Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2008, Article ID 153285, 13 pages
http://dx.doi.org/10.1155/2008/153285
Research Article

Variational Methods for Almost Periodic Solutions of a Class of Neutral Delay Equations

Laboratoire Marin Mersenne, Université Paris 1 Panthéon-Sorbonne, Centre P.M.F., 90 rue de Tolbiac, Paris Cedex 13 75634, France

Received 14 June 2007; Accepted 27 November 2007

Academic Editor: H. Bevan Thompson

Copyright © 2008 M. Ayachi and J. Blot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide new variational settings to study the a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend Shu and Xu (2006) variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions, results of existence of a.p. solutions, and also a density result for the forced equations.

1. Introduction

The aim of this paper is the study of a.p. (almost periodic) solutions of neutral delay equations of the following form:𝐷1𝐿𝑥(𝑡𝑟),𝑥(𝑡2𝑟),𝑥(𝑡𝑟),𝑥(𝑡2𝑟),𝑡𝑟+𝐷2𝐿𝑥(𝑡),𝑥(𝑡𝑟),𝑥(𝑡),𝑥=𝑑(𝑡𝑟),𝑡𝐷𝑑𝑡3𝐿𝑥(𝑡𝑟),𝑥(𝑡2𝑟),𝑥(𝑡𝑟),𝑥(𝑡2𝑟),𝑡𝑟+𝐷4𝐿𝑥(𝑡),𝑥(𝑡𝑟),𝑥(𝑡),𝑥,(𝑡𝑟),𝑡(1.1) where 𝐿(𝑛)4× is a differentiable function, 𝐷𝑗 denotes the partial differential with respect to the 𝑗th vector variable, and 𝑟(0,) is fixed. We will consider the almost periodicity in the sense of Corduneau [1], and in the sense of Besicovitch [2].

A special case of (1.1) is the following forced neutral delay equation:𝐷1𝐾𝑥(𝑡𝑟),𝑥(𝑡2𝑟),𝑥(𝑡𝑟),𝑥(𝑡2𝑟)+𝐷2𝐾𝑥(𝑡),𝑥(𝑡𝑟),𝑥(𝑡),𝑥𝑑(𝑡𝑟)𝐷𝑑𝑡3𝐾𝑥(𝑡𝑟),𝑥(𝑡2𝑟),𝑥(𝑡𝑟),𝑥(𝑡2𝑟)+𝐷4𝐾𝑥(𝑡),𝑥(𝑡𝑟),𝑥(𝑡),𝑥(𝑡𝑟)=𝑏(𝑡),(1.2) where 𝐾(𝑛)4 is a differentiable function, and 𝑏𝑛 is an a.p. forcing term. To see (1.2) as a special case of (1.1), it suffices to take𝐿𝑥1,𝑥2,𝑥3,𝑥4𝑥,𝑡=𝐾1,𝑥2,𝑥3,𝑥4𝑥1𝑏(𝑡+𝑟),(1.3) where the point denotes the usual inner product in 𝑛.

Another special case of (1.1) is the following forced second-order neutral delay equation:𝑥(𝑡𝑟)+𝐷1𝐹𝑥(𝑡𝑟),𝑥(𝑡2𝑟)+𝐷2𝐹𝑥(𝑡),𝑥(𝑡𝑟)=𝑏(𝑡),(1.4) where 𝑏𝑛 and 𝐹(𝑛)2. To see that this last equation is a special case of (1.1), it suffices to take 𝐿(𝑥1,𝑥2,𝑥3,𝑥4,𝑡)=(1/2)𝑥32𝐹(𝑥1,𝑥2)+𝑥1𝑏(𝑡+𝑟), where the norm is the usual Euclidian norm of 𝑛. In their work [3], Shu and Xu study the periodic solutions of this last equation by using a variational method. We want to extend such a view point to the study of the a.p. solutions.

And so our approach to the study of the a.p. solutions of (1.1) consists to search critical points of a functional Φ defined on suitable Banach spaces of a.p. functions byΦ(𝑥)=lim𝑇+12𝑇𝑇𝑇𝐿𝑥(𝑡),𝑥(𝑡𝑟),𝑥(𝑡),𝑥(𝑡𝑟),𝑡𝑑𝑡.(1.5) At this time, we give some historical elements. Recall that the work [4] of Elsgolc treats the calculus of variations with a retarded argument on a bounded real interval. This work was followed by these ones of Hughes [5] and Sabbagh [6]. Since the variational problems can be seen as optimal control problems, recall also the existence of the theory of the periodic optimal control with retarded argument as developed by Colonius [7]. For instance, we consider a periodic optimal control problem with a criterion of the form (1/𝑇)𝑇0𝑔(𝑥(𝑡),𝑢(𝑡),𝑡)𝑑𝑡 and with an equation of motion of the form 𝑥(𝑡)=𝑓(𝑥(𝑡),𝑥(𝑡𝑟),𝑢(𝑡),𝑡), where 𝑥(𝑡) is the state variable and 𝑢(𝑡) the control variable. In the special case, where 𝑓(𝑥(𝑡),𝑥(𝑡𝑟),𝑢(𝑡),𝑡)=𝑓1(𝑥(𝑡),𝑥(𝑡𝑟),𝑡)+𝑢(𝑡), the previous optimal control problem can be transformed into a calculus of variations problem with the criterion (1/𝑇)𝑇0𝑔(𝑥(𝑡),𝑓1(𝑥(𝑡),𝑥(𝑡𝑟),𝑡)𝑥(𝑡),𝑡)𝑑𝑡, which is a special case of (1.5). Note that the Euler-Lagrange equation of such a variational problem is a special case of (1.1).

On another hand, calculus of variations in mean time was developed to study the a.p. solutions of some (nonretarded) differential equations [813]. Here, we extend this approach to treat equation like (1.1).

Now we describe the contents of this paper. In Section 2, we precise the notations about the function spaces used later. In Section 3, we establish a variational formalism suitable to the Bohr a.p. solutions; we give a variational principle and a result on the structure of the set of the a.p. solutions of (1.1) in the convex case. In Section 4, we establish a variational formalism suitable to the Besicovitch a.p. solutions, we give a variational principle, results of existence, and a result of density for the a.p. forced equations.

2. Notations

AP0(𝑛) is the space of the Bohr almost periodic (Bohr a.p.) functions from in 𝑛; endowed with the supremum , it is a Banach space [1].

AP1(𝑛)={𝑥𝒞1(,𝑛)AP0(𝑛)𝑥AP0(𝑛)}; endowed with the norm 𝑥𝒞1=𝑥+𝑥, it is a Banach space.

When 𝑘{}, AP𝑘(𝑛)={𝑥𝒞𝑘(,𝑛)𝑗𝑘,𝑑𝑗𝑥/𝑑𝑡𝑗AP0(𝑛)}.

When 𝑥AP0(𝑛), its mean value is𝑥(𝑡)𝑡=lim𝑇+12𝑇𝑇𝑇𝑥(𝑡)𝑑𝑡existsin𝑛.(2.1)

The Fourier-Bohr coefficients of 𝑥AP0(𝑛) are the complex vectors𝑎(𝑥;𝜆)=lim𝑇+12𝑇𝑇𝑇𝑒𝑖𝜆𝑡𝑥(𝑡)𝑑𝑡(2.2) and Λ(𝑥)={𝜆𝑎(𝑥,𝜆)0}.

When 𝑝[1,),𝐵𝑝(𝑛) is the completion of AP0(𝑛) (in 𝐿𝑝loc(,𝑛)) with respect to the norm 𝑢𝑝={|𝑢|𝑝}1/𝑝. When 𝑝=2, 𝐵2(𝑛) is a Hilbert spaces and its norm 2 is associated to the inner product (𝑢𝑣)={𝑢𝑣} [2]. The elements of these spaces 𝐵𝑝(𝑛) are called Besicovitch almost periodic (Besicovitch a.p.) functions.

Recall the useful following fact: if (𝑢𝑚)𝑚 is a sequence in AP0(𝑛) and if 𝑢𝐿𝑝loc(,𝑛) (Lebesgue space), which satisfy𝑢𝑚𝑢𝑝1/𝑝=(limsup𝑇12𝑇𝑇𝑇𝑢𝑚𝑢𝑝𝑑𝑡)1/𝑝0(𝑚0)(2.3) then 𝑢𝐵𝑝(𝑛) and we have 𝑢𝑚𝑢𝑝0(𝑚0).

We use the generalized derivative 𝑢𝐵2(𝑛) of 𝑢𝐵2(𝑛) (when it exists) defined by 𝑢(1/𝑠)(𝑢(+𝑠)𝑢)20(𝑠0), and we define 𝐵1,2(𝑛)={𝑢𝐵2(𝑛)𝑢𝐵2(𝑛)}; endowed with the inner product 𝑢𝑣=(𝑢𝑣)+(𝑢𝑣), 𝐵1,2(𝑛) is a Hilbert space [11, 13].

If 𝐸 and 𝐹 are two finite-dimensional-normed spaces, APU(𝐸×,𝐹) stands for the space of the functions 𝑓𝐸×𝐹,(𝑥,𝑡)𝑓(𝑥,𝑡), which are almost periodic in 𝑡 uniformly with respect to 𝑥 in the classical sense given in [14].

To make the writing less heavy, we sometimes use the notations𝑢(𝑡)=𝑢(𝑡),𝑢(𝑡𝑟),𝑢(𝑡),𝑢(𝑡𝑟),(2.4) when 𝑢𝐵1,2(𝑛), and𝑥(𝑡)=𝑥(𝑡),𝑥(𝑡𝑟),𝑥(𝑡),𝑥(𝑡𝑟),(2.5) when 𝑥AP1(𝑛).

3. A Variational Setting for the Bohr a.p. Functions

We consider the following condition:𝐿APU(𝑛)4×,,forall(𝑋,𝑡)𝑛4×,thepartialdierential𝐷𝑋𝐷𝐿(𝑋,𝑡)exists,and𝑋𝐿APU𝑛4×,𝑛4.,(3.1)

Lemma 3.1. Under [9] the functional ΦAP1(𝑛) defined by (1.5) which is of class 𝒞1, and for all 𝑥,AP1(𝑛), then 𝐷𝐷Φ(𝑥)=1𝐿𝑥(𝑡),𝑡(𝑡)+𝐷2𝐿𝑥(𝑡),𝑡(𝑡𝑟)+𝐷3𝐿𝑥(𝑡),𝑡(𝑡)+𝐷4𝐿𝑥(𝑡),𝑡(𝑡𝑟)𝑡.(3.2)

Proof. We introduce the linear operator 𝒯AP1(𝑛)(AP0(𝑛))4 by setting 𝒯(𝑥)(𝑡)=𝑥(𝑡). The four components of 𝒯 are continuous linear operators that imply the continuity of 𝒯, and therefore 𝒯 is of class 𝒞1, and for all 𝑥,AP1(𝑛) we have 𝐷𝒯(𝑥)=𝒯().
Under (3.1), the Nemytski operator 𝒩𝐿(AP0(𝑛))4AP0(𝑛), defined by 𝒩𝐿(𝑋)(𝑡)=𝐿(𝑋(𝑡),𝑡), is of class 𝒞1 (cf. [15, Lemma 7]) and we have, for all 𝑋,𝐻AP0(𝑛)4, (𝐷𝒩𝐿(𝑋)𝐻)(𝑡)=𝐷𝑋𝐿(𝑋(𝑡),𝑡)𝐻(𝑡).
The linear functional AP0(𝑛) is continuous, therefore it is of class 𝒞1 and we have, for all 𝜙,𝜓AP0(𝑛), 𝐷{𝜙}𝜓={𝜓}.
And so Φ=𝒩𝐿𝒯 is of class 𝒞1. Furthermore, we have𝒩𝐷Φ(𝑥)=𝐷𝐿𝒩𝒯(𝑥)𝐿𝒯(𝑥)𝒯(𝑥)=𝐷𝒩𝐿𝐷𝒯(𝑥)𝒯()=𝑋𝐿𝑥(𝑡),𝑡(𝑡)𝑡(3.3) and expressing 𝐷𝑋𝐿 in terms of 𝐷𝑗𝐿, we obtain the announced formula.

Note that in the case without delay, when 𝐿 is autonomous, that is, 𝐿(𝑋,𝑡)=𝐿(𝑋), in [9], it is established that the functional 𝑥{𝐿(𝑥,𝑥)} is of class 𝒞1 when 𝐿 is of class 𝒞1. In [16], we can find a proof of the differentiability of the Nemytski operator on AP0(𝑛) which is different to this one of [9].

Theorem 3.2 (variational principle). Under (3.1), for 𝑥AP1(𝑛), the following two assertions are equivalent.
(i)𝐷Φ(𝑥)=0, that is, 𝑥 is a critical point of Φ in AP1(𝑛).(ii)𝑥 is a Bohr a.p. solution of (1.1).

Proof. First we assume (i). Since the mean value is translation invariant, we have𝐷2𝐿𝑥(𝑡),𝑡(𝑡𝑟)𝑡𝐷=2𝐿𝑥(𝑡+𝑟,𝑡+𝑟(𝑡)𝑡,𝐷4𝐿𝑥(𝑡),𝑡(𝑡𝑟)𝑡𝐷=4𝐿𝑥(𝑡+𝑟,𝑡+𝑟(𝑡)𝑡,(3.4) and so by using Lemma 3.1 we obtain, for all AP1(𝑛),𝐷0=1𝐿(𝑥(𝑡),𝑡+𝐷2𝐿𝑥(𝑡+𝑟),𝑡+𝑟)(𝑡)𝑡𝐷+3𝐿(𝑥(𝑡),𝑡+𝐷4𝐿𝑥(𝑡+𝑟),𝑡+𝑟)(𝑡)𝑡.(3.5)
Setting 𝑞(𝑡)=𝐷1𝐿(𝑥(𝑡),𝑡)+𝐷2𝐿(𝑥(𝑡+𝑟),𝑡+𝑟), denoting by 𝑞𝑘(𝑡) its coordinates for 𝑘=1,,𝑛, setting 𝑝(𝑡)=𝐷3𝐿(𝑥(𝑡),𝑡)+𝐷4𝐿(𝑥(𝑡+𝑟),𝑡+𝑟), and denoting by 𝑞𝑘(𝑡) its coordinates for 𝑘=1,,𝑛, we deduce from the previous equality that, for all 𝜙AP(), we have {𝑞𝑘(𝑡)𝜙(𝑡)}𝑡={𝑝𝑘(𝑡)𝜙(𝑡)}𝑡. Then by reasoning like in the proof of Theorem 1 in [8], we obtain that 𝐷𝑝𝑘=𝑞𝑘 in the sense of the a.p. distributions of Schwartz [17], and by using the proposition of the Fourier-Bohr series, we obtain that 𝑝𝑘 is 1 and that 𝑝𝑘=𝑞𝑘 in the ordinary sense. From this, we obtain that 𝑝(𝑟) is 1 and that 𝑝(𝑡𝑟)=𝑞(𝑡𝑟) which is exactly (ii).
Conversely by using the formula {𝑙𝑦}={𝑙𝑦}, for all 𝑙AP1((𝑛,)) and , and by translating the time, we obtain from (ii), for all the following relation: and so we have (i).

Theorem 3.2 is an extension to the nonautonomous case in presence of a delay of [8, Theorem 1]. Now we use Theorem 3.2 to provide some results on the structure of the set of the Bohr a.p. solutions of (1.1) in the case, where is autonomous and convex.

Theorem 3.3 (structure result). Assume that , and that is convex. Then the following assertions hold.
(i) The set of the Bohr a.p. solutions of (1.1) is a closed convex subset of .(ii) If is a -periodic nonconstant solution of (1.1), if is a -periodic nonconstant solution of (1.1), and if is no rational, then is a Bohr a.p. nonperiodic solution of (1.1), for all .(iii) If is a Bohr a.p. solution of (1.1), then is a constant solution of (1.1).(iv) If is a Bohr a.p. solution of (1.1), if is such that , then there exists a nonconstant -periodic solution of (1.1).

Proof. Since is convex, the functional of (1.5) is also convex on . Since is autonomous and of class , satisfies (3.1), and so is of class . Therefore, we have which is closed and convex, and (i) becomes a consequence of Theorem 3.2. The assertion (ii) is a straightforward consequence of (i).
We introduce , when is a Bohr a.p. solution of (1.1), for all . By using a theorem of Besicovitch (see [2, page 144]), there exists a -periodic continuous function, denoted by , such that
We easily verify that
Since is autonomous, is a Bohr a.p. solution of (1.1). Since is a convex combination of Bohr a.p. solutions of (1.1), is a Bohr a.p. solution of (1.1), and also by using the closeness of the set of Bohr a.p. solutions. And so is a -periodic solution of (1.1). By using a straightforward calculation, we see that and consequently . When , then is not constant that proves (iv).
To prove (iii) it suffices to choose such that , and then all the Fourier-Bohr coefficients of are zero except (perhaps) the mean value of which is equal to .

The assertions (i) and (ii) are extensions of [8, Theorems 3 and 4]; the assertions (iii) and (iv) are extension to neutral delay equations of [10, Theorem 2].

The space does not possess good topological properties like to be a reflexive space. It is why in the following section we extend our variational formalism to the Hilbert space .

4. A Variational Setting for the Besicovitch a.p. Functions

and are Euclidean finite-dimensional spaces.

Lemma 4.1. Let be a function which satisfies the following Hölder condition: Let be such that .
Then the following two assertions hold.
(i)If then .(ii)The Nemytski operator on , defined by satisfies , for all .

Proof. We set , and so we have and the Hölder assumption implies for all . If then we have for all , and since is continuous, we have (the Lebesgue space), and since , we have Since , there exists a sequence in such that . By using [14, Theorem 2.7 page 16], setting , we have , and a straightforward calculation gives us the following inequality: and consequently we obtain that implies , and so (i) is proven; moreover the last inequality becomes the one of (ii) when we replace by .

This lemma is an extension to the nonautonomous case of [13, Theorem 1].

Lemma 4.2. Let be a function such that the partial differential exists, for all , such that . We assume the following condition fulfilled.
(C) There exist , such that, for all , and for all ,
Then the Nemytski operator , defined by , is of class and, for all

Proof. First step: we show that there exist , , such that, for all , . The following hold:
By using the mean value theorem (see [18, page 144]), we have, for all , Setting , then . Since , and , we have .
Second step: we show that when .
Let . Then the inequality implies that By using Lemma 4.1 with , and , we have . Let be a sequence in such that . By using the mean value theorem (see [18, page 144]), we have, for all , and consequently we obtain Since and since , we have By using (see [14, Theorem 2.7 page 16] ), we have and so, by setting we have . The last inequality implies and therefore we have .
Third step: we show that, for all , the operator , defined by , is linear continuous. We have yet seen that . The linearity of is easy to verify. By using a Cauchy-Schwartz-Bunyakovsky inequality, we have that proves the continuity of .
Fourth step: we show the differentiability of .
Let and . By using the mean value inequality (see [18, page 144]), we have, for all , and by using the monotonicity of , we obtain that is, that implies that is differentiable at and that .
Fifth step: we show that is of class .
Let . By using (C), for all , such that , for all , we have That implies, by using the Cauchy-Schwartz-Bunyakovsky inequality, the following majorization holds: Therefore we have that implies the continuity of .

Note that Lemma 4.2 is an extension to the nonautonomous case of [13, Theorem 2].

Theorem 4.3 (variational principle). Let , be a function and let . Assume the following conditions fulfilled: Then the functional , defined by is of class , and the two following assertions are equivalent.
(i), that is, is a critical point of .(ii)

Definition 4.4. When satisfies the equation of (ii) in Theorem 4.3, we say that is a weak Besicovitch a.p. solution of (1.1).

Proof. We consider the operator , defined by . is clearly linear continuous, therefore is of class and we have .
We consider the Nemytski operator By using Lemma 4.2, is of class and, for all we have The mean value is linear continuous, therefore it is of class , and , for all .
Consequently is of class as a composition of three mappings of class .
Let . If (i) is true then, for all , we have and then we obtain (ii) by using [13, Proposition 10].
Conversely, if (ii) is true, then , and for all we have therefore by using [13, Proposition 9], we obtain Since is dense in , we have , for all , therefore .

Note that the Theorem 4.3 is an extension to the nonautonomous case of [13, Theorem 4].

Theorem 4.5 (existence, uniqueness). Let be a function which satisfies (4.21) and also satisfies the following two conditions: Then there exists a function which is a weak Besicovitch a.p. solution of (1.1).
Moreover, if in addition the following condition fulfilled: then the weak Besicovitch a.p. solution of (1.1) is unique.

Proof. By using Theorem 4.3, the functional is of class and, by using (4.28), is a convex functional. Assumption (4.29) ensures that, for all , we have Since the mean value is translation invariant, consequently is coercive on , and so (see [19, page 46]) there exists such that . Therefore we have and by using Theorem 4.3, is a weak Besicovitch a.p. solution of (1.1). The existence is proven.
To treat the uniqueness, we note that, under (4.30), the functional , defined by , is convex and since is of class , is also of class . Note that we have . By using the Mint monotonicity of the differential of a convex functional, for all we have Now if and are two weak Besicovitch a.p. solutions of (1.1), by using Theorem 4.3 we have , and consequently , therefore .

Theorem 4.6 (existence and density). Let be a function which satisfies the following conditions: Then the following conclusions hold.
(i) For all , there exists a unique which is a weak Besicovitch a.p. solution of (1.2).(ii) The set of the for which there exists a Bohr a.p. solution of (1.2) is dense in with respect to the norm

Proof. We introduce the functionals and from in setting and . They are special cases of the functional of the Theorem 4.3, and consequently they are of class . Note that . By using the F. Riesz isomorphism , , for all , we can define the gradients and . By using the Minty-monotonicity of (due to the convexity of ) we have, for all , that implies that is strongly monotone and consequently (see [20, page 100]) the following property holds: From each we define the linear functional by setting
Therefore we have and by using (4.36), there exists such that , that is, which means that, for all , and by using [13, Proposition 10], we obtain that is a weak Besicovitch a.p. solution of (1.2).
About the uniqueness, note that if is a weak Besicovitch a.p. solution of (1.2), then we verify that , for all , and consequently , that is, , and by using (4.36), we have . And so (i) is proven.
Now we introduce the nonlinear unbounded operator defined by And so means that is a weak Besicovitch a.p. solution of (1.2). By using the assertion (i), is bijective. We verify that for all , and by using (4.36) we see that is a homeomorphism from on . Since is dense in , is dense in with respect to the norm , and since , we have proven (ii).

This result is an extension to the neutral delay equations of [13, Theorem 5].

Acknowledgment

The authors thank an anonymous referee for several remarks which have permitted them to improve Theorems 4.3 and 4.5.

References

  1. C. Corduneau, Almost Periodic Functions, Chelsea, New York, NY, USA, 2nd edition, 1989.
  2. A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, Cambridge, UK, 1932.
  3. X.-B. Shu and Y.-T. Xu, “Multiple periodic solutions for a class of second-order nonlinear neutral delay equations,” Abstract and Applied Analysis, vol. 2006, Article ID 10252, 9 pages, 2006. View at Publisher · View at Google Scholar
  4. L. È. Èl'sgol'c, Qualitative Methods in Mathematical Analysis, Translations of Mathematical Monographs, Vol. 12, American Mathematical Society, Providence, RI, USA, 1964.
  5. D. K. Hughes, “Variational and optimal control problems with delayed argument,” Journal of Optimization Theory and Applications, vol. 2, no. 1, pp. 1–14, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. L. D. Sabbagh, “Variational problems with lags,” Journal of Optimization Theory and Applications, vol. 3, pp. 34–51, 1969. View at Publisher · View at Google Scholar
  7. F. Colonius, Optimal Periodic Control, vol. 1313 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988.
  8. J. Blot, “Calculus of variations in mean and convex Lagrangians,” Journal of Mathematical Analysis and Applications, vol. 134, no. 2, pp. 312–321, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Blot, “Une approche variationnelle des orbites quasi-périodiques des systèmes hamiltoniens,” Annales des Sciences Mathématiques du Québec, vol. 13, no. 2, pp. 7–32, 1990. View at Google Scholar
  10. J. Blot, “Calculus of variations in mean and convex Lagrangians. II,” Bulletin of the Australian Mathematical Society, vol. 40, no. 3, pp. 457–463, 1989. View at Google Scholar
  11. J. Blot, “Une méthode hilbertienne pour les trajectoires presque-périodiques,” Comptes Rendus de l'Académie des Sciences. Série I, vol. 313, no. 8, pp. 487–490, 1991. View at Google Scholar
  12. J. Blot, “Almost-periodic solutions of forced second order Hamiltonian systems,” Annales de la Faculté des Sciences de Toulouse Mathématiques, vol. 12, no. 3, pp. 351–363, 1991. View at Google Scholar
  13. J. Blot, “Oscillations presque-périodiques forcées d'équations d'Euler-Lagrange,” Bulletin de la Société Mathématique de France, vol. 122, no. 2, pp. 285–304, 1994. View at Google Scholar
  14. T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol. 14 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1975.
  15. J. Blot, P. Cieutat, and J. Mawhin, “Almost-periodic oscillations of monotone second-order systems,” Advances in Differential Equations, vol. 2, no. 5, pp. 693–714, 1997. View at Google Scholar
  16. P. Cieutat, Solutions presque-périodiques d'équations d'évolution et de systèmes nonlinéaires, Doctorat Thesis, Université Paris 1 Panthéon-Sorbonne, Paris, France, 1996.
  17. L. Schwartz, Théorie des Distributions, Hermann, Paris, France, 1966.
  18. V. Alexéev, V. Tikhomirov, and S. Fomine, Commande Optimale, Mir, Moscow, Russia, French edition, 1982.
  19. H. Brezis, Analyse Fonctionnelle, Théorie et applications, Masson, Paris, France, 1983.
  20. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.