RECURRENT EQUATIONS WITH SIGN AND FREDHOLM ALTERNATIVE

. We prove that a Fredholm–type Alternative holds for recurrent equations with sign, extending a previous result by Cieutat and Haraux in [3]. Moreover, we show that this can be seen a particular case of [1] and we provide a solution to an interesting question raised by Hale in [6]. Finally we characterize the existence of exponential dichotomies also in the nonrecurrent case.


1.
Introduction. Consider a linear differential equation in R N : (1) where the vector f is bounded continuous on time and the matrix A is recurrent.
That is, A is bounded and uniformly continuous and its hull: is compact minimal with respect to the compact-open topology and the flow induced by translations (Aτ )(t) = A(t + τ ). A necessary condition in order (1) to admit bounded solutions can be easily obtained integrating by parts, namely: for every bounded solution v of the adjoint equation: Here BP stands for having bounded primitive. In [1] we discussed whether or not this condition gives rise to a Fredholm Alternative in the recurrent framework, as implicitly suggested in [14]. To be more precise the question in [1] is to decide if, under the additional assumption that f is jointly recurrent with A, condition (2) is also sufficient for equation (1) to admit a solution x which is not only bounded, but even recurrent and with the same recurrence properties of A and f . This can be expressed by saying that a continuous flow epimorphism exits: It is worth to stress that recurrent solutions to (1) always exist when bounded solutions do, but having solutions satisfying (4) is much more restrictive and, in general, is not for free. If for instance A and f are almost periodic, then this restriction implies that x is almost periodic too.
The main result of [1] is that (2) is sufficient to get such x satisfying (4), as soon as A and f are jointly recurrent and A fulfills the additional condition: The above quantities are respectively called the Favard dimension and the Sacker-Sell dimension of A. To introduce them one has to look at the hull of the homogeneous equations:u = A(t)u (6) that is the class of equations: The number d B of independent bounded solutions may vary with B, and the Favard dimension of A is the smallest possible value: The Sacker-Sell spectrum σ(A) of A is defined as the set of real λ's such that the equation:u = A(t) − λ u does not admit an exponential dichotomy. The spectrum is the union of a finite number of closed intervals and, roughly speaking, the Sacker-Sell dimension d S (A) is the number of independent solutions of (6) having Lyapunov exponents in the spectral interval containing zero. See Section 2 for a precise definition and a description of the consequences of (5). Here we just recall that the same condition has been already used by Sacker and Sell in [12], to obtain a special case of their successive Spectral Theorem [13]. The aim of the present paper is to test the result of [1] in some concrete situations, showing that it gives new insights even in cases already studied in the literature. The first concrete result concerns a matrix A which has a sign, say for instance it is nonnegative in the classical sense: Here A is possibly asymmetric and the sign is that of its symmetric part: This situation has been already considered by Cieutat and Haraux in [3], when A(t) and f (t) are both almost periodic and the antisymmetric part of A(t), that is: is purely periodic. Under these assumptions, the authors obtain a Fredholm-type Alternative which looks quite different from that of [1]. Precisely, they prove that (1) has an almost periodic solution if and only if: for every solution v of (3) which is almost periodic and satisfies the following pair of conditions: the second of which is not even differential. The point is that, contrarily to the appearance, the result of [3] can be seen as a particular case of that proved in [1]. In Section 3 we show indeed that condition (5) is satisfied regardless of the possible periodicity of K A , obtaining the following improvement. In Section 4 we present a scalar example showing that the recurrence of A is optimal for the result, in spite of the fact that the conclusion refers to bounded solutions with no prescribed recurrence properties. In order to find solutions with prescribed recurrence properties, we need to strengthen a little bit the assumptions. Because condition (5) is satisfied, the Corollary follows from the mentioned result of [1]. In Section 3 we give however and independent proof of Theorem 1.1 and Corollary 1 for the following reasons: the paper [1] is not yet published and, moreover, the way to prove (5) gives the full result in few additional steps.
The comparison of Corollary 1 with the aforementioned result of [3] open the doors to a natural question. Imagine indeed that A and f are both almost periodic with A satisfying the sign condition (8), and we are interested in almost periodic solutions to equation (1): is it enough testing condition (2) on those v which are almost periodic too? In Section 4 we show that the answer is negative, by using a well know example by Lillo [7] and a special case of Fredholm Alternative already published in [14].
The last contribution of the present work concerns the real extent of condition (5). Beside showing that it is optimal for the validity of the Fredholm Alternative in the recurrent framework, in [1] we proved that is is also necessary for small Sacker-Sell dimensions: In Section 4 we show that such restriction can be dropped in the periodic case, does not matter A has a sign or not.
Proposition 1. Let A be continuous and periodic. Assume moreover that, for every recurrent f , equation (1) has a solution satisfying (4) if and only if condition (2) holds for every bounded solution to (3). Then condition (5) is satisfied.
We believe that this fact and the results of [1] can be considered as a reasonable answer to a question raised by Hale in its textbook [6]: see Exercise 1.1 at page 147, asking for the appropriate form of the Fredholm Alternative when A is purely periodic and f is almost periodic.
Notations. The symbols |x| and x, y stand for Euclidean norm and inner product in R N . With L(N ) we denote the N×N matrices with real entries, while GL(N ) and O(N ) stand for the subsets of the invertible and the orthogonal matrices respectively.
The symbol BP (R), AP (R) and AP 1 (R) stand respectively for the space of the continuous functions having bounded primitive, that of the almost periodic function and that the almost periodic functions having almost periodic derivative. When f ∈ AP (R) we denote by f its mean value.
2. Preliminaries. We start assuming that the function A and f are just continuous and we denote by φ A and φ * A the Cauchy operators of the homogeneous equations (6) and (3) respectively. Moreover we set: for the initial data giving rise to bounded solutions of (6). We use B * (A) for the same set when referred to the adjoint equation (3). A straightforward integration by parts gives: for every x solving (1) and every v solving the adjoint homogeneous equation (3). Entering now the boundedness of v(t) we find that the following condition must be satisfied: f, φ * A (·)ζ ∈ BP (R) ∀ζ ∈ B * (A) (10) in order (1) may admit a bounded solution.
This necessary condition becomes trivially sufficient when the equation (6) admits an exponential dichotomy. This means that there exist a time independent projector P in R N and constants K, α > 0 such that: Sometimes we say that A has an exponential dichotomy too. When this is true, also the adjoint equation admits an exponential dichotomy and hence B * (A) = {0} and condition (10) is empty. On the other hand it is easy to check that, for every bounded and continuous f , the function: is a bounded solution to (6), actually the only one. A kind of opposite situation has been considered in [14]. There the author assumes that all the solutions to (6) are not only bounded but also separated from zero, namely: and proves the following result. Actually, the boundedness of A and f is explicitly requested in [14] but not used in the proof. The proof depends on the classical variation of constants formula, once one has checked that φ * A also satisfies condition (12). The above proposition is at the core of the result of [1] for the recurrent framework, which we now introduce. We assume from now on that A is bounded and uniformly continuous: the hull H(A) is then compact metrizable and connected, and both spectral theory and Favard theory behave well. The Sacker-Sell spectrum σ(A) has been introduced in [13] as the set of the real λ's such that: does not admit an exponential dichotomy. Since having an exponential dichotomy transfers from a matrix to its hull, when A is recurrent the previous condition is independent of B ∈ H(A) in the sense that σ(B) = σ(A) for every B ∈ H(A). When on the contrary A is not recurrent, it survives only the inclusion σ(B) ⊂ σ(A) for every B ∈ H(A). The spectrum is always nonempty and is the union of at most N compact intervals. Each spectral interval s canonically associated to a spectral vector subbundle of H(A)×R N . Taken any interval (λ, µ) which contains the desired spectral interval and whose closure avoids all the others, the fiber over B ∈ H(A) is given by the initial data ξ such that: In [13] it is also proved that the all these subbundles are linearly independent and their sum is the whole H(A)×R N . We are mainly interested in the spectral boundle associated to the interval containing 0. We denote its fibres by V(B) and we agree that V(B) = {0} when 0 ∈ σ(A). The dimension being independent of B ∈ H(A) due to the connectedness of H(A), we set: and we call it the Sacker-Sell dimension of A. From the definition is clear that V(B) ⊃ B(B) for every B ∈ H(A). Contrarily to V(B) however, the dimension of B(B) may vary with B. We set: and we call it the Favard dimension of A. Automatically the following inequality: holds for all B ∈ H(A). The event that this inequality becomes an equality is particularly relevant for Fredholm Alternative: indeed in [1] the following statement is proved for the recurrent framework.
Theorem 2.1. Let A be recurrent and f bounded and continuous, and assume moreover that: If in addition f is jointly recurrent with A, then one of these solutions satisfies condition (4).
Actually, only the second claim is stated in [1] but the first one can be easily deduced from the proof. The bridge between the two claims in the statement is of course Favard theory, introduced by Favard in [5] for the almost periodic framework and extended to the recurrent framework by Palmer in [9]. The central assumption in this theory is a separation condition on the bounded solutions of the equations (7), namely: inf When this is true for every B ∈ H(A), one says that the Favard condition (F A ) is satisfied. If in addition H(A, f ) is minimal, then the following result is proved:

admits a recurrent solution satisfying (4) if and only if it admits a bounded solution.
The relationship between Favard dimension and Favard condition has been investigated in [15] and [2]. In the latter the authors prove that a given B ∈ H(A) satisfies the separation condition (15) if and only if: which moreover defines a residual subset of H(A) itself. Thus Favard condition (F A ) holds if and only if the previous dimensional equality is satisfied in the whole H(A). This conclusion was already proved in [15]. By taking into account the inequality (13), it is now manifest that (14) implies the validity of (F A ) and hence that the second part of Theorem 2.1 is a consequence of the first one. It is also worth noticing that condition (14) actually implies a much stronger equality, that is: and that moreover the Favard condition (F * A ) for the adjoint equation is satisfied too. This is proved in [1], where it is also shown that the same conclusions are false in general: the two Favard conditions (F A ) and (F * A ) may indeed be not equivalent, and both can be true with different Favard dimensions. The symmetric role of the direct and the adjoint equations is at the basis of a partial inverse of Theorem 2.1, showing that condition (13) is much more than optimal for the Fredholm Alternative. Indeed in [1] the following result is proved.
are satisfied. Assume moreover that, for every f jointly recurrent with A, condition (10) is sufficient to get a solution of (1) satisfying (4).
We will show in Section 4 that, as already anticipated in the Introduction, the restriction d S (A) ≤ 2 can be removed in the class of purely periodic A's.
In the next section we will show that all the results stated in the Introduction can be seen as particular instances of the above theorem. The proof makes use of some specialized normal forms for the equation, obtained via some suitable change of variables. Actually, these normal forms allow to give an independent and short proof of these results. We end the present section by commenting what happens when we do a change of variables. By that we precisely mean a differentiable function Q : R → GL(N ) such that Q, Q −1 andQ are bounded and uniformly continuous on time. It is not difficult to check that, in this case, all functions in H(Q) preserve all these properties. Setting x = Q(t)x transforms the equation (1) where:

This defines an equivalence relation
it is clear that (6) has an exponential dichotomy if and only if equation: Passing to the adjoint equations, it is easy to check (Q −1 ) T is again a change of variables, transforming the equation (3) so that the conclusions we obtained for the direct equations extend to their adjoint equations. Finally, notice that: (19) if and only if v is for (3). Thus condition (10) is satisfied for the equation (1) if and only if the analogous condition is satisfied for equation (17). Summing up, the whole problem of Fredholm Alternative is totally unaffected by changes of variables. Nevertheless, there is a point that deserves some more attention, which concerns recurrence properties. Indeed, in general the recurrence of A does not implies that of A, not even if Q is recurrent: it can be easily checked that this is however true when A and Q are jointly recurrent, in which case A is jointly recurrent with them too.
3. Sign and normal forms. In this section we enter the sign condition (8) and discuss his consequences: the devices we use are essentially the same of [3] but we are able to strengthen the conclusions there. Start supposing that u is a solution of and notice that: By integrating we get the following inequality: As a consequence, if λ = 0 then |u(t)| is not decreasing in time. When on the contrary λ < 0 we have u(−∞) = 0 and the above equation admits an exponential dichotomy, which yields: If now 0 / ∈ σ(A) then equation (6) has an exponential dichotomy with P ≡ 0 as projector, namely it has the whole R N as unstable subspace and trivial stable subspace. Next proposition uses this fact to characterize the existence of an exponential dichotomy.
Theorem 3.1. Assume A is bounded uniformly continuous, symmetric and satisfying (8). Then 0 ∈ σ(A) if and only if for every recurrent B 0 ∈ H(A) one has: Notice that a recurrent B 0 exists in any closed invariant subset of H(A) due to the compactness of H(A) itself.
Proof. Since exponential dichotomy passes from A to H(A), the only if part is trivial. To prove the if part we use Theorem 2 in [11], saying that 0 ∈ σ(A) if we have two ingredients. The first one is that, for every B ∈ H(A), the equation: does not admit nontrivial bounded solutions. To show that this is true, suppose by contradiction that for some B we have a nontrivial bounded solution u(t) and define: lim Let now B 0 be a recurrent element in the ω-limit of B and choose a sequence τ n → +∞ such that Bτ n → B 0 . Standard compactness arguments show that uτ n → u 0 up to subsequences, whereu 0 = B 0 (t)u 0 . Finally: |u 0 (t)| = β ∀t holds by the very construction and hence: Since B 0 ≥ 0 and is symmetric, this implies: In other words u 0 (t) = ξ for some suitable ξ ∈ R N in the kernel of each B 0 (t).
Since |ξ| = β > 0, we conclude that the intersection of these kernels is nontrivial, contradicting (21). The second ingredient for Theorem 2 in [11] is that the stable and unstable subspaces of (22), that is the class of ξ ∈ R N such that: respectively, have complementary dimensions which are moreover independent of B ∈ H(A). Here φ B is the Cauchy operator of (22). We claim that the stable subspace of (22) is always trivial while the unstable subspace is the whole R N . Due indeed to B ≥ 0 we know that |u(t)| in nondecreasing in time for every solution to (22). The triviality of the stable subspace is then manifest. Concerning the unstable subspace, suppose by contradiction that, for some B ∈ H(A), the equation (22) has a solution u with: lim t→+∞ |u(t)| = α > 0 .
Then the failure of (21) follows again from the very same arguments already used in the first part of the proof.
Condition (21) means that the corresponding equation (7) has no nontrivial constant solutions, an occurrence which is very easy to check and then gives rise to an useful test for exponential dichotomy. Notice also that, due to the symmetry of A and hence of every B ∈ H(A), these solutions coincide with the constant solutions of the adjoint equation:v Later on we will see that a related property holds even in the asymmetric case: see Corollary 4.
Next results say what happens when recurrence is added.
Corollary 2. Let A be recurrent, symmetric and satisfying (8). Then 0 ∈ σ(A) if and only if: Proof. Just use recurrence to prove that actually t ker B(t) = t ker A(t) holds for every B ∈ H(A).

Remark 1.
The corollary extends to the recurrent case an analogous result by [3] for the case of an almost periodic A. In fact, there the authors state that exponential dichotomy is equivalent to: This result can be also extended to the recurrent case: we don't provide here such extension since it is not relevant for our aims.
We already know that Fredholm Alternative holds, though trivially, when 0 ∈ σ(A). To understand what happens in the opposite case, we change the variables and put the equation in a convenient normal form. Set: and choose a P ∈ O(N ) straightening the orthogonal decomposition: namely realizing an isomorphism of the first m coordinates of R N onto the first factor in the decomposition, and the second N − m coordinates onto the second factor. Here m is the codimension of V . The time-independent change of variables u = P u transforms equation (6) into: and next proposition shows why the new equation is more convenient to deal with.
Proposition 3. Assume A is recurrent and symmetric and (8) holds. Then: with blocks of dimension m and N − m respectively. Moreover A * is recurrent, symmetric and satisfies (8) together with 0 ∈ σ(A * ).
Proof. Since P is orthogonal, the matrix funtion P T A P is again symmetric and satisfying (8), while recurrence follows trivially from the fact that P is timeindependent. Consider now the the block-decomposition driven by P that is: The definition of V and the construction of P say that B * (t) and D * (t) are identically zero. The same is actually true also for C * (t), since the symmetry of P T A(t)P implies C * (t) = B * (t) T . The remaining block A * (t) is then the only nontrivial one, and hence is recurrent, symmetric and fulfills (8).
As a trivial consequence of Proposition 3 we have a first partial result about Fredholm Alternative.
Notice that (26) is just a specialized form of the more general condition (10). That is, Fredholm Alternative has not changed.
Proof. Because of the normal form (25) the new equation (24) splits into the two independent equations:u 1 = A * (t)u 1u2 = 0 where u 1 ∈ R m and u 2 ∈ R N −m . From Proposition 3 we know that 0 ∈ σ(A * ). This implies that: B(P T AP ) = V(P T AP ) = {0} × R k = P T V and that these constant are indeed the only bounded solutions of (24). Exactly the same conclusions hold for the adjoint equation, which in the new variables writes: . The first part of the Corollary the follows from the final part of Section 2.
To prove the second part, define x = P T f and write x = (x 1 , x 2 ). Then consider the equations:ẋ Again due to 0 ∈ σ(A * ), the first equation has one and only one bounded solution, whatever the bounded and continuous f is. On the other hand, the second equation has bounded solutions if and only if x 2 ∈ BP (R). But for every ζ 2 ∈ R N −m we have: x 2 , ζ 2 = (x 1 , x 2 ), (0, ζ 2 ) = f, P (0, ζ 2 ) so that condition (26) is obtained by construction of P .
We now see what happens when symmetry is removed from the assumptions. Next result shows that symmetry can can be recovered by means of a suitable change of variables. In the proof, we use the decomposition: into the symmetric and the antisymmetric parts of A(t).
Proposition 4. Assume A is recurrent and satisfying (8). Then there exists a change of variables Q : R → O(N ) which is jointly recurrent with A and such that: is again recurrent and satisfying (8) but in addition is symmetric.
Proof. The rule: Consider now the other canonical projection q : M → O(N ). It is a continuous map whose multiplicative inverse: is also continuous. Moreover it admits a derivative along the flow: which is also continuous on the whole M . By taking into account the compactness of M , all the mentioned continuities are indeed uniform continuities. Thus it can be easily checked that the map: is a change of variables in the sense of Section 2. Since K A is antisymmetric we know that φ K A (t) ∈ O(N ) and hence the same is true for Q(t) for every t. Due to the minimality of M , such Q is moreover jointly recurrent with the flow line t → p (A, R A )t = At in H(A) and hence with A itself.
It remains to show that the matrix A defined by (27) has the properties claimed in the statement. This is standard fact, also proved in [3]. IndeedQ = K A Q and hence: so that symmetry and sign of A are the same of A, due to the orthogonality of Q. That A is recurrent follows trivially from the joint recurrence of Q and A.
We are finally ready to describe Fredholm Alternative in the general, possibly asymmetric case. It remain to prove the claim about the bounded solutions. Again due to Corollary 3, we know that the bounded solutions of (28) and its adjoint equation coincide, and are exactly the constants belonging to: Thus the bounded solutions to (6) are: On the other hand, we explained in Section 2 that the adjoint equations to (6) and to (28) are connceted by the change of variables (Q T ) −1 . Thus, by using the orthogonality of Q, we can conclude that the bounded solutions to the adjoint equation to (28) are: that is, they are exactly the same of (28). That these solutions have constant norm, it follows once again from the orthogonality of Q.
The rest of the section is devoted to compare the previous results with the existing literature. Comparison with [1] is rather trivial: Corollaries 3 and 4 are particular cases of Theorem 2.1, since condition (14) is obtained in both cases.
To do the comparison with [3], we need to strengthen recurrence by requiring that A and f are both almost periodic. Notice that joint almost periodicity is never a problem. The analogous of our Corollary 3 and Corollary 4 in the paper [3] are Proposition 3.3 and Theorem 3.6 respectively. We start comparing conclusions which, apparently, are stronger in [3] than here: the authors in [3] grant indeed the existence of almost periodic solutions of (1) instead of bounded ones only. However, Favard theory applies to pass from bounded solutions to solutions satisfying (4), which are almost periodic since A and f are.
Concerning now the assumptions of the symmetric case, Proposition 3.3 in [3] differs from our Corollary 3 for the class of functions involved in condition (26).
In fact, since f, ξ is almost periodic, its primitive is almost if and only if it is bounded.
In the possibly asymmetric case, the comparison between the assumptions of Theorem 3.6 in [3] and our Corollary 4 is more delicate. Indeed the assumptions in [3] are expressed in a completely different form, which does not involve the adjoint equation. More precisely, in [3] the authors assume that: and the test condition for the Fredholm Alternative is that: for every v solving the following system of conditions: the second one not even being differential. Notice however that such v is bounded from the first equation, since K A is antisymmetric, and hence also almost periodic due to (29) and classical Floquet theory: see also the final part of Section 4. Thus again the primitive of f, v in (30) is almost periodic if and only if it is bounded.
The following lemma then close the comparison. Proof. It is trivial to check that (31) is equivalent to the system of differential where the adjoint equation plays now a role. The first part of Corollary 4 says that the bounded solutions of the two equations in the system (32) are exactly the same.

4.
Counterexamples and other results. We start showing that the recurrence of A is optimal for the validity of Theorem 1.1 in the Introduction or, which is equivalent, Corollary 4 in the previous section. Let indeed a > 0 be a bounded and uniformly continuous function satisfying: where we set: Since 0 is a minimal subset of H(a), the function a is not recurrent. Moreover it is clear that 0 ∈ σ(a). Consider now the scalar equation: The adjoint equationv = −a(t)v has no nontrivial bounded solutions, so that each continuous function f passes the test (2). We claim that, on the contrary, equation (33) does not admit bounded solutions as soon as f is such that: Indeed the general solution of (33) is: Thus applying Hôpital's rule we have: which explodes, showing that x is always unbounded. As a final comment, notice that: so that the example works also for the same type of optimality in Theorem 2.1 and Proposition 2.
Next we focus on condition (2), discussing whether or not the class of the test function v is appropriate to obtain for the Fredholm Alternative in some special recurrent frameworks. When for instance A is purely periodic, Floquet theory says that the bounded solutions of the adjoint equation (3) are almost periodic, actually even quasi periodic: testing condition (2) only on the almost periodic v's is then enough a posteriori, in this case. The results of [3] seem to suggest that the same restriction may work when A and f are both almost periodic and the sign condition (8) is satisfied. Indeed, we already proved that this is true when, in addition, A is symmetric: see the final part of the previous section. To show that, on the contrary, the conclusion may be false in the asymmetric case, consider a scalar almost periodic function a and introduce the antisymmetric matrix: The sign condition (8) holds trivially, even if the sign is reversed. Moreover the direct equation (6) and the adjoint equation (3) coincide, and their solutions all have constant norm. In particular, all of them satisfy condition (12) and then Proposition 2 provides a working Fredholm-type Alternative, does not matter the almost periodic a we take. Assume now a satisfies the following additional conditions: Lillo proved in [7] that, in this case, none of the solutions to (6) or, which is equivalent, to (3) is almost periodic. Thus the weak form of condition (2) is trivially satisfied whatever f is. Next result closes the question, showing that there exists almost periodic f 's for which things go wrong.
Proposition 5. Asssume a ∈ AP (R) satisfies (35) and define A as in (34). Then there exists an almost periodic f such that equation (1) does not admit bounded solutions.
Proof. Arguing by contradiction suppose that, for every f ∈ AP (R), equation (1) admits bounded solutions. Classical Favard theory applies to show it admits also an almost periodic solution: see for instance [14]. Such almost periodic solution is unique, since (1) has no almost periodic solutions. Summing up, the map: is a bijection and then a Banach space isomorphism AP 1 (R) ∼ = AP (R). Consider now a sequence a n of trigonometric polynomials such that: a n = 0 a n − a ∞ → 0 .
Since the class of isomorphisms is open, eventually the map L an must be also an isomorphism. On the other hand, the solutions of the equation: u 1 u 2 = 0 −a n a n 0 are all almost periodic, inasmuch they write: cos a n − sin a n sin a n cos a n u 10 u 20 where u 10 , u 20 are the initial data and: a n (t) = t 0 a n (s) ds is almost periodic. Thus the kernel of L an is nontrivial, contradicting the previous conclusion.
In the last part of the section we remove the sign condition on the coefficients' matrix A, but we restrict to the case where: A is purely periodic with the aim of discussing the relevance of condition (5) for the solvability of equation (1) in some suitable class of recurrent functions. We stress that we are not interested in the standard case, where the nonhomogeneous term f and the solution x are also required to be periodic of the same period of A: the solvability condition in this case is the classical Fredholm Alternative, where condition (5) plays no role. For a proof see for instance Hale's textbook [6]. Just after that proof, Hale asks the reader the following question: which form does it take the Fredholm Alternative, when A is again purely periodic but f and hence x are allowed to be almost periodic?
In spite of the fact that this is actually the text of Exercise 1.1 at page 147 in [6], we believe the answer is nontrivial: as for the general recurrent case, condition (2) seems to be the only reasonable Fredholm-type Alternative also for the mixed framework proposed by Hale. To be more consistent, Theorem 2.1 in Section 2 says this is the case, as soon as the additional condition (5) is satisfied by A. It has been already proved in [1] that this additional condition is optimal for the validity of our Fredholm-type Alternative. Next we provide a sharper result: Proposition 7 says that condition (5) is even necessary, in order our Fredholm-type Alternative works fine in the above considered mixed framework. To show it, we start constructing a suitable normal form: we suspect that the result may be known, but we sketch the proof since we are unable to provide any reference. Proposition 6. Let A be continuous and periodic. Then there exists a quasiperiodic change of variables Q such that: in independent of time and each diagonal block D k is a real Jordan cell: for some suitable real eigenvalue λ k .
It is not difficult to check that the λ k are exactly the Lyapunov exponents of the solutions of the homogeneous equation (6). Notice however that to each exponent can correspond many different blocks.
Proof. Real Floquet theory applies to show that A is kinematically similar to a constant matrix C, by means of a periodic change of variables whose period is twice that of A. We can now such C into its real Jordan canonical form, by means of a time independent change of variable: see for instance [4]. To complete the proof, we must have the better of the real Jordan cells corresponding to complex eigenvalues of C. Since different cells give rise to uncoupled equations, this can be done cell by cell. Consider then the equation associated to the real Jordan cell, corresponding to a complex eigenvalue a + ib repeated m times. That is: Let us now consider the rotation: and the periodic change of variables u = P (t)v where: With some lengthy but straightforward computations one proves that the new equation is: To conclude is now sufficient to reorder the components of v, separating those with odd index from those with even index: where of course z = (z 1 , . . . , z 2m ) and D * is the m × m Jordan cell: Summing up, we replaced the initial 2m × 2m complex block, with two m × m real blocks. To obtain such result we have composed four changes of variables: two which are time independent and two which are periodic, but with possibly unrelated periods. The composition is then quasi-periodic.
Next step is to discuss condition (5). Since A is kinematically similar to D as in Proposition 6, the arguments of the final part of Section 2 apply to say that this condition can be equivalently discussed for D itself. From the proof it will be clear that d F (D) is the geometric multiplicity of zero and d S (D) the algebraic multiplicity. We can express their equality in a more traditional language, by saying that the unitary Floquet multipliers of (6) have equal algebraic and geometric multiplicity. We are finally ready for the main, and already announced, result.
Proposition 7. Let A be continuous and periodic. If: then there exists a quasi-periodic f such that condition (2) is satisfied for every bounded solution to (3), but nevertheless equation (1) does not admit any bounded solution.
Since it is well known that the Favard separation conditions (F A ) and (F * A ) are automatically satisfied in view of the periodicity of A, we obtain that the conclusions of Theorem 2.2 in Section 2 hold, independently of the assumption d S (A) ≤ 2.
Proof. Let D be the matrix considered in Proposition 6. Since condition (36) is equivalent to d F (D) < d S (D), in particular we have 0 ∈ σ(D) and hence 0 is an eigenvalue of D. Moreover we can use Lemma 4.1 to find a Jordan cell for D with zero eigenvalue and dimension m > 1. We fix now the attention on such cell and consider the equation: Assume that this condition is satisfied. This implies that the primitives of g m are quasi-periodic: we denote by g m the primitive satisfying g m = 0. The point is that the existence of bounded solutions for equation (37) requires much stronger conditions, inasmuch it involves other m − 1 conditions which can be described recursively. The first one is that determining the boundedness of z m−1 and writes: If we now take for g m−1 any quasi-periodic function with mean value zero but unbounded primitive, the previous condition fails and hence equation (37) does not admit any bounded solutions.
Define now x to be the quasi-periodic function obtained by completing to zero outside the block (g 1 , . . . , g m ). It is clear that: does not admit bounded solutions but fulfills the test condition (2). To conclude it enough to transfer the counterexample to the original equation (1) by setting: where Q is the change of variables involved in Proposition 6. Since such Q is quasi-periodic, the same is true for f .