MULTIPLE PERIODIC SOLUTIONS FOR PERTURBED SECOND-ORDER IMPULSIVE HAMILTONIAN SYSTEMS

The existence of three distinct periodic solutions for a class of perturbed impulsive Hamiltonian systems is established. The techniques used in the proofs are based on variational methods. AMS Subject Classification: 34B15

As is well known, a Hamiltonian system is a system of differential equations that can model the motion of a mechanical system.An important and interesting question is under what conditions will the Hamiltonian system possess periodic solutions.Background information and applications of Hamiltonian systems can be found for example in [16,28,31,37].The monographs [29,32] have inspired a great deal of work on the existence and multiplicity of periodic solutions for Hamiltonian systems using variational techniques; for example, see [9,10,11,13,14,15,18,19,24,25,26,36,38,40,42,43,45] and the references therein.
Impulsive differential equations provide a general framework for modeling many real world phenomena; they too have been studied extensively in the literature.Background information and applications of impulsive differential equations can be found in [2,3,23,27,33].Recently, using critical point theory, several authors have studied the existence and multiplicity of solutions of impulsive problems; see, for example, [1,7,20,30,39,41].
The existence and multiplicity of solutions for second-order impulsive Hamiltonian systems have attracted a good deal of attention in the literature, and we refer the reader to [12,34,35,44] and the included references for recent results.In [12,35], using variational methods and critical point theory, the existence of multiple solutions for second-order impulsive Hamiltonian systems was studied.In [21], using different variational techniques from the ones used in this paper, the present authors obtained the existence of infinitely many classical periodic solutions to problem (1); in [22], using variational methods and critical point theory different from those in [21] and this paper, they investigated the existence of nontrivial periodic solutions to problem (1) in case µ = 0.
Motivated by the results in [12,35] and using two kinds of three critical points theorems (Theorems 1 and 2 below), in this paper we are able ensure the existence of at least three classical periodic solutions to problem (1); see Theorems 5 and 6 below.Theorems 1 and 2 have been successfully employed to establish the existence of at least three solutions for perturbed boundary value problems in the papers [5,6,17].

Preliminaries
Our main tools are the three critical point theorems that we recall here in convenient forms.The first has been obtained in [4], and it is a more precise version of Theorem 3.2 of [8].The second has been established in [8].We will use the notation that if X is a Banach space then X * is its dual space.Theorem 1. ([4, Theorem 2.6]) Let X be a reflexive real Banach space, Φ : X −→ R be a coercive continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a con-tinuous inverse on X * , Ψ : X −→ R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact and such that Φ(0) = Ψ(0) = 0.
Assume there exist r > 0 and x ∈ X with r < Φ(x) such that , the functional Φ − λΨ is coercive.
Then, for each λ ∈ Λ r , the functional Φ − λΨ has at least three distinct critical points in X.
Theorem 2. ([8, Corollary 3.1]) Let X be a reflexive real Banach space, Φ : X −→ R be a convex, coercive, and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X * , Ψ : X −→ R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact and such that Assume that there are two positive constants r 1 , r 2 and x ∈ X, with 2r 1 < Φ(x) < r 2 2 , such that: and for every x 1 , x 2 ∈ X, which are local minima for the functional Φ−λΨ and such that Ψ(x 1 ) ≥ 0 and Ψ(x 2 ) ≥ 0, we have Then, for each λ ∈ Λ ′ r 1 ,r 2 , the functional Φ − λΨ has at least three distinct critical points that lie in Φ −1 (−∞, r 2 ).
We assume that A satisfies the following conditions: Next, we recall some basic concepts.Let where (•, •) denotes the inner product in R N .The corresponding norm is defined by For every u, v ∈ E, we define and we observe that, by assumptions (A1) and (A2), this defines an inner product in E.Then, E is a separable and reflexive Banach space with the norm for all u ∈ E.
Clearly, E is a uniformly convex Banach space.
A simple computation shows that for every t ∈ [0, T ] and x ∈ R N , and this, along with condition (A2), yields where m = min{1, κ} and M = max{1, N k,l=1 a kl ∞ }.This means that the norms • and • E are equivalent.
If u ∈ E, then u is absolutely continuous and u ∈ L 2 ([0, T ], R N ).In this case, ∆ u(t) = u(t + ) − u(t − ) = 0 is not necessarily valid for every t ∈ (0, T ), and the derivative u may possess some discontinuities that lead to the impulsive effects.
Next we define what is meant by a solution of (1).
)) N , j = 0, 1, 2, . . ., p} is said to be a classical solution of the problem (1) if u satisfies (1).By a weak solution of problem (1), we mean any u ∈ E such that The following lemma should come as no surprise.
We assume throughout that

Main Results
We begin by letting D > 0 be the constant For notational purposes, for any two positive constants θ and η such that , and set and We will use the convention that ρ/0 = +∞ for ρ ∈ R + ; hence, and G η = G θ = 0. We now formulate our main result.
Theorem 5. Assume that there exist two positive constants θ and η with Then, for each λ ∈ Λ and for every function G : [0, T ] × R N → R that is measurable with respect to t for all u ∈ R N , continuously differentiable in u for almost every t ∈ [0, T ], and satisfies (2), and there exists δ λ,G > 0 given by (7) such that, for each µ ∈ [0, δ λ,G ), the problem (1) admits at least three distinct classical periodic solutions in E.
Proof.Fix λ and µ as in the conclusion of the theorem.Set X = E and define the functionals Φ, Ψ : for every u ∈ X.It is well known that Ψ is a Gâteaux differentiable functional whose Gâteaux derivative at the point u ∈ X is the functional Ψ ′ (u) ∈ X * given by for every v ∈ X, and Ψ ′ : X → X * is a compact operator.Moreover, Φ is a Gâteaux differentiable functional whose Gâteaux derivative at the point u ∈ X is the functional Φ ′ (u) ∈ X * given by for every v ∈ X.Also, [22, Proposition 2.4] ensures that Φ ′ admits a continuous inverse on X * .To show that Φ is sequentially weakly lower semicontinuous, let u n ∈ X with u n → u weakly in X.We then have lim inf n→+∞ ||u n || ≥ ||u|| and That is, lim inf n→+∞ Φ(u n ) ≥ Φ(u), which means that Φ is sequentially weakly lower semicontinuous.From (3) and the fact that H(0, . . ., 0) = 0, we have |H(ξ)| ≤ L|ξ| 2 for all ξ ∈ R N .This, in conjunction with the fact that −L ij |s| 2 ≤ I ij (s)s ≤ L ij |s| 2 for every s ∈ R for all i = 1, 2, . . ., N , j = 1, 2, . . ., p, and inequality (5), we have It is easy to see that w ∈ X = E and w 2 E = Dη 2 .Hence, in view of (4), and this together with the condition θ c √ Dm < η, ensures that 0 < r < Φ(w).From ( 5) and ( 10), we see that for each u ∈ X, and it follows that sup On the other hand, from condition (A 1 ), we have Therefore, we have sup and so Since µ < δ λ,G , we have Moreover, since and G η ≤ 0, we see that Therefore, Hence, from ( 13)-( 15), we see that condition (a 1 ) of Theorem 1 is satisfied.Finally, since µ < δ λ,G , by (9), we can fix l > 0 such that lim sup for every t ∈ [0, T ] and ξ ∈ R N .Now, for λ > 0, choose 0 From (A 3 ), there is a function for every t ∈ [0, T ] and ξ ∈ R N .From (5) ( 10), (16), and ( 17), it follows that, for each u ∈ X, and so lim which means the functional Φ − λΨ is coercive.Now ( 13)-( 15) imply , so condition (a 2 ) of Theorem 1 is satisfied.Clearly, weak solutions of problem (1) are precisely the solutions of the equation Φ ′ (u) − λΨ ′ (u) = 0. Therefore, in view of Lemma 4, the conclusion of the theorem follows from Theorem 1 with x = w.
Next, we present a variant of Theorem 5 in which no asymptotic condition on the nonlinear term G is required, but F and G are assumed to be nonnegative.
For positive constants θ 1 , θ 2 , and η with 3 2 , we introduce the notation We then have the following existence result.
Theorem 6.Let F : [0, T ]× R N → R be a non-negative function.Assume that there exist three positive constants θ 1 , θ 2 , and η with such that condition (A 1 ) in Theorem 5 holds.In addition, assume that Then, for each λ ∈ Λ ′ and for every nonnegative function G : [0, T ] × R N → R that is measurable with respect to t for all u ∈ R N , continuously differentiable in u for almost every t ∈ [0, T ], and satisfies (2), there exists δ * λ,G > 0 given by min such that, for each µ ∈ [0, δ * λ,G ), the problem (1) admits at least three distinct classical periodic solutions Proof.Fix λ, G, and µ as in the conclusion of the theorem and take X, Φ, and Ψ as in the proof of Theorem 5. Note that the regularity assumptions in Theorem 2 on Φ and Ψ, and condition (b 1 ) are satisfied.We need to show that (b 2 ) and (b 3 ) hold, so choose w as in (11) and set , and recalling (10), we see that 2r 1 < Φ(w) < r 2 2 .Since µ < δ * λ,G and G η = 0, we have sup Therefore, (b 2 ) and (b 3 ) of Theorem 2 are satisfied.
Now it is easy to see that all assumptions of Theorem 6 are satisfied, and so the conclusion follows.

Theorem 7 .
Let F : R N → R be a continuously differentiable function such that