Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.


Introduction
The use of lower and upper solutions in boundary value problems dates back to the pioneering papers of Peano [20] in 1885 and Picard [21] in 1893. Later, Scorza-Dragoni [23] in 1931 and Nagumo [18] in 1937 were those who provided the main contributions toward a modern theory for scalar second-order ordinary differential equations with separated boundary conditions. The first results for the periodic problem were obtained by Knobloch [15] in 1963. There is nowadays a large literature on this subject, dealing with different types of boundary conditions for ordinary and partial differential equations of elliptic or parabolic type (see, e.g., [5,7] and the references therein).
for every t ∈ [0, T ], and e.g., in [22]. In Sect. 5 we will discuss on these and other extensions and generalizations of our results, possibly also to partial differential equations of elliptic or parabolic type.

Well-Ordered Lower and Upper Solutions for Systems
In this section and the next one, we consider the problem where f : [0, T ] × R N → R N is a continuous function. We are thus in a finitedimensional setting. Let us recall a standard procedure to reduce the search of solutions of (P ) to a fixed point problem in Banach space. We define the set Step 2. Let us show that (1) holds for every solution of (P ), thus proving the theorem. By contradiction, assume that there is a j ∈ {1, . . . , N} and a leading to a contradiction. Case 2t j = 0 ort j = T . Assume for instance thatt j = 0 (the other situation being similar). Then, so that, with v j (T ) = v j (0) being the maximum value of v j (t) over [0, T ], it has to be thatv j (T ) = 0, and hence alsov j (0) = 0. Now, since v j (0) > 0, there is a small δ > 0 such that v j (s) > 0, for every s a contradiction, since 0 is a maximum point for v j (t). The proof is thus completed.
We now provide some illustrative examples.
, for some constants a j > 0, and assume that there is a c > 0 such that Then, taking the constant functions α j = − 3 c/a j , β j = 3 c/a j , we see that Theorem 2 applies, and hence (P ) has a solution.
To work with Leray-Schauder degree, we need to introduce the notions of strict lower/upper solutions.
When we have a well-ordered pair of strict lower/upper solutions, the previous theorem provides some additional information.  Proof. Arguing as in Step 1 of the proof of Theorem 2, we can introduce the modified problem (P ) and we know, by Schauder Theorem, that where B R is an open ball in C([0, T ], R N ) centered at the origin with a sufficiently large radius R > 0. In particular, we may assume that Ω ⊆ B R . By the argument in Step 2 of the same proof and the fact that the pair of lower/upper solutions is strict, we have that all the solutions of (P ) belong to Ω. In other words, there are no zeroes of I − L −1 N in the set B R \ Ω. Then, by the excision property of the degree, Finally, since N and N coincide on the set Ω, the conclusion follows.

Non-well-Ordered Lower and Upper Solutions for Systems
In this section, we still consider problem (P ) in the finite-dimensional space R N . We will treat the case in which we can find lower and upper solutions which are not well-ordered. To this aim, we need to distinguish the components which are well-ordered from the others. We will say that the couple (J , K) is a partition of the set of indices Here, #J and #K denote, respectively, the cardinality of the sets J and K.
Similarly, every function F : A → R N can be written as

Definition 7.
Given two C 2 -functions α, β : [0, T ] → R N , we will say that (α, β) is a pair of lower/upper solutions of (P ) related to the partition (J , K) of {1, . . . , N} if the following four conditions hold: for every (t, x) ∈ E, where Definition 8. The pair (α, β) of lower/upper solutions of (P ) is said to be strict with respect to the j-th component, with j ∈ J , if α j (t) < β j (t) for every t ∈ [0, T ], and for every solution x of (P ) we have it is said to be strict with respect to the k-th component, with k ∈ K, if for every solution x of (P ) we have The following proposition provides a sufficient condition to guarantee the strictness property of a pair of lower/upper solutions of (P ) with respect to a certain component.
The proof can be easily adapted from the corresponding scalar result in [5, Proposition III-1.1] and is omitted.
We are able to prove the existence of a solution of (P ) in presence of a pair of lower/upper solutions (α, β) provided that we ask the strictness property when the components α k , β k are non-well-ordered.
Then, (P ) has a solution x with the following property: for any (j, k) ∈ J ×K, ). In Sect. 3.2 we will provide a generalization of the above result, removing the strictness assumption on one of the components κ ∈ K. Let us now present two illustrative examples. Example 11. Assume J = ∅ and let, for every k ∈ K, for some a k > 0, with Then, taking the constant functions we see that Theorem 10 applies. The same would be true if J = ∅, assuming for j ∈ J , e.g., a situation like in Examples 3 and 4.
Example 12. Let with a n > 0 and h n satisfying (8) Then, for each equation we have both well-ordered and non-well-ordered pairs of lower/upper solutions. Let us fix, e.g., Choosing ι = (ι 1 , . . . , ι N ) ∈ {−1, 1} N , and defining (α, β) with β n = β ιn n , by Theorem 10 we get the existence of at least 2 N solutions x ι of problem (P ), whose components are such that We notice that, even if the function h(t, x 1 , . . . , x n ) is 2π-periodic in each variable x n , the solutions we find are indeed geometrically distinct. We thus get a generalization of a result obtained for the scalar equation in [17].

Proof of Theorem 10
Notice that the case K = ∅ reduces to Theorem 2. We thus assume K = ∅ and, without loss of generality, we take either J = ∅, or J = {1, . . . , M} and K = {M + 1, . . . , N} for a certain M ∈ {1, . . . , N}. Indeed, mixing the coordinates of x = (x 1 , . . . , x N ), we can always reduce to such a situation. We continue the proof in the case J = ∅. (The case J = ∅ can be treated essentially in the same way.) We need to suitably modify problem (P ). For every r > 0, we consider the problem where g r : is defined as follows.
We first introduce the functionsf : Now we define, for every index j ∈ J , and for every index k ∈ K, Notice that, for the indices j ∈ J , the value r > 0 does not affect the definition of the components g r,j .
The proof follows from a classical reasoning and can be easily adapted from Step 2 of the proof of Theorem 2.

Proposition 14.
There is a constant K > 0 such that, if x is a solution of (P r ), for any r > 0, which satisfies (NW k ) for a certain index k ∈ K, then Fix any k ∈ K. If x(t) is a solution of (P r ), multiplying the k-th equation bỹ x k and integrating, we have that So, by a classical reasoning, there is a constant C 1 > 0 such that x k H 1 ≤ C 1 , and there is a constant C 0 > 0 such that x k ∞ ≤ C 0 , for every solution x of (P r ). Define Then, if x is a solution of (P r ), Then, thus proving the proposition.
From now on, we fix r > max{K, α ∞ , β ∞ }, where K is given by Lemma 14. Problem (P r ) is equivalent to the fixed point problem where we have introduced the Nemytskii operator Since we are looking for zeros of we compute the Leray-Schauder degree on a family of open sets. Let us define the constant functionsα as well as the functionš Proof. Assume by contradiction that there is x ∈ ∂Ω μ such that T r x = 0, i.e., x is a solution of (P r ). All the several different situations which may arise lead back to the following four cases.
so that arguing as in Step 2 of the proof of Theorem 2 we obtain a contradiction.
we easily get a contradiction as before.
. Such a situation cannot arise since (7) holds by assumption.
. Such a situation cannot arise since (6) holds by assumption.
Proof. In this case, it can be verified by the arguments of the previous proof that the definition of the set Ω μ provides us a well-ordered pair of strict lower/upper solutions of problem (P r ). The conclusion is then an immediate consequence of Theorem 6.
Arguing similarly, we can prove by induction the following result.
Proof. We proceed by induction. The validity of the statement for K = 1 follows by Proposition 17. So, we fix K ≥ 2 and assume that We then see that yielding d(T r , Ωμ4 ) = (−1) K . The proof is complete.
By the previous proposition, we conclude that As a consequence, there is a solution x of problem (P r ) in the set Ω (4,...,4) . Recalling the a priori bounds in Propositions 13 and 14, we see that the solution x is indeed a solution of problem (P ) and satisfies (W j ) and (NW k ), for every j ∈ J and k ∈ K. The proof is thus completed.

An Extension of Theorem 10
The existence of a solution of (P ) can be obtained also removing from the assumptions of Theorem 10 the strictness assumption on one of the components. (α, β) be a pair of lower/upper solutions of (P ) related to the partition (J , K) of {1, . . . , N}. Fix κ ∈ K and assume that (α, β) is strict with respect to the k-th component, for every k ∈ K \ {κ}. Assume moreover the existence of a constant C > 0 such that Then, (P ) has a solution x such that (W j ) and (NW k ) hold for every (j, k) ∈ J × (K \ {κ}), and

Theorem 19. Let
Proof. Without loss of generality, we can choose J = {1, . . . , M}, K = {M + 1, . . . , N} and κ = N . We can follow the proof of Theorem 10 step by step in the first part, noticing that Proposition 14 holds with the same constant when we assume ( NW N ). Moreover, since we do not ask the strictness assumption with respect to the N -th component, when we introduce the sets Ω μ as in (12) However, we cannot conclude the proof saying that the Leray-Schauder degree is different from zero in Ω (4,...,4) as in (14), since we cannot ensure that it is well defined in the sets Ω (4,...,4, ) with = 2, 3, 4.
Anyhow, at this step of the proof, we can follow the classical reasoning adopted in the scalar case in presence of non-well-ordered lower/upper solutions, cf. [5, Theorem III-3.1]. If there exists x ∈ ∂Ω (4,...,4,2) such that T r x = 0, then we can easily see that x must be a solution of (P r ) such that x N (t) ≤ β N (t) for every t ∈ [0, T ] and x N (τ ) = β N (τ ) for a certain τ ∈ [0, T ]. Since the components α N , β N are non-well-ordered, we have So ( NW N ) holds, thus giving us that x is a solution of (P r ) satisfying all the required assumptions.

Lower and Upper Solutions for Infinite-Dimensional Systems
We now focus our attention on a system defined in a separable Hilbert space H with scalar product ·, · and corresponding norm | · |. We study the problem Similarly, for the function f , we will write We will sometimes identify H with 2 .
As in the finite-dimensional case, we will say that the couple (J , K) is a partition of N + if and only if J ∩K = ∅ and J ∪K = N + . Correspondingly, we can decompose the Hilbert space as H = H J ×H K , where every x ∈ H can be written as Similarly, every function F : A → H can be written as We rewrite Definition 7 in this context.
Moreover, it is said to be strict with respect to the n-th component, with n ∈ N + , if the conditions of Definition 8 hold.
We recall the definition of the set Here is our result in this infinite-dimensional setting. • there exists a constant C > 0 such that • for every bounded set B ⊂ E, the set f K (B) is precompact.
Then, (P ) has a solution x with the following property: for any (j, k) ∈ J ×K, The proof of the theorem is carried out in Sect. 4.2.
Remark 22. As in Theorem 19, we can drop the strictness assumption for a certain index κ ∈ K.
As an immediate consequence of Theorem 21, taking α and β constant functions, we have the following.

Corollary 23.
Let there exist two sequences (p n ) n∈N+ and (q n ) n∈N+ in 2 , with p n < q n for every n ∈ N + , and a partition (J , K) of N + , such that, for

(19)
Furthermore, let there exist a sequence (C k ) k∈K ∈ 2 such that, for every k ∈ K, Then, (P ) has a solution x(t) such that, for every j ∈ J , k ∈ K, We now give some examples of applications, with H = 2 , where we implicitly assume all functions to be continuous.
Example 24. Let, for every j ∈ N + , x), and assume that there is a c > 0 such that Then, f : [0, T ] × 2 → 2 is well defined and taking q j = −p j = 3 √ c/j, we see that both (p j ) j , (q j ) j belong to 2 , and (18) is satisfied, so that Corollary 23 applies with K = ∅.
and assume that there is a c > 0 such that (23) holds. Then, f : 2c/j, we see that both (p j ) j , (q j ) j belong to 2 , and (18) is satisfied, so that Corollary 23 applies with K = ∅.
Furthermore, for every ∈ Z with | | sufficiently large, we can see that the constants p = −π/2 + 2 π, q = π/2 + 2 π satisfy (18), for every j ∈ N + . Thus, we can replace a finite number of couples (p j , q j ) with some couples (p , q ). Such a replacement must be performed only for a finite number of indices j ∈ N + , since we need to guarantee that the new sequences (p j ) j and (q j ) j remain in 2 . Recalling that the so found solution of problem (P ) must satisfy (22), then we conclude that (P ) admits an infinite number of solutions.
Example 26. Let, for every k ∈ N + , and assume that there is a c ∈ ]0, 1[ such that Then, f : [0, T ]× 2 → 2 is well defined and taking q k = −p k = c (1−c)k , we see that both (p k ) k , (q k ) k belong to 2 , and (19) is verified, so that Corollary 23 applies with J = ∅.
Example 27. Let (a n ) n and (σ n ) n be sequences of positive numbers in 2 and let, for every n ∈ N + , f n (t, x) = −a n sin 2πx n σ n + h n (t, x).
Then, for each equation, we have both well-ordered and non-well-ordered pairs of lower/upper solutions. Applying Corollary 23 we thus get the existence of infinitely many solutions of problem (P ). By the same argument in Example 12, we notice that, even if the function h(t, x 1 , x 2 , . . . ) is σ n -periodic in each variable x n , the solutions we find are indeed geometrically distinct. Remark 28. This result should be compared with the ones in [4,11], where one or two geometrically distinct solutions were found assuming a Hamiltonian structure of the problem, i.e., for some function V(t, x 1 , x 2 , . . . ) which is σ n -periodic in each variable x n . It was said in the final section of [11] that it remained an open problem to know if the existence of more than two T -periodic solutions could be proved, and in [4] that "it would be natural to conjecture the existence of infinitely many T -periodic solutions". It is interesting to notice that even in [4,11], to recover some compactness, it was assumed that the sequence of the periods (σ n ) n belongs to 2 .
Remark 29. For any choice of a partition (J , K) of N + , we can consider functions f satisfying the requirements of Examples 24, 25 or 27 for every j ∈ J and of Examples 26 or 27 for every k ∈ K. Corollary 23 applies also in this case.
In the next section, we provide some preliminary lemmas, which will be used in order to prove Theorem 21. We will need the following extension of [11, Lemma 3.2]. ([0, T ], H). As a consequence, the set

Lemma 30. Let E ⊆ C([0, T ], H) be such that the set
Let V = Span(v 1 , . . . , v m ), and denote by Q : H → V the corresponding orthogonal projection. We first prove that the set by the Hölder Inequality and the use of the Monotone Convergence Theorem, recalling (26), and then On the other hand, for every u ∈ E, τ ∈ [0, T ] N+ and every t 1 , Now, for every u ∈ E, τ ∈ [0, T ] N+ and t ∈ [0, T ], by (25), and so On the other hand, since P τ (Qu) ∈ R, by (27) there existsῑ such that We have thus shown that, given ε > 0, there are hence proving that Σ is precompact. The fact that Ξ is precompact in H now follows again from the Ascoli-Arzelà Theorem, recalling that this theorem gives a necessary and sufficient condition for precompactness. Proof. By contradiction, let there exist an ε > 0 such that, for every M ≥ 1, there is a M = (a M n ) n∈N+ ∈ A such that ∞ n=M |a M n | 2 > ε 2 . By compactness, the sequence (a M ) M ∈N+ has a subsequence, for which we keep the same notation, such that a M → a * , for some a * ∈ A. Let M * be any positive integer. Then, taking M ≥ M * sufficiently large, We thus get a contradiction with the fact that a * ∈ H.
As an immediate consequence, we find the following compactness property.

Lemma 32. Let A be a compact subset of H. Then, the set
Proof. Let us consider a sequence (x n ) n∈N+ contained in A P .
If there exists N 0 ∈ N + and a subsequence (x n ) such that x n ∈ Π N0 A for every , then the conclusion is reached since Π N0 A is compact.
If the previous situation does not arise, then we can find a diverging sequence (N ) ⊂ N + and a subsequence (x n ) such that x n ∈ Π N A for every . So, there is a sequence (y n ) ⊆ A such that x n = Π N y n . Since A is compact, then, up to a subsequence, we have y n →ȳ ∈ A. Hence, |x n −ȳ| ≤ |x n − y n | + |y n −ȳ| ≤ |(Π N − Id)y n | + |y n −ȳ| → 0, where Lemma 31 has been applied.
Remark 33. The above statements have been formulated for a Hilbert space H. We will apply them also treating the previously introduced Hilbert spaces H K and H J .

Proof of Theorem 21
We consider, for every N ∈ N + , the auxiliary system . . We recall the projections Π N , introduced in (28), and define the function The auxiliary problem can then be written as Notice that By Theorem 10, for every N ∈ N + , there is a solution x N (t) of ( P N ) such that (W j ) and (NW k ) hold for every j ∈ J ∩ [1, N] and k ∈ K ∩ [1, N]. We impose x N n (t) = 0, for every n > N and t ∈ [0, T ].
(32) Arguing as in the proof of Proposition 14, cf. (10) and (11), we conclude that , for every k ∈ K and j ∈ J . Concerning the indices j ∈ J , we thus have for every N ∈ N + and t ∈ [0, T ]. Now, we repeat the arguments of Proposition 14 with a slight modification. Given the solution x N of ( P N ), we can compute so that x N K H 1 ≤ C 1 and x N K ∞ ≤ C 0 for some constants C 1 and C 0 . Recalling the validity of (33), we can find a sequence τ N Then, we can prove that the sequence (x N K ) N ∈N+ is uniformly bounded. Indeed, we deduce from Lemma 30 that the set {ẋ N Finally, we prove that also the set {x N (34), we can write using the notation of Sect. 4.1, , so that, by Lemma 30, we conclude that both the addenda are in a compact set. Hence there is a compact set D K such that We can now prove similar properties for the components of x N (t), and their derivatives, with indices j ∈ J . At this step, the continuity of f J is sufficient. Indeed, from (33)  By the above arguments, the sequence (u N ) N ∈N+ takes its values in a compact set, and it is equi-uniformly continuous. By the Ascoli-Arzelà Theorem there exists a subsequence, for which we keep the same notation, which uniformly converges to some u * : [0, T ] → H ×H. Writing u * (t) = (x * (t), y * (t)), we have that (x N ,ẋ N ) uniformly converges to (x * , y * ). In particular x * (0) = x * (T ), y * (0) = y * (T ). Rewriting the differential equation in ( P N ) as a planar system, we have where F N (t, x, y) = (y, Π N f (t, Π N x(t))). The corresponding integral formulation is then System ( Q N ) has a solution u N = (x N ,ẋ N ) such that u N (0) = u N (T ) for every N ∈ N + . We want to show that F N (t, u N (t)) → F (t, u * (t)), where F (t, x, y) = (y, f (t, x)). Fix ε > 0; for N sufficiently large, we have x * (t))|.

Final Remarks
In this final section, we briefly outline some possible extensions of the previous results.
1. The boundedness assumption on the function f K (t, x) could be replaced by a nonresonance condition with respect to the higher part of the spectrum of the differential operator −ẍ with T -periodic conditions. For instance, denoting by λ 2 the first positive eigenvalue (2π/T ) 2 , one could assume that where γ K (t, x) ≤ c < λ 2 and r K (t, x) is bounded. Or, more generally, one could assume an asymmetric behaviour of the type where (μ K (t, x), ν K (t, x)) lie below the first curve of the Fučík spectrum (here, as usual, x + = max{x, 0} and x − = max{−x, 0}). 2. One could deal with nonlinearities of the type f (t, x,ẋ), depending also on the derivative of x, assuming some type of Nagumo growth condition (see [5]). Such a situation has already been studied in the infinitedimensional setting, e.g., in [22], in the well-ordered case. 3. In this paper we defined the lower and upper solutions as C 2 -functions.
However, this regularity could be weakened, and different definitions could be adopted. We do not enter into the details, for briefness, and we refer to the book [5] for further possible developments. 4. The results of this paper hold the same for the Neumann problem ẍ = f (t, x), x(0) = 0 =ẋ(T ), with almost identical proofs. Concerning the Dirichlet problem ẍ = f (t, x), x(0) = 0 = x(T ), some modifications are needed in the non-well-ordered case. Both problems have their partial differential equations analogues. We will provide in [10] an extension of Theorem 10 in a finite-dimensional abstract setting including the case of elliptic and parabolic type systems with different types of boundary conditions, thus generalizing the results in [6,7,12]. However, an infinite-dimensional extension in the PDE case remains an open problem.