Periodic Solutions for Autonomous (q, p)-Laplacian System with Impulsive Effects

Theorem 1.1. Let q^′ and p^′ be such that   1/q + 1/q^′ = 1 and 1/p + 1/p^′ = 1. Suppose F satisfies assumption (A) and the following conditions:

•  

  (F1) there exist

  $$ ()
  such that
  $$ ()
  where
  $$ ()

•  

  (F2) F(x) → +∞,     as    |x| → ∞,   where    x = (x₁, x₂),

•  

  (I1) there exists β ∈ ℝ such that

  $$ ()

Then, system (1.1) has at least one solution in $$, where $$ is absolutely continuous,  u(0) = u(T) and$$.

Furthermore, if I_j ≡ 0  (j ∈ B), K_m ≡ 0  (m ∈ C) and the following condition holds:

•  

  (F3) there exist δ > 0, $$ and $$ such that

  $$ ()
  then system (1.7) has at least two nonzero solutions in $$.

Corollary 1.2. Suppose F₁ satisfies the following conditions:

•  

  (A)^′ F₁(z) is continuously differentiable in z and there exists a₁ ∈ C(ℝ⁺, ℝ⁺) such that

  $$ ()
  for all z ∈ ℝ^N.

•  

  (F1)^′ there exists 0 < r < 6/T² such that

  $$ ()

•  

  (F2)^′ F₁(z) → +∞,     as    |z| → ∞,     z ∈ ℝ^N;

•  

  (I1)^′   there exists β ∈ ℝ such that

  $$ ()

Then, system (1.6) has at least one solution in $$. Furthermore, if I_j ≡ 0  (j ∈ B) and the following condition holds:

•  

  (F3)^′ there exist δ > 0 and a ∈ [0, (3/T²)) such that

  $$ ()
  then system (1.4) has at least two nonzero solutions in $$.

Theorem 1.3. Suppose F₁ satisfies assumption (A)′, (F2)′, (I1)′ and

•  

  (F1)^′′ there exists 0 < r < 4π²/T² such that (1.15) holds.

Then, system (1.6) has at least one solution in $$. Furthermore, if I_j ≡ 0  (j ∈ B) and the following condition holds:

•  

  (F3)^′′ there exist δ > 0 and a ∈ [0, (2π²)/T²) such that

  $$ ()
  then system (1.4) has at least two nonzero solutions in $$.

Theorem 1.4. Suppose F₁ satisfies assumption (A)′, (F1)^{′ ′} and the following conditions:

•  

  (F4) F₁(z) is (λ, μ)-quasiconcave on ℝ^N,

•  

  (F5) F₁(z) → −∞ as |z| → +∞, z ∈ ℝ^N,

•  

  (I2) there exist d_j > 0  (j ∈ B) such that

  $$ ()

•  

  (I3) there exist b_j > 0,   c_j > 0, γ_j ∈ ℝ, α_j ∈ [0,   2)(j ∈ B) such that

  $$ ()

Then, system (1.6) has at least one solution in $$.

Remark 1.5. In [17], Yang considered the second-order Hamiltonian system with no impulsive effects, that is, system (1.4). When I_j ≡ 0  (j ∈ B), our Theorems 1.3 and 1.4 still improve those results in [17]. To be precise, the restriction of r is relaxed, and some unnecessary conditions in [17] are deleted. In [17], the restriction of r is 0 < r < T/12, which is not right. In fact, from his proof, it is easy to see that it should be 0 < r < 12/T². Obviously, our restriction 0 < r < 4π²/T² is better. Moreover, in our Theorem 1.4, we delete such conditions (of in [17, Theorem  1]): ∇F₁(0) = 0, and there exist positive constants M, N such that

Finally, it is remarkable that Theorems 1.3 and 1.4 are also different from those results in [1–16]. We can find an example satisfying our Theorem 1.3 but not satisfying the results in [1–21]. For example, let where $$. We can also find an example satisfying our Theorem 1.4 but not satisfying the results in [1–21]. For example, let where 12/T² < r < 4π²/T².

Lemma 2.1 (see [31] or [32].)Each $$ and each $$ can be written as $$ and $$ with

Then, where

Definition 2.2. We say that a function $$ is a weak solution of system (1.1) if

holds for any $$.

Define the functional $$ by

where $$, By assumption (A) and [33], we know that $$. The continuity of I_j(j ∈ B) and K_m(m ∈ C) implies that $$. So, $$, and for all $$, we have Definition 2.2 shows that the critical points of φ correspond to the weak solutions of system (1.1).

We will use the following lemma to seek the critical point of φ.

Lemma 2.3 (see [3], Theorem  1.1.)If φ is weakly lower semicontinuous on a reflexive Banach space X and has a bounded minimizing sequence, then φ has a minimum on X.

Lemma 2.4 (see [34].)Let φ be a C¹ function on X = X₁ ⊕ X₂ with φ(0) = 0, satisfying (PS) condition, and assume that for some ρ > 0,

Assume also that φ is bounded below and inf _Xφ < 0, then φ has at least two nonzero critical points.

Lemma 2.5 (see [35], Theorem  4.6.)Let X = X₁ ⊕ X₂, where X is a real Banach space and X₁ ≠ {0} and is finite dimensional. Suppose that φ ∈ C¹(X, ℝ) satisfies (PS)-condition and

•  

  (φ1) there is a constant α and a bounded neighborhood D of 0 in X₁ such that φ∣_∂D ≤ α,

•  

  (φ2) there is a constant β > α such that $$.

Then, φ possesses a critical value c ≥ β. Moreover, c can be characterized as

where,

Lemma 3.1. Under assumption (A), φ is weakly lower semicontinuous on $$.

Proof. Let

Since then ϕ₁ is convex. Moreover, by [33], we know that ϕ₁ is continuous, and so, it is lower semicontinuous. Thus, it follows from [3, Theorem  1.2] that ϕ₁ is weakly lower continuous. By assumption (A), it is easy to verify that ϕ₂(u₁, u₂) is weakly continuous. We omit the details. Let Next, we show that ψ₁ and ψ₂ are weakly continuous on $$ and $$, respectively. In fact, if then by in [3, Proposition  1.2], we know that So, there exists M₁ > 0 such that ∥u₁∥_∞ ≤ M₁ and ∥u_1n∥_∞ ≤ M₁, for all n ∈ ℕ. Thus, we have Hence, ψ₁ is weakly continuous on $$. Similarly, we can prove that ψ₂ is also weakly continuous on $$. Thus, we complete the proof.

Proof of Theorem 1.1. It follows from (F1) and (2.7) that

Hence, by (I1), (3.7), and (3.8), we have Note that for $$, and for $$, So, (F2) and (3.9) imply that Thus, by Lemma 2.3, we know that φ has at least one critical point which minimizes φ on W.

Furthermore, if I_j(u₁(t_j)) ≡ 0  (j ∈ B) and K_m(u₂(s_m)) ≡ 0  (m ∈ C), then system (1.1) reduces to (1.7). When (F3) also holds, we will use Lemma 2.4 to obtain more critical points of φ. Let X = W,   X₂ = ℝ^N × ℝ^N and $$.

By (3.9), we know that φ(u₁, u₂) → +∞ as ∥(u₁, u₂)∥_W → ∞. So, φ satisfies (PS) condition and is bounded below. Take ρ = δ/c₁, where c₁ is a positive constant such that $$ and $$ for all (u₁, u₂) ∈ W. It follows from (F3) and Lemma 2.1 that

Since $$ and $$, (3.13) implies that φ(u₁, u₂) ≥ 0 for all (u₁, u₂) ∈ X₁ with ∥u∥_W ≤ ρ. By (F3), it is easy to obtain that φ(u₁, u₂) ≤ 0, for all (u₁, u₂) ∈ X₂ with ∥u∥_W ≤ ρ.

If inf {φ(u₁, u₂) : (u₁, u₂) ∈ W} = 0, then from above, we have φ(u₁, u₂) = 0 for all (u₁, u₂) ∈ X₂ with ∥(u₁, u₂)∥_W ≤ ρ. Hence, all (u₁, u₂) ∈ X₂ with ∥(u₁, u₂)  ∥_W ≤ ρ are minimal points of φ, which implies that φ has infinitely many critical points. If inf {φ(u₁, u₂)   : (u₁, u₂)   ∈ W} < 0, then by Lemma 2.4, φ has at least two nonzero critical points. Hence, system (1.7) has at least two nontrivial solutions in W. We complete our proof.

Proof of Theorem 1.3. We only need to use (1.18) and (1.19) to replace (2.6) and (2.7) in the proof Theorem 1.1 with p = q = 2, F(t, u₁, u₂) = F₁(u₁) and K_m(u₂) ≡ 0  (m ∈ C). It is easy. So, we omit it.

Lemma 3.2. Under the assumptions of Theorem 1.4, the functional φ₁ defined by

satisfies (PS) condition.

Proof. Suppose that {u_1n} is a (PS) sequence for φ₁; that is, there exists D₁ > 0 such that

Hence, for n large enough, we have ∥φ^′(u_1n)∥   ≤ 1. It follows from (F1) ^′′, (I2), and (1.18) that for n large enough. By (1.18), we have and (3.16), (3.17), and r < 4π²/T² imply that there exists D₂, D₃ > 0 such that It follows from (F4), (3.15), (I3), (1.18), and (3.18) that for all n and (3.19) and (F5) imply that $$ is bounded. Combining (3.18), we know that {u_1n} is a bounded sequence. Similar to the argument in [25], it is easy to obtain that φ satisfies (PS) condition.

Example 4.1. Let q = 4, p = 2, T = π, t₁ = 1, and s₁ = 2. Consider the following system:

where $$, x₁ = (x₁₁, x₁₂, …, x_1N), x₂ = (x₂₁, x₂₂, …, x_2N), $$, $$, x ∈ ℝ^N. It is easy to verify that all conditions of Theorem 1.1 hold so that system (4.1) has at least one weak solution. Moreover, if $$, x₂ = (x₂₁, x₂₂, …, x_2N), I₁(x) = 0 and K₁(x) = 0, x ∈ ℝ^N, then system (4.1) has at least two nonzero solutions.

Example 4.2. Let T = 2, t₁ = 1. Consider the following autonomous second-order Hamiltonian system with impulsive effects:

where $$,$$, $$. It is easy to verify that all conditions of Theorem 1.3 hold so that system (4.2) has at least one weak solution. Moreover, if $$ and I₁(z) = 0, z ∈ ℝ^N, then system (4.2) has at least two nonzero solutions.

Example 4.3. Let T = π, t₁ = 2. Consider the following autonomous second-order Hamiltonian system with impulsive effects:

where F(z)   = −|z|²,   I₁(z)   = 2sin z₁, $$. It is easy to verify that all conditions of Theorem 1.4 hold so that system (4.3) has at least one weak solution.