON-AUTONOMOUS SECOND-ORDER SYSTEMS WITH IMPULSIVE EFFECTS∗

In this paper, we establish the existence of infinitely many solutions for a class of non-autonomous second-order systems with impulsive effects. Our technique is based on the Fountain Theorem of Bartsch and the Symmetric Mountain Pass Lemma due to Kajikiya.

As a special case of dynamical systems, Hamiltonian systems are very important in the various applications in mechanics, electronics, economics and so on. In 1978, Rabinowitz [20] published his pioneer paper for the existence of periodic solutions for Hamiltonian systems via the critical point theory. From then on, there has been a vast literature on the study of existence and multiplicity of periodic solutions for Hamiltonian systems, see [4,8,13,14,21,24,25] and the references therein.
In recent years, some classical methods and techniques such as the coincidence degree theory, upper and lower solutions method, iterative technique, fixed point theory have been applied to study the impulsive differential equations by many researchers, due to its widely application in various problems of technology and science, see [3,7,9,10,12,16,17,22,23,26,[28][29][30] and the references therein. Especially, Nieto and O'Regan [16] presented a new approach via variational methods and critical point theory to study the existence of solutions to impulsive problems.
In the present paper, we are interested in the existence of infinitely many solutions of problem (1.1) under some new conditions. With the aid of variational methods, we get the multiplicity results for both superquadratic and subquadratic cases, which generalize and sharply improve the results in [7,23]. Moreover, our proofs are much simpler.

The superquadratic case
By applying a variant of Fountain Theorem (see [31]), Sun, Chen and Nieto [23] proved the existence of infinitely many solutions for system (1.1), where W (t, x) is even in x. The following theorem was obtained. Theorem 1.1 (Theorem 1.1, [23]). Assume the following conditions hold: (V 2 ) There exists a positive constant ν such that V (t)u · u ≥ ν|u| 2 for every u ∈ R N and a.e. in [0, T ].
In [7], Chen and He established the following result for system (1.1), which improves Theorem 1.1.
hold. Assume the following conditions hold: (H ′ 1 ) There exists L 0 > 0 such that Then problem (1.1) has infinitely many solutions.
Here, applying the Fountain Theorem due to Bartsch (see [2, Theorem 2.5] and [27, Theorem 3.6]), we obtain the existence of infinitely many solutions for problem (1.1) with some more general conditions, which generalizes and improves upon the results mentioned above. The following theorem is established.
for all x ∈ R N and a.e. t ∈ [0, T ]. Then, W does not satisfy the result mentioned above. But W satisfies Theorem 1.3 with a 1 = 1.

The subquadratic case
Sun, Chen and Nieto studied systems (1.1) that are asymptotically case and subquadratic case. Applying the minimax technique, they obtained the two following theorems. Theorem 1.4 (Theorem 1.2, [23]). Assume that the following conditions are satisfied: Then problem (1.1) has infinitely many solutions. Theorem 1.5 (Theorem 1.3, [23]). Suppose that (S 1 ), (S 2 ) and (H 6 ) hold. Assume that W satisfies (A) and the following conditions: Then problem (1.1) has infinitely many solutions.
Here, by means of the new version of the Symmetric Mountain Pass Lemma established in [11], we obtain the following theorem, which unifies and significantly improves upon Theorems 1.4 and 1.5. Theorem 1.6. Assume that the following conditions hold: Then problem (1.1) possesses infinitely many solutions.

Proof of main results
Let us consider the functional ϕ on H 1 T given by It follows from assumption (A) that the functional ϕ is continuously differentiable on H 1 T . Moreover, one has It is well known that the solutions of problem (1.1) correspond to the critical points of ϕ. Since the embedding [14]). Letting one has that The compact imbedding of H 1 T into C(0, T ; R N ) (see [14]) implies that K is compact. Using classical spectral theory, the following decomposition holds Let X be a reflexive and separable Banach space. It is well known that there exist {w n } n∈N ⊂ X, {ψ n } n∈N ⊂ X * such that (1) ⟨ψ n , w m ⟩ = χ n,m , where χ n,m = 1 for n = m and χ n,m = 0 for n ̸ = m.
Palais-Smale condition (P S-condition) was introduced by Palais and Smale in [18]. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais-Smale-type conditions. It is clear that the P S-condition implies the (P S) c condition for each c ∈ R. Cerami condition (condition (C)) was given by Cerami [6]. It is well known that condition (C) is weaker than P S-condition. In [2], Bartsch established the Fountain Theorem (Theorem 2.5 in [2], Theorem 3.6 in [27]) under the (P S) c condition. As shown in [1,15,19], the deformation lemma can be proved with the weaker condition (C) replacing the usual P S-condition, and it turns out that the Fountain Theorem is true under the condition (C). So, we have the following Fountain Theorem. Theorem 2.1 (Fountain Theorem, [2,27]). Assume that ϕ ∈ C 1 (X, R) satisfies the condition (C), ϕ(−u) = ϕ(u). If for almost every k ∈ N, there exist ρ k > r k > 0 such that then ϕ has an unbounded sequence of critical values.
To prove Theorem 1.6, we need the following Symmetric Mountain Pass Theorem due to Kajikiya (see [11]). Before stating it, we first recall the definition of genus. Let X be a Banach space and let A be a subset of X. A is said to be symmetric if u ∈ A implies that −u ∈ A. For a closed symmetric set A, which does not contain the origin, we define a genus γ(A) of A by the smallest integer k such that there exists an odd continuous mapping from A to R k \{0}. If such a k does not exist, we define γ(A) = ∞. Moreover, we set γ(∅) = 0. Let Γ k denote the family of closed symmetric subsets A of X such that 0 ̸ ∈ A and γ(A) ≥ k. For the convenience of the readers, we summarize the property of genus that will be used in the proof of Theorem 1.6. We refer the readers to [21, proposition 7.5] for the proof of the next proposition. A and B be closed symmetric subsets of X that do not contain the origin. Then the following hold. A to B, then γ(A) ≤ γ(B).

(i) If there exists an odd continuous mapping from
(iv) The n-dimensional sphere S n has a genus of n+1 by the Borsuk-Ulam theorem.
We now state the Symmetric Mountain Pass Lemma.
(2) For each k ∈ N, there exists an A k ∈ Γ k such that sup u∈A k ϕ(u) < 0.
Now, we can demonstrate the proof of our results.
Proof of Theorem 1.3. First of all, we will prove that ϕ satisfies condition (C). Let {u n } be a sequence in H 1 T such that {ϕ(u n )} is bounded and (1 + ∥u n ∥)∥ϕ ′ (u n )∥ → 0 as n → ∞.
Then there exists a positive constant M 0 such that By a standard argument, we only need to prove that {u n } is a bounded sequence in H 1 T . Otherwise, we can assume that ∥u n ∥ → +∞ as n → ∞. Let v n = un ∥un∥ , one then has ∥v n ∥ = 1. Going if necessary to a subsequence, we can suppose that as n → ∞.
We conclude from (2.10) and Fatou Lemma, one has which contradicts to (2.5). So, ∥u n ∥ is bounded. And, the condition (C) holds.
Let {e j } j∈N be a basis for H 1 T and define Y k and Z k as in Theorem 2.1. Since dim(Y k ) < ∞, all the norms are equivalent. For each u ∈ Y k , there exists constant C k > 0 such that ∥u∥ ≤ C k ∥u∥ L 2 . (2.11) From condition (H 3 ), there exists L 3 > 0 such that for all |x| ≥ L 3 and a.e. t ∈ [0, T ]. From assumption (A), one gets for all x ∈ R N with |x| ≤ L 3 and a.e. t ∈ [0, T ], where a 3 = max 0≤s≤L3 a(s). Hence, we obtain from (2.12) and (2.13) that for all x ∈ R N and a.e. t ∈ [0, T ].
Set r k = β −1 k , one has Thus, for k large enough such that Z k ⊂ H + and r 2 k ≥ 4δ −1 a 4 ∥b∥ L 1 , where a 4 = max 0≤s≤1 a(s). Then, for u ∈ Z k with ∥u∥ = r k , one sees ∥u∥ ∞ ≤ 1. So, by (H 0 ) and (2.2), we have Therefore, it follows from (2.15) and the above expression that And, relation (A 2 ) is proved. Hence, the proof is completed by using the Fountain theorem. □ Proof of Theorem 1.6. We consider the following truncated functional Obviously, ψ ∈ C 1 (H 1 T , R) and ψ(0) = 0. Since Hence, if we can get that ψ possesses a sequence of critical points {u k } such that ψ(u k ) ≤ 0, u k ̸ = 0 and u k → 0 as k → ∞, then for the k large enough, the critical points of ψ satisfying ∥u k ∥ ≤ δ 3 /(2C) are just critical points of ϕ. So, the conclusion of Theorem Hence, ψ is bounded from below and satisfies the P S-condition.

Now, taking
by Proposition 2.1 one sees that So, we get A k ∈ Γ k and Then, Theorem 1.6 follows from Theorem 2.2 and the proof is complete. □