Periodic Solutions of Asymptotically Linear Autonomous Hamiltonian Systems with Resonance

In this paper we define the index at infinity of an asymptotically linear autonomous Hamiltonian system. We use this index to prove the existence and bifurcation from infinity of periodic solutions of the system. We apply the degree for G-invariant strongly indefinite functionals defined by Gołȩbiewska and Rybicki in (Nonlinear Anal 74:1823–1834, 2011).


Introduction
Consider the problem of existence of periodic solutions of the systeṁ where H ∈ C 2 (R 2N , R) is such that H is asymptotically linear at infinity, i.e.
One of the ideas of studying such a system is to consider an associated functional defined on an appropriate Hilbert space. Using this functional one can define an index of the stationary solution and of the infinity. Comparing these indices we can prove the existence of solutions. Such an idea has been used by many authors, see for example [1,10,15,16,21]. The methods used to define the indices include theories of Morse index and the Conley index. In the paper [12] we have defined the indices, using the degree of S 1 -invariant strongly indefinite functionals. Namely, we have considered the system (1.1) with assumptions (1) (H ) −1 (0) = {p 1 , . . . , p q }, (2) σ (J H (∞)) ∩ iZ = ∅.
Assumption (2) implies the system is nonresonant at the infinity. On the other hand, the stationary solutions can be resonant. For p being such a solution and for the infinity we have defined indices (or almost all their coordinates) I H ( p) and I H (∞) being elements of the Euler ring U (S 1 ). Comparing the sum of the indices I H ( p) with the index I H (∞) we proved the existence of solutions, see Theorems 3.1, 3.2 of [12].
The main aim of our paper is to define the index I H (∞) in the resonant case. To this end, following the method of Su (see [20], also [4,17]), we introduce the additional assumptions, see conditions (H4) and (H5) of Sect. 3, and obtain the so called strong angle conditions on the associated functional.
Note that the index I H (∞) can be also used for studying other problems, for instance the bifurcation from infinity, i.e. the problem of the existence of unbounded closed connected sets of periodic solutions of the family of systems: where H ∈ C 2 (R 2N × R, R) is such that H (x, λ) is asymptotically linear at infinity. It is known that if the difference of the indices computed on some levels λ − , λ + is nontrivial, then there exists an unbounded continuum of solutions. The proof of this fact in the case of the operator being completely continous perturbation of the identity can be found in [8], the proof in the general case is analogous.
After this introduction this paper is organized in the following way: in Sect. 2 we fix notation and remind the definitions of degrees used in the next part of the paper. Moreover we compute the index at the infinity for the asymptotically linear operator. To this end we introduce the so called strong angle conditions. In Sect. 3 we study periodic solutions of autonomous Hamiltonian systems. We formulate main results of this paper, namely Theorems 3.1 and 3.3. In the former one we prove the existence of solutions in the resonant case, while in the latter one we prove the existence of a connected set of solutions bifurcating from the infinity.

Preliminaries
In this section we collect basic facts from the S 1 -equivariant degree theory. We remind, for G = S 1 , the properties of the degree for G-equivariant gradient maps defined by Gȩba in [9]. We also recall for G = S 1 the generalisation of this degree defined in [13], namely the degree for S 1 -invariant strongly indefinite functionals.

Degree for S 1 -Equivariant Gradient Maps
Let V be a finite-dimensional, orthogonal S 1 -representation and ⊂ V an open, bounded and , as a special case of the degree for G-equivariant gradient maps defined by Gȩba in [9]. The coordinates of the degree can be written in the following way: Remark 2.1 Note that the definition of the degree for S 1 -equivariant, orthogonal maps has been given also in [19]. Since every gradient map is an orthogonal one, we can use this definition instead of the one mentioned above. However, formulas defining degree in those two approaches differ by sign. The general summary of the equivariant degree theory can be found in [2,3].
The properties of the degree are formulated in the following theorem (see [9]): Theorem 2.1 (1) Let and f satisfy the above assumptions. Then: To compute the degree of a product map we use for G = S 1 the following fact, proven in the general case in [14].
Let us now remind the classification of the equivalence classes of finite-dimensional S 1 -representations. Recall that two representations V, V are equivalent (briefly V ≈ V ) if there exists an equivariant linear isomorphism L : V → V . For k ∈ N define an S 1action on R 2 by γ t · (x, y) = (x cos(kt) − y sin(kt), x sin(kt) + y cos(kt)) where γ t = cos t + i sin t ∈ S 1 , (x, y) ∈ R 2 . Denote by R[1, k] the two-dimensional S 1 -representation with this action and put R[ It is known that any finite-dimensional S 1 -representation is equivalent to Using this fact and the definition of the degree, we obtain the following computational formulas for the degree of a self-adjoint, S 1 -equivariant linear isomorphism (see [9]). By m − (L) we denote the Morse index of L, i.e. the sum of multiplicities of negative eigenvalues of L.

Degree for S 1 -Invariant Strongly Indefinite Functional
We briefly recall, in the special case G = S 1 , the definition of the degree for G-invariant strongly indefinite functionals given in [13].
Let (H, ·, · H ) be an infinite-dimensional, separable Hilbert space, which is an orthogonal S 1 -representation. Denote by = {π n : H → H : n ∈ N ∪ {0}} an S 1 -equivariant approximation scheme on H, i.e. a sequence of S 1 -equivariant orthogonal projections satisfying (1) H n = π n (H) is a finite-dimensional subrepresentation, for all n ∈ N, (2) there exists a subrepresentation H n of H n+1 such that H n+1 = H n ⊕ H n and H n ⊥ H n for all n ∈ N, Assume that: where n is sufficiently large and ε sufficiently small.
It was shown in [13] that such the degree is well-defined and it has the same properties as the degree for S 1 -equivariant gradient mappings, i.e. properties of existence, additivity, excision, linearisation and homotopy invariance. In the fact below we formulate the slightly different version of the last of those properties, the so called generalized homotopy invariance property. The proof of this fact carries over from the Leray-Schauder degree case, see [5]. We put an assumption: where L satisfies (a2) and ∇ u K is an S 1 -equivariant completely continuous operator.
Using this property we can study the bifurcation from infinity of solutions of the equation Theorem 2.2 Let satisfy condition (a6) and let λ ± ∈ R, γ > 0 be such that assumption The proof of this theorem in the case of the SO(2)-equivariant operators of the form compact perturbation of identity can be found in [8]. The authors use the properties of the SO(2)-degree, especially the generalized homotopy invariance property and the fact that the set ∇ u −1 (0) is compact in H × R. Using the counterparts of these facts in the case of the degree for G-invariant strongly indefinite functionals, we obtain our result.

Index at the Infinity
In this section, under some additional assumptions, we compute the degree of an asymptotically linear operator, which is a gradient of a strongly indefinite, S 1 -invariant functional at the sufficiently large disc centered at the origin. Let (H, ·, · H ) be an infinite dimensional Hilbert space, which is an orthogonal S 1 -representation. Let ∈ C 2 S 1 (H, R) be such that where L satisfies condition (a2) of Sect. 2.2 and moreover Denote by ∇ 2 (∞) = L + L ∞ the linearization of ∇ at the infinity. In the case when ∇ 2 (∞) is an isomorphism, analogously to the linearization property at the origin, one can prove the property of the linearization at the infinity.
In the rest of this section we will assume ∇ 2 (∞) is not an isomorphism, i.e.
Denote by V ∞ and W ∞ the kernel and the image of ∇ 2 (∞), respectively and put To compute the degree of at the sufficiently large disc centered at the origin we put additional assumptions, so called strong angle conditions.
The above conditions has been introduced by Li and Su in [17], see also [20]. Using the method introduced by Bartsch and Li in [4] for a similar type of assumptions (namely the angle conditions (AC ± ∞ ) they have computed the critical groups of the functional at the infinity. Combining some arguments from [4] and the degree theory for S 1 -invariant strongly indefinite functionals, we can compute the degree ∇ S 1 -deg (∇ , B γ (H)).
Choose M > 0 and α ∈ (0, π 2 ) as in (SAC + ∞ ) and note that for and note that from (b3) without loss of generality we can assume that for We will show that this homotopy satisfies assumptions of Fact 2.3. Using arguments as in the proof of Proposition 2.5 in [4], one can show that for γ sufficiently large and u ∈ H satisfying u H > γ : Hence we obtain that ∇ u H ∞ (u, t) = 0 for every u ∈ H satisfying u H > γ and t ∈ [0, 1]. Moreover, . From assumptions (b1), (b2) and the fact that t · I d V ∞ is a finite dimensional mapping for every t ∈ [0, 1], we obtain that ∇ψ is S 1 -equivariant and completely continuous. Therefore assumptions of the homotopy invariance property (see Fact 2.3) are satisfied. Hence: (2.4) Note that functional : H → R defined by (v, w) = (v, w) + 1 2 v 2 H satisfies assumptions of the property of linearization at the infinity (Fact 2.4). Using this fact and Definition 2.1 we have (2.5) where, according to the definition of the degree, (B γ (H)) ε is an ε-neighbourhood of the set For n sufficiently large, from (a2) and (b1), V ∞ ⊂ H n . Therefore, using the product formula, (2.6) From (2.4)-(2.6), we obtain the assertion.
To prove (b) it is enough to consider the homotopy H ∞ ∈ C 2 (H × [0, 1], R) defined by the formula H ∞ ((v, w) The rest of the proof is analogous to the proof of (a).

Remark 2.3
Note that if L(V ∞ ) ⊂ V ∞ , from the excision property of the degree and the product formula we have Therefore, using again the definition of the degree, we have We also study the problem of existence of closed connected sets of periodic solutions bifurcating from infinity for the family of autonomous Hamiltonian systems:

Periodic Solutions of Autonomous Hamiltonian Systems
where H (∞, λ) is a real, symmetric matrix for all λ ∈ R.
Remark 3.1 Note that if H ∈ C 2 (R 2N × R, R) satisfies (H3), then for a fixed λ the potential We start with recalling the definitions of the appropriate Hilbert space and the functional corresponding to this system. Put E := H 1/2 (S 1 , R 2N ), the Sobolev space of functions It is known that E is a separable Hilbert space with an inner product defined by the formula Moreover, if we consider an S 1 -action given by γ ·u(t) = u(t +ϕ) for γ = cos ϕ+i sin ϕ ∈ S 1 , u ∈ E, it is easy to show that E is an orthogonal representation of the group S 1 .
Define a sequence of projections = {π n : E → E; n ∈ N ∪ {0}} by π n (a 0 + ∞ k=1 (a k cos kt + b k sin kt)) = a 0 + n k=1 (a k cos kt + b k sin kt) and put E n = π n (E). Then is an S 1 -equivariant approximation scheme. Under the assumption (H1) (or (H3) respectively) one can prove (see [18]) that u(t) ∈ C 2 (R, R 2N ) is a 2π-periodic solution of (3.1) ((3.2) respectively) if and only if u is a critical point with respect to u of the functional (3.4) or respectively˜ Let us summarize the properties of these functionals. Define From the Riesz Theorem we obtain the existence of a unique, bounded, S 1 -equivariant, self-adjoint Fredholm operator of index 0, L : Using the definition of L and the inner product formula (3.3) we obtain an explicit formula for this operator. Namely for u = a 0 + ∞ k=1 (a k cos kt + b k sin kt) we have: From the above we obtain (1) ker L = π 0 (E), (2) π n • L = L • π n for all n ∈ N ∪ {0}.

Note that using (H1) and (H3) we can rewrite H and˜ H as
From the Riesz theorem it follows that there exists a unique, bounded linear operator L ∞ : E → E defined by (3.7) Additionally we put η(u) = 2π 0 g(u(t))dt. It is easy to check that L ∞ and ∇η are S 1equivariant and, since H (∞) is symmetric, L ∞ is self-adjoint. Moreover, it is known (see [18]) that L ∞ and ∇η are completely continuous and that condition g (x) = o(|x|) for |x| → ∞ implies ∇η(u) = o( u E ) as u E → ∞ (see [16]).
Let A be a real, symmetric, (2N × 2N )-matrix. Consider the functional associated to a linear systemẋ = J Ax. (3.8) According to (3.4) we obtain the functional A : E → R given by From the above definition and (3.3) we can compute the explicit formula for B. Namely, for u = a 0 + ∞ k=1 (a k cos kt + b k sin kt) ∈ E we have Remark 3.2 Note that from (3.9) it follows that π n • B = B • π n for n ∈ N ∪ {0}.

Corollary 3.1 From the above considerations, taking A = H (∞), we obtain that the functional H is of the form (2.3) with conditions (a2),(b1)-(b3) satisfied. The same statement is valid if we consider˜ H (·, λ) with λ fixed.
For . Note that from (3.6) and (3.9) it follows that the restriction of ∇ A to E k can be represented by the matrix T k (A) for k ≥ 1 and by A for k = 0. From the above considerations we obtain the following properties of the operator ∇ A .

Lemma 3.1 (1) ∇ A is an isomorphism if and only if σ (J
where m k is a geometric multiplicity of an eigenvalue ik of the matrix J A.
In the case of an isomorphism, we have the following formula for the degree of ∇ A , see [12].

Lemma 3.2 If σ (J A) ∩ iZ = ∅, then
For a nonlinear Hamiltonian system with potential H satisfying (H1) we define an index I H (∞) ∈ U (S 1 ) depending on the matrix H (∞). We start with a nonresonant case.
In the following we define an index I H (∞) in the case when σ (J H (∞)) ∩ iZ = ∅. We assume that one of the following additional conditions on potential H is satisfied. Remind that H (x) = H (∞)x + g (x).
(H4) There exist R, c 1 , c 2 > 0, and 0 < s < 1 such that for all x ∈ R 2N (g (x), x) ≤ 0 and moreover for all x ∈ R 2N with |x| ≥ R we have (H5) There exist R, c 1 , c 2 > 0 and 0 < s < 1 such that for all x ∈ R 2N with |x| ≥ R we have Note that I H (∞) is a well-defined element of U (S 1 ).

Lemma 3.4 Let H satisfy condition (H1) and σ (J H (∞)) ∩ iZ = ∅. Moreover assume that (H4) is satisfied. Then, for γ sufficiently large, I H
where V ∞ is the kernel of the operator ∇ 2 H (∞) associated to the linear equatioṅ where m k is a geometric multiplicity of an eigenvalue ik of the matrix J H (∞). Therefore, from Fact 2.2, Reasoning as in the proof of the previous lemma, we obtain To end the proof we use the fact that the pair of the eigenvalues ik, −ik of the matrix J H (∞) corresponds to an eigenvalue 0 of matrix T k (H (∞)). Therefore 2m k is the multiplicity of the eigenvalue 0 and m − (T k (H (∞))) + 2m k =m − (T k (H (∞))).

Existence of Solutions
In the following we use the definition of the index I H (∞) to formulate conditions sufficient to the existence of solutions of (3.1). 3. Therefore, we can choose α p i , α ∞ > 0 for i = 1, . . . , q, such that B α p i (E, p i ) are disjoint neighborhoods of p i ∈ E satisfying cl(B α p i (E, p i )) ⊂ B α ∞ (E) for i = 1, . . . , q. Suppose, contrary to our claim, that p 1 , . . . , p q are the only 2π-periodic solutions of (3.1). From the additivity and the excision properties of the degree, we have Taking into consideration Remark 3.4, Lemmas 3.4 and 3.5, we obtain Then there exists at least one non-stationary 2π-periodic solution of (3.1).

Corollary 3.3
The above corollary remains valid if we replace (H4) by (H5) and conditions (1), (2) by: Note that we can omit the assumption σ (J H ( p)) ∩ iZ = ∅ for p ∈ (H ) −1 (0) by defining the index I H ( p) (or almost all its coordinates) as in Remark 3.4. In such a situation we obtain the following theorem: Proof Observe, that we can assume p 1 , . . . , p q are isolated critical points of the functional H . If not, we obtain an infinite sequence of 2π-periodic solutions of (3.1) and hence the assertion of the theorem.
Therefore the degrees (E, p)). The rest of the proof is analogous to the proof of Theorem 3.1.