Bounded and almost periodic solutions of nonlinear

Bounded and almost periodic solutions of nonlinear differential equations : variational vs nonvariational approach∗ Jean Mawhin

1

Introduction

The relation between the theory of ordinary differential equations and the calculus of variations is essentially as long as the history of the calculus of variations itself. Before the middle of the XVIIIth century, it was made clear by Euler, through an approximation method, and soon later by Lagrange in a purely analytical way, that the problem of finding the critical points of functionals of the form Z b ϕ(y) = F (t, y(t), y ′ (t)) dt (1) a

over functions y with fixed values at a and b was “reduced” to solving the associate Euler-Lagrange second order differential equation d ∂F ∂F − = 0. (2) ′ dt ∂y ∂y A lot of efforts were devoted to solve explicitely this equation in special problems, but the time rapidly came where, using a quotation of Painlevé, “all differential equations which could be integrated were integrated”. Hence an idea slowly grew up that it could be more easy to find a critical point of (1) than to solve (2). In the XIXth century, under the influence of Gauss, Bolzano and Cauchy, existence proofs gained their importance in mathematics, and proving the existence of a minimum to (1) in order the get the existence of some solution to (2) became the basis of the so-called Dirichlet principle. After some pioneering work of Arzel´ a, Hilbert’s paper of 1900 made the argument rigorous and marked the beginning of the direct method of the calculus of variations. ∗ Calculus of variations and differential equations (Technion 1998), A. Ioffe, S. Reich and I. Shafir eds., Chapman & Hall/CRC, Boca Raton, 1999, 167-184

1

During the XXth century, this approach, and more generally the search of critical points of ϕ, has led to significant progress in the study of various boundary value problems over a finite interval [0, T ], for the corresponding Lagrangian or Hamiltonian equations or systems. The setting in the ordinary differential case was already fixed around the first World War by Lichtenstein, who introduced Sobolev spaces for functions of one variable instead of the classical C k spaces used by Hilbert. Besides finding a T-periodic solution (y(t) = y(t + T ) for all t ∈ R) of the second order equation y ′′ = f (t, y, y ′ )

(3)

when f is T-periodic with respect to t, (which is equivalent to finding a solution y satisfying the periodic boundary conditions y(0) = y(T ), y ′ (0) = y ′ (T ), it was natural to consider, when the T-periodic assumption upon f is dropped, the existence of solutions y such that y and y ′ are bounded over R (bounded solutions of (3)). An intermediate class, the almost periodic solutions, was introduced in the twenties by Harald Bohr, Almost periodic functions can be defined as the completion, Pn for the uniform norm over R, of the space of trigonometric polynomials k=−n ck eiλk t , where the λk are real. They are characterized by the property that (∀ǫ > 0)(∃L > 0)(∀a ∈ R)(∃τ ∈ [a, a + L])(∀t ∈ R) : |y(t + τ ) − y(t)| < ǫ. Clearly, if y is continuous and T-periodic (and hence kT-periodic for each nonzero integer k), one can take L = T and, for τ, the multiple kT of T contained in [a, a + T ]. Ordinary differential equations with almost periodic coefficients and their possible almost periodic solutions were considered from the beginning of the theory of almost periodic functions, starting with linear equations. The formula for the general solution of a linear differential equation with constant coefficients shows that the possible bounded solutions of such an equation are nothing but its almost periodic ones, showing a first link between the existence of bounded solutions and of almost periodic solutions for such equations. Bohr and Neugebauer considered almost periodically forced linear differential equations with constant coefficients in 1926, and showed that all bounded solutions of such equations are also almost periodic. It was only in 1962 that Bochner obtained the extension of this result to linear equations with almost periodic coefficients, using a new characterization of almost periodic functions. For nonlinear equations, it is known since the pioneering work of Amerio in the fifties that the link between the existence of bounded solutions and that of almost periodic solutions is fairly delicate. We refer to [16, 18, 17] for details and references. Bounded and almost periodic solutions seems to have remained untouched by the methods of the calculus of variations until the last decade of this century. It is therefore a rather recent history, that I will shorty describe, concentrating on specific equations like Duffing and pendulum ones, but with the hope of giving a fair description of the state of the art, as well as a comparison with the 2

results based upon nonvariational approaches. Only the case of general almost periodic solutions will be considered, leaving aside the search of quasi-periodic solutions or of special classes of almost periodic solutions recently treated, using variational methods, by Berger and Zhang [6, 9, 10, 11] in the case of Duffing equation, and by Belley, Fournier, Saadi Drissi, and Hayes, in the case of the forced pendulum equation [3, 4, 5, 2]. Our analysis shows that, presently, the variational approach does not seem to provide sharper results or easier proofs than nonvariational ones, and its use in getting sharper results, like in the case of boundary value problems on compact intervals, remains a challenge for the next century.

2

Bounded solutions of forced linear equations

Let C = {h : R → R : h is continuous over R}, BC = {h ∈ C : h is bounded over R},

BC k = {h ∈ C : h ∈ C k , h ∈ BC, h′ ∈ BC, . . . , h(k) ∈ BC}, BP 1 = {h ∈ C : h has a bounded primitive over R}

BP 2 = {h ∈ C : h has its first two primitives bounded over R}

BC are BC k are normed vector spaces for the norms respectively defined by khk∞ = sup |h(t)|, t∈R

khkk,∞ = max{khk∞ , . . . , kh(k) k∞ }.

We also set, for h ∈ BC, oscR h = sup h − inf h. R

R

We recall in this section a few results over the solutions bounded over R of the first order linear differential equation u′ + cu = h(t),

(4)

and of the second order linear equation y ′′ + cy ′ = h(t),

(5)

where c ∈ R and h ∈ C. A detailed treatment can be found, for example, in [23]. The following existence and uniqueness result for a first order linear equation is standard.

3

Lemma 1 If c > 0, then, for each h ∈ BC, the differential equation (4) has a unique solution u ∈ BC, given by Z t e−c(t−s) h(s) ds. (6) u(t) = −∞

Furthermore, the following inequalities hold for all t ∈ R, inf R h supR h ≤ u(t) ≤ . c c

(7)

The existence part of the following result for a second order linear equation is due to Ortega [26]. Lemma 2 If c > 0 and h ∈ C, then the differential equation (5) has a solution y ∈ BC 1 if and only if h ∈ BP 1 . If it is the case, all those solutions verify the inequality oscR y ≤

oscR H , c

(8)

Rt where H(t) = 0 h(s) ds. Furthermore, for each h ∈ BP 1 there exists a unique solution Hc ∈ BC 1 such that sup Hc = − inf Hc = R

R

1 oscR Hc . 2

(9)

Proof. The easy necessary condition is left to the reader. For the sufficient condition, let us consider the linear equation u′ + cu = H(t) + b,

(10)

where b ∈ R. Using Lemma 1, one easily finds that the unique solution ub ∈ BC of (10) is given by Z t b b ub (t) = e−c(t−s) H(s) ds + := Hc# (t) + . c c −∞ Consequently, ub ∈ BC 1 and satisfies equation (5). Furthermore, taking c supR Hc# + inf R Hc# , b=− 2 one easily finds that the corresponding solution Hc (t) = Hc# (t) − is such that sup Hc = R

supR Hc# + inf R Hc# 2

oscR Hc = − inf Hc . R 2

The following slight extension of Landau’s inequality is proved in [14]. Here E denotes a Banach space with norm k · k. 4

Lemma 3 Let A ∈ C 0 (R, L(E, E)) and y ∈ C 2 (R, E) be such that kyk∞ ≤ α, ky ′′ + A(·)y ′ k∞ ≤ β, kAk∞ ≤ γ. Then ky ′ k∞ ≤ δ, where δ is the largest root (δ is non negative) of the equation x2 − 4αγx − 4αβ = 0. We also need, in some uniqueness problems, a result over second order differential inequalities, proved in [14]. Lemma 4 Let α > 0, β ∈ R and γ ∈ BC. If the function r ∈ BC 2 verifies the differential inequality r′′ ≥ γ(t)r′ + αr − β, then sup r ≤ R

β . α

An easy consequence is the following uniqueness condition for a bounded solution of a linear equation [24]. Lemma 5 Let c ∈ R and f ∈ BC be such that f (t) ≤ −δ < 0 for all t ∈ R. Then the unique solution y ∈ BC 1 of the differential equation y ′′ + cy ′ + f (t)y = 0,

(11)

is the trivial one.

3

Bounded solutions of Duffing’s type systems

In their pioneering work dedicated to proving the existence of almost periodic solutions through the calculus of variations, Berger and Chen [7, 8] have considered, as an intermediate tool, the existence of bounded solutions of second order systems of the form y ′′ − Ay − ∇U (y) = h(t),

(12)

when A is a symmetric positive definite matrix, U : RN → R is of class C 2 , its Hessian matrix U ′′ (y) is semi-positive definite, and U (y) ≥ U (z) whenever all the components are such that |yj | ≥ |zj | and sufficiently large. Following the idea of [14], we want to show that a substantially more general result follows 5

from standard techniques in ordinary differential equations. We present here a generalization of the conditions of [14]. The space RN is endowed with its usual inner product x · y, and | · | denotes the associated Euclidean norm. For continuous functions F : R × RN → RN , with F (·, x) bounded on compact x-sets, and bounded continuous mappings b : R → R, B : R → L(RN , RN ), we consider the following second order ordinary differential system of Duffing’s type: y ′′ (t) + [b(t)I + B(t)]y ′ (t) − F (t, y(t)) = 0.

(13)

We denote by B ∗ (t) the transposed matrix of B(t), and introduce the following Condition (H): There exists R > 0 such that, for each y satisfying |y| = R and each t ∈ R, one has 1 ∗ F (t, y) − B(t)B (t)y · y ≥ 0. 4 For linear two-point boundary value problems, a similar condition was introduced by Picard in the scalar case, and by Hartman and Wintner in the vector case (see [20], ch.XII, Th. 3.3). Theorem 1 If condition (H) holds, then (13) has at least one solution y ∈ BC 1 (R, RN ) such that |y(t)| ≤ R for all t ∈ R. Proof. . We define X : R × RN → RN as follows: F (t, x) if t ∈ [−T, T ] X(t, x) := F (2T − t, x) if t ∈ [T, 3T ] and X(t+4kT, x) = X(t, x) for every (t, x) ∈ [−T, 3T ]×RN and k ∈ Z. It is easy to see that X ∈ C 0 (R × RN , RN ). Also we see that, for every x ∈ RN , X(·, x) is 4T -periodic. We define C : R → L(RN , RN ) as follows: b(t)I + B(t) if t ∈ [−T, T ] C(t) := b(2T − t)I + B(2T − t) if t ∈ [T, 3T ] and C(t+ 4kT ) = C(t) for all t ∈ [−T, 3T ] and all k ∈ Z. So C is continuous and 4T -periodic. We first prove that, for each T > 0, there exists uT ∈ C 2 (R, RN ) such that: i) uT is a solution of the equation (13) on [−T, T ]. ii) uT (−T ) = uT (T ), u′T (−T ) = u′T (T ). iii) ∀t ∈ [−T, T ], |uT (t)| ≤ R. iv) ∃c1 ∈ (0, +∞), ∀T > 0, ∀t ∈ R, |u′T (t)| ≤ c1 . 6

For this, we use a Leray-Schauder argument [21] and consider T-periodic solutions of the family of equations y ′′ + C(t)y ′ − (1 − λ)ay − λX(t, y) = 0,

(14)

where λ ∈ [0, 1] and a > 0 is such that 1 a > sup max |B ∗ (t)v|2 . t∈R |v|=1 4 If y : R → RN is a possible 4T-periodic solution of (14) for some λ ∈ [0, 1[, and 2 reaches its maximum at some τ ∈ [−2T, 2T ], then if r(t) := |y(t)| 2 0 = r′ (τ ) = y(τ ) · y ′ (τ ), 0 ≥ r′′ (τ ) = |y ′ (τ )|2 + y(τ ) · [(1 − λ)ay(τ ) + λF (σ, y(τ )) − b(σ)y ′ (τ ) − B(σ)y ′ (τ )] 2 ′ B ∗ (σ)y(τ ) |B ∗ (σ)y(τ )|2 2 = y (τ ) − + (1 − λ) a|y(τ )| − 2 4 B(σ)B ∗ (σ)y(τ ) +λy(τ ) · F (σ, y(τ )) − 4 ∗ 2 B(σ)B ∗ (σ)y(τ ) |B (σ)| |y(τ )|2 + λy(τ ) · F (σ, y(τ )) − , ≥ (1 − λ) a − 4 4 where σ := τ if τ ∈ [−T, T ], σ := 2T − τ if τ ∈ [T, 2T ], σ := −τ − 2T if τ ∈ [−2T, −T ]. Using condition (H) and our choice of a, we see that we cannot have |y(τ )| = R. Thus, if B4T (0, R) denotes the open ball of center 0 and radius R in the space C4T (R, RN ) of 4T -periodic continuous functions with values in RN , the family of equations (14) with λ ∈ [0, 1[ has no solution y ∈ ∂B4T (0, R). It follows from Leray-Schauder theorem that equation y ′′ + C(t)y ′ − X(t, y) = 0, admits a 4T-periodic solution yT such that |yT (t)| ≤ R for all t ∈ R. For such a solution we have |yT′′ (t) + C(t)yT′ (t)| ≤ |X(t, yT (t))| ≤ R1 , kCk∞ ≤ kbI + Bk∞ ≤ R2 .

Using Lemma 3, we obtain a constant R′ > 0 such that kyT′ k∞ ≤ R′ . Now, by the construction of C and X, it is clear that yT is a solution of (13) on [−T, T ]. We can now use a standard diagonal argument based upon Ascoli-Arzela theorem (Krasnosel’skii’s lemma; see e.g. [23]) to extract from the family {yT } a solution y of (13) such that |y(t)| ≤ R and |y ′ (t)| ≤ R′ for all t ∈ R. 7

An easy consequence of Theorem 1 is a slight generalization of a result of Opial [25] (see [19] for a generalization to elliptic problems), dealing with the existence of solutions y ∈ BC 1 of the scalar differential equation y ′′ + b(t)y ′ = g(t, y)

(15)

where b ∈ BC(R, R) and g : R×R → R is continuous and bounded on R×[−r, r] for each r > 0. Corollary 1 If there exist r− < r+ such that g(t, r− ) ≤ 0 ≤ g(t, r+ ),

(16)

for all t ∈ R, then equation (15) has at least one solution y ∈ BC 1 such that r− ≤ y(t) ≤ r+ for all t ∈ R. Proof. Setting r=

r+ + r− , 2

R=

r+ − r− , 2

f (t, x) = g(t, r + x),

and making the change of unknown y(t) = r + x(t) in (15), we obtain the equivalent equation x′′ + b(t)x′ = f (t, x),

(17)

which is such that f (t, −R) = g(t, r − R) = g(t, r− ) ≤ 0,

f (t, R) = g(t, r + R) = g(t, r+ ) ≥ 0,

for all t ∈ R, which is equivalent to condition (H) when N = 1. We now introduce a sharpening of Condition (H) which insures the uniqueness of the bounded solution given by Theorem 1. Condition (M): There exists c∗ > 0 such that for all t ∈ R, and all y, z ∈ RN , one has 1 F (t, y) − F (t, z)) − B(t)B ∗ (t)(y − z) · (y − z) ≥ c∗ |y − z|2 . 4 Condition (M) is in particular satisfied if (F (t, y) − F (t, z)) · (y − z) ≥ c|y − z|2 for all t ∈ R, y, z ∈ RN and some c >

kBk2∞ 4 .

8

Theorem 2 Under Condition (M ), there exists a unique y ∈ C 2 (R, RN ) ∩ BC 1 (R, RN ) which is a solution of equation (13) on R. Moreover we have y ∈ BC 2 (R, RN ). Proof. Condition (M) implies that, for all t ∈ R, and for all y ∈ RN , one has 1 F (t, y) − F (t, 0)) − B(t)B ∗ (t)y · y ≥ c∗ |y|2 , 4 and hence

1 F (t, y) − B(t)B ∗ (t)y · y ≥ c∗ |y|2 − kF (·, 0)k∞ |y| ≥ 0, 4

∞ whenever |y| ≥ R := kF (·,0)k . By using Theorem 1, we can assert that there c∗ 2 N 1 exists y ∈ C (R, R ) ∩ BC (R, RN ), solution of the equation (13) on R, and the differential equation implies that y ∈ BC 2 (R, RN ). To prove the uniqueness, we consider z ∈ C 2 (R, RN ) ∩ BC 0 (R, RN ) a solution of equation (13) on R, and we set r(t) := 12 |y(t)−z(t)|2 . Since y, z ∈ BC 2 (R, RN ), the function r ∈ BC 2 (R, R). Easy estimates and Lemma 4 imply that supt∈R r(t) ≤ 0. Hence r(t) = 0 for all t ∈ R.

4

Almost periodic solutions of Duffing’s type systems

We now consider the case of system (13) when F is almost periodic with respect to t in the following sense. We say that f : R × K → Rp , where K ⊂ Rn is compact, is almost periodic, uniformly with respect to x ∈ K, if it satisfies one of the two following conditions: 1. For each real sequence (tk )k∈N , the sequence (f (· + tk , ·))k∈N contains a sequence which converges uniformly over R × K (Bochner). 2. For each ǫ > 0, there exists l > 0 such that each interval in R of length l contains at least a point τ with the property that, for each (t, x) ∈ R × K, one has kf (t + τ, x) − f (t, x)k < ǫ (Bohr). The case of a function f (t) corresponds to replacing R × K by R, and the above conditions are equivalent to assuming that f (·, x) is almost periodic for each x and f (t, ·) is continuous in x on K uniformly in t ∈ R (see [18]). When b, B F are almost periodic, and system (13) admits a bounded solution y, it is natural to raise the question of its almost periodicity. This difficult problem can be attacked through the following special case of a theorem of Amerio [1], which can be found in [18]. It requires the introduction of the concept of hull of an almost periodic vector field: let K ⊂ Rn be compact and let f : R × K → Rn be almost periodic uniformly with respect to x ∈ K; we say that g : R × K → Rn belongs to the hull H(f ) of f if one can find a sequence (tk )k∈N such that (f (· + tk , ·))k∈N converges uniformly over R × K to g. 9

Lemma 6 If, for each g ∈ H(f ), equation x′ = g(t, x) has a unique solution defined over R and with values in K, then this solution is almost periodic. Lemma 6 reduces to existence of an almost periodic solution for an almost periodic system to the existence and uniqueness of a suitable class of bounded solutions. A consequence of Lemma 6 and Theorem 2 is the following existence and uniqueness result for the almost periodic solutions of equation (13). Theorem 3 Assume, in addition to the conditions of Theorem 2, that b and B are almost periodic, and that F is uniformly almost periodic on compact x-sets. Then system (13) has a unique almost periodic solution. Proof. It essentially consists in verifying, using Bochner’s characterization of almost periodicity, that any equation in the hull of (13) has the same structure as (13) and satisfies condition (M). See [14] for the details. The following example show that Condition (M) is sharp with respect to the uniqueness of the almost periodic solution of (13). Consider the following linear system, related to gyroscopic stabilization, y ′′ + By ′ − y = 0, where y = col(y1 , y2 ) and, for some ω > 0, 0 ω2 B= . −ω 2 0 The conditions for the existence of a unique almost periodic (here the solution ω4 trivial one) given by Theorem 3 are fulfilled if the matrix 1 − 4 I is positive √ definite, i.e. if ω < 2. Now the characteristic equation associated to y ′′ +By ′ − y = 0 is λ4 + (ω 4 − 2)λ2 + 1 = 0, and y ′′ + By ′ − y = 0 has nontrivial almost periodic solutions if and only if the corresponding quadratic equation µ2 + (ω 4 − 2)µ + 1 = 0, has nonpositive roots, which is the case if and only if ω 2 ≥

10

√ 2.

We conclude this section with a few remarks concerning the comparison of our results and those of Berger-Chen [8], which correspond to the special case of (13) where b = 0, B = 0, and F (t, y) = Ay + ∇U (y) + h(t).

(18)

1. Theorem 3 adds the possibility to have some linear dissipation, and in particular to add an isotropic dissipation (of the form b(t)I) without modifying the assumption on the nonlinear term. 2. Even for the equation (12), Theorem 3 provides an improvement on Theorem 4 and Theorem 6 of Berger and Chen. In [8], A is a symmetric positive definite matrix, and U ∈ C 2 (RN , R) is such that U ′′ (x) is semi positive definite. Therefore there exists c∗ > 0 (the smallest eigenvalue of A) such that Aξ · ξ ≥ c∗ |ξ|2 , for every ξ ∈ RN . Furthermore, U necessarily is convex, and therefore its gradient ∇U is monotone. Consequently, for every x, y ∈ RN , we have, for the F given by (18) (F (t, y) − F (t, z)) · (y − z) = A(y − z) · (y − z) + (∇U (y) − ∇U (z)) · (y − z) ≥ c∗ |y − z|2 . In [8], the authors use the following additional condition: ∃M > 0, ∀x = (x1 , ..., xN ), y = (y1 , ..., yN ) ∈ RN , (∀i = 1, ..., N, M ≤ |xi | ≤ |yi |) =⇒ (U (x) ≤ U (y)), which is not needed to prove Theorem 3. Notice that a recent paper of Carminati [15] shows, by a better analysis, that this growth condition is also superfluous in the Berger-Chen’s type variational argument.

5

Bounded solutions of the forced pendulum equation

Let us now consider the forced pendulum equation y ′′ + cy ′ + a sin y = h(t),

(19)

where h : R → R is continuous, and where, without loss of generality, we can suppose that c ≥ 0 and a > 0. A first direct application of Corollary 1 to equation (19) provides the following existence theorem. Theorem 4 If c ≥ 0 and if h ∈ C is such that, for all t ∈ R, −a ≤ h(t) ≤ a,

11

(20)

then equation (19) has at least one solution y ∈ BC 1 such that π 3π ≤ y(t) ≤ 2 2

for all t ∈ R. Proof. Take r− =

π 2

and r+ =

3π 2

in Corollary 1.

If we restrict slightly condition (20), we obtain a uniqueness result playing an important role in the study of almost periodic solutions. Theorem 5 If c ≥ 0 and if h ∈ C is such that khk∞ < a,

(21)

then there exists ǫ > 0 such that equation (19) has one and only one solution y ∈ BC 1 satisfying, for all t ∈ R, π 3π + ǫ ≤ y(t) ≤ − ǫ. 2 2

(22)

Proof. By (21), we can find ǫ > 0 small enough so that π 3π + ǫ ≥ khk∞ , a sin − ǫ ≤ −khk∞ . a sin 2 2 Hence, r− = π2 + ǫ and r+ = 3π 2 − ǫ verify the conditions of Corollary 1 and the existence conclusion follows. On the other hand, there exists δ > 0 such that cos u ≤ −δ for all u ∈ [r− , r+ ]. Consequently, if y1 and y2 are two solutions in BC 1 of equation (19), one has, letting z = y1 − y2 , Z 1 ′′ ′ z + cz + a cos [y2 (t) + s(y1 (t) − y2 (t))] ds z = 0. 0

1

Thus z ∈ BC is solution of a linear equation of type (11) with Z 1 f (t) = a cos [y2 (t) + s(y1 (t) − y2 (t))] ds ≤ −aδ < 0. 0

It follows from Lemma 5 that z = 0. For extensions of those results to elliptic equations, see [19]. Condition (20) requires that h ∈ BC has a sufficiently small norm, independently of the value of c. When c > 0, one can prove other existence conditions with allow unbounded forcing terms h, and show that the nonlinearity may allow, with respect to the linear case considered in Lemma 2, forcing terms which are not necessarily in BP. Those conditions were first given in [24]. We use the notations of Section 2. 12

Theorem 6 If c > 0, h = h∗ + h∗∗ ∈ BP + BC is such that oscR Hc∗ ≤ π,

(23)

where Hc∗ is the unique solution in BC 1 of equation y ′′ + cy ′ = h∗ (t),

(24)

and if ∗∗

kh k∞ ≤ a cos

oscR Hc∗ 2

,

(25)

then equation (19) has at least one solution y ∈ BC 1 such that, for all t ∈ R, π 3π + Hc∗ (t) ≤ y(t) ≤ + Hc∗ (t). 2 2

(26)

Proof. By Lemma 2, Hc∗ ∈ BC 1 exists and is such that oscR Hc∗ ≤ where H ∗ (t) =

Rt 0

oscR H ∗ , c

(27)

h∗ (s) ds, and sup Hc∗ = − inf Hc∗ = R

R

1 oscR Hc∗ . 2

(28)

If we set y(t) = z(t) + Hc∗ (t), then y ∈ BC 1 is a solution of (19) if and only if z ∈ BC 1 is a solution of equation z ′′ + cz ′ + a sin (z + Hc∗ (t)) = h∗∗ (t).

(29)

Taking again r− = π2 , r+ = 3π 2 , the conditions of Corollary 1 applied to (29) become h∗∗ (t) − a cos Hc∗ (t) ≤ 0 ≤ h∗∗ (t) + a cos Hc∗ (t),

i.e.

−a cos Hc∗ (t) ≤ h∗∗ (t) ≤ a cos Hc∗ (t),

for all t ∈ R. Such a condition is only possible when (23) holds, in which case it is equivalent to (25). Remark 1 Because of inequality (8), we see that if h = h∗ + h∗∗ ∈ BP + BC is given, then inequality (23) holds for each c≥

oscR H ∗ . π

Thus equation (19) with h ∈ BP always has a solution y ∈ BC 1 when c is sufficiently large. 13

Example 1 Equation y ′′ + cy ′ + a sin y = 2t cos t2 , with c > 0 et a > 0 has a bounded solution when c ≥

(30) 2 π.

The uniqueness of the bounded solution given by Theorem 6 is important for the study of almost periodic solutions. It requires a condition which is substantially more restrictive than (23). Theorem 7 If c > 0, h = h∗ + h∗∗ ∈ BP + BC is such that oscR Hc∗
0,.

Proof. It is identical to that of Theorem 6 till the obtention of the equivalent 5π problem (29). One then takes r− = 3π 4 , r+ = 4 in Corollary 1, applied to (29). 1 The existence of a solution y ∈ BC satisfying (26) follows, and hence, by (31), there exists ǫ > 0 such that (22) holds. The uniqueness follows like in the proof of Theorem 6.

With similar proofs, one can obtain an existence result when c = 0, if H 1 1 denotes the second primitive of h such that supR H 1 = − inf R H 1 = osc2R H . Theorem 8 If c = 0 and if h ∈ BP 1 is such that oscR H 1 ≤ π,

(33)

equation (19) has at least one solution y ∈ BC 1 satisfying inequality 3π π + H 1 (t) ≤ y(t) ≤ + H 1 (t). 2 2 If oscR H 1
0 such that equation (19) has one and only one solution y ∈ AP 1 satisfying inequality (22). Proof. Again the application of Theorem 5 and Lemma 6 only requires the verification that any equation in the hull of (19) has the same structure as (19) and satisfies the conditions of Theorem 5. See [24] for details. Similarly, Lemma 6 and Theorem 7 provide another existence condition for an almost periodic solution when c > 0. Theorem 10 If c > 0, h = h∗ +h∗∗ ∈ AP P +AP and satisfies inequalities (31) and (32), then equation (19) has one and only one solution y ∈ AP 1 satisfying inequality (22). Proof. It follows the same lines as that of Theorem 9. See [24] for details.

Example 2 Equation √ √ √ y ′′ + cy ′ + a sin y = cos t sin 2t + 2 sin t cos 2t has an almost periodic solution for each c >

4 π.

Indeed,

√ √ √ √ cos t sin 2t + 2 sin t cos 2t = (sin t sin 2t)′ , √ and oscR sin(·) sin 2(·) ≤ 2.

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When c = 0, Theorem 8 provides similarly an existence condition which generalizes a result of Blot [13], who proved uniqueness for each h satisfying (34) and existence only for a dense subset of those forcing terms h. Theorem 11 If c = 0 and h ∈ C has an almost periodic second primitive H 1 satisfying (34), then there exists ǫ > 0 such that equation (19) has one and only one solution y ∈ AP 1 satisfying (22). Acknowledgement. The authors thanks the referee for his/her careful reading of the manuscript leading to various improvements.

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