Subharmonic solutions with prescribed minimal period for a class of second order impulsive systems

Based on variational methods and critical point theory, the existence of subharmonic solutions with prescribed minimal period for a class of second-order impulsive systems is derived by estimating the energy of the solution. An example is presented to illustrate the result.

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time.There have been many approaches to study impulsive problems, such as method of upper and lower solutions with the monotone iterative technique, fixed point theory and topological degree theory.In recent years, variational method was employed to consider the existence of solutions for impulsive problems (see e.g.[1-5, 8, 9, 11, 13, 14]).
Corresponding author.Email: tj_bailiang@126.comWhen D = 0 and all I ij ≡ 0, (1.1) is reduced to the Hamiltonian system, which has been studied extensively on subharmonic solutions (see e.g.[10,15,17,18]).Recently, Luo, Xiao and Xu [6] established the conditions for the existence of subharmonic solutions for the following impulsive differential equation where f (t, x) : R × R → R and I k ∈ C(R, R + ∪ {0}).After that, Xie and Luo [16] investigated subharmonic solutions for the following forced pendulum equation with impulsive effects where f : R → R and However there are cases which are not possible to satisfy I k ≥ 0 or (1.2).For example, impulsive functions I k (s) = −s/9.Thus it is valuable to further improve conditions on impulsive functions.One thing to be noted is that a problem with impulsive functions −s/9 is considered in this paper (see Example 4.1 in Section 4).What is more, to the best of our knowledge, the existence of subharmonic solutions for impulsive systems has received considerably less attention.
Inspired by the aforementioned facts, we consider the impulsive system (1.1) under different assumptions on the impulsive function from [6] and [16].It will be shown that v satisfies (1.1) if and only if After that, subharmonic solutions of (1.3) will be obtained by estimating the energy of the solution in terms of minimal period.Finally, an example is given to illustrate the result, and a corollary concerning the equation (1.1a) is presented.

Preliminaries
Let us recall some basic concepts.
is a Hilbert space with the inner product where (•, •) denotes the inner product in R N , and the corresponding norm is where • L 2 is the norm of L 2 (0, pT; R N ).Assume that orthogonal matrix Q = (q rj ) N satisfies Lemma 2.1.v satisfies the impulsive system (1.1) if and only if u = Q −1 v satisfies the impulsive system (1.3).
Proof.Multiplying both sides of (1.1a) by Q −1 results (1.3a).In view of (1.1b) and we have Solutions of the above nonhomogeneous linear equations are (1.3b).
If u(t) is a pT-periodic solution of (1.3), following the ideas of [9], we have By (1.3b), the first term of the above equation is (2. 3) Consider the functional Φ : where Thanks to So critical points of Φ correspond to weak pT-periodic solutions of (1.3) (but pT might not be the minimal period).It follows from (2.3) and (2.5) that Consider the restriction of Φ on a subspace X of H 1 pT , where For any u ∈ X, by Wirtinger's inequality, where ω = 2π/T.For convenience, we introduce some assumptions.
(H2) There exist constants A ≥ A > −λ and B > 0 such that (H3) For each i = 1, 2, . . ., N, j = 1, 2, . . ., l, there exist constants a ij ≥ 0 such that Thanks to (2.1) and Hölder's inequality, we have which combined with (H3) yields to (2.8) For any u ∈ H 1 pT and each k = 1, 2, . . ., N, it follows from the mean value theorem that for some τ ∈ (0, pT).Hence, for t ∈ [0, pT], using Hölder's inequality, which combined with the discrete version of Minkowski's inequality yields to In view of this inequality and (2.8), we find where := ∑ l j=1 ∑ N r=1 a rj .The following fact is important in the proof of our main result.

Lemma 2.3 ([12, Theorem 1.2]
). Suppose V is a reflexive Banach space with norm • , and let M ⊂ V be a weakly closed subset of V. Suppose E : M → R ∪ {+∞} is coercive and (sequentially) weakly lower semi-continuous on M with respect to V. Then E is bounded from below on M and attains its infimum in M. Lemma 2.4.Suppose the assumption (H1) holds.If u is a critical point of Φ on X, then u is also a critical point of Φ on H 1 pT .And the minimal period of u is an integer multiple of T.
Proof.If u is a critical point of Φ on X, that is, Φ (u), v = 0 holds for any v ∈ X and u is odd, then Q −1 ∇F(t, Qu(t)) is pT-periodic and odd in t by (H1).Thus for any even w ∈ H 1 pT , we have So, in view of (2.6), we have Φ (u), w = 0.That gives us that Φ (u), v = 0 holds for any v ∈ H 1 pT , which implies that the equation (1.3a) holds by (2.6).Assume that the minimal period of u is pT/q for some integer q > 1, it follows from (1.3a) that ∇F(t, Qu(t)) is pT/q-periodic, then ∇F(t, Qu(t)) = ∇F t + pT q , Qu t + pT q = ∇F t + pT q , Qu(t) .
Thus p/q must be an integer by (H1), which completes the proof.

Main results
In this section, main results of this paper are obtained.
Then the impulsive system (1.1) has at least one weak periodic solution with minimal period pT.
Proof.We will complete the proof in three steps.
Let {u n } be a weakly convergent sequence to u 0 in H 1 pT , then {u n } converges uniformly to u 0 on [0, pT] (see Proposition 1.2 in [7]) and there exists a constant I rj (y)dy It follows from X is a closed convex space that {v k } ∈ X and u ∈ X.Thus, X is a weakly closed subset of H 1 pT .By (3.2), there exist 0 < ε 0 < (0.5 − p 2 T)ω 2 /p 2 and W > 0 such that which combined with (2.1) yields to where , we have So for any u ∈ X, Φ(u) → +∞ as u → ∞.Thus it follows from Lemma 2.3 that the result of Step 1 holds.
Step 2. Under the assumptions of Theorem 3.1, we have where for integer q ≥ 1.
Step 3. The critical point u * has minimal period pT.Assume the contrary; minimal period of u * is pT/q for some integer q > 1.By Lemma 2.4, q is a factor of p and q ≥ s p .By Fourier expansion, where {e 1 , e 2 , . . ., e N } denotes the canonical orthogonal basis in R N .By (H2), (2.1) and Hölder's inequality, which combined with (2.9) yields to as we find by minimizing with respect to u * L 2 .This contradicts with (3.5) since B q ≥ B s p for q ≥ s p .
Thus it follows from Lemma 2.4 that Φ has a critical point u * on H 1 pT and u * has minimal period pT.Therefore Qu * is a weak periodic solution of (1.1) with minimal period pT by Lemma 2.1.

Examples and corollaries
In this section, an example is given to illustrate Theorem 3.1, and a corollary of Theorem 3.1 concerning the equations (1.1a) is presented.

4.2. Assume
that F satisfies (H1), (H2) and there exists an integer p > 1 such that − A , where s p is the least prime factor of p. Then the equation (1.1a) has at least one weak periodic solution with minimal period pT.When prime integer p → ∞, the following is deserved by Corollary 4.2.