Near an equilibrium we study the existence of asymptotically a.p. (almost periodic), asymptotically a.a. (almost automorphic), pseudo a.p., pseudo a.a., weighed pseudo a.p. and weighed pseudo a.a. solutions of Liénard differential equations in the form , where the forcing term possesses a similar nature, and where p is a parameter in a Banach space. We use a perturbation method around an equilibrium. We also study two special cases of the previous family of equations that are and .
MSC: 34C27, 34C99, 47J07.
Keywords:asymptotically almost periodic functions; asymptotically almost automorphic functions; pseudo almost periodic functions; pseudo almost automorphic functions; weighted pseudo almost periodic functions; weighted pseudo almost automorphic functions; Liénard equations
We consider the following family of forced Liénard equations:
When is almost periodic in the Bohr sense (respectively almost automorphic), in , Theorem 3.1 (respectively Theorem 3.2), we have proven the existence of an almost periodic (respectively almost automorphic) solution near an equilibrium, by using the perturbation method in the setting of Nonlinear Functional Analysis.
In the present paper, we extend this result to the frameworks of asymptotically almost periodic, asymptotically almost automorphic, pseudo almost periodic, pseudo almost automorphic, weighted pseudo almost periodic and weighted pseudo almost automorphic functions.
Martínez-Amores and Torres in , then Campos and Torres in  described the dynamics of equation () in the periodic case, namely the forcing term e is periodic. Then Cieutat extended these results to the almost periodic case in , Ait Dads, Cieutat and Lhachimi did to the pseudo almost periodic case in , and Cieutat, Fatajou and N’Guérékata did to the almost automorphic case in .
For almost periodic solutions of second-order differential systems, results on the differentiable dependence were established in  by Blot, Cieutat and Mawhin.
Our approach of the problem is to transform it into a nonlinear equation with parameters in Banach function spaces, and to apply the implicit function theorem of the differential calculus in Banach spaces. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. Then we establish, for a linear differential equation in a Banach space, a result on the existence and uniqueness of the solutions described above. We also extend the well-know result on the almost periodicity of the derivative of an almost periodic function to the weighted pseudo almost periodic and weighted pseudo almost automorphic cases.
Now we give a brief description of the contents of the paper. In Section 2, we fix our notation and we recall some definitions. In Section 3, we establish some results on the differentiation of weighted pseudo almost periodic functions and weighted pseudo almost automorphic functions, then on a linear differential equation and on the Nemytskii operators. In Section 4, we state the main theorem (Theorem 4.1) and we give the proof of this theorem. In Section 5, we establish two corollaries (Corollary 5.1 and Corollary 5.2) of the two special cases of ().
When X and Y are Banach spaces, stands for the space of all bounded linear operators from X into Y, (respectively , respectively ) stands for the space of continuous (respectively Fréchet continuously differentiable, respectively twice Fréchet continuously differentiable) functions from X into Y.
A function is called an almost periodic function (in the Bohr sense) when it satisfies the following criterion (due to Bochner): is relatively compact in . denotes the space of almost periodic functions from ℝ into X. Endowed with the norm , is a Banach space which is invariant by translation , that is to mean that for all , when .
A function is called an almost automorphic function (in the Bochner sense) when, for all real sequence , there exists a subsequence of such that for all , exists in X, and for all , exists. denotes the space of almost automorphic functions from ℝ into X. Endowed with the norm , is a Banach space which is invariant by translation .
We also consider the following other function spaces which one can find in :
• the space of asymptotically almost periodic functions .
• the space of asymptotically almost automorphic functions .
We define the following spaces:
• , where (H) is the following condition due to :
• the space of weighted pseudo almost automorphic functions .
Let μ be a positive measure on ( is a Lebesgue σ-field on ℝ) satisfying and for all . A function u is called μ-pseudo almost periodic (respectively μ-pseudo almost automorphic) if , where (respectively ) and ϕ is a function satisfying
The space (respectively ) is a special case of the space (respectively ) in the following sense: (respectively ), where the measure μ is absolutely continuous with respect to the Lebesgue measure and its Radon-Nikodym derivative is . The function ρ satisfies hypothesis (H) if and only if the measure μ satisfies condition (C), cf. Remark 3.4 in  (respectively Remark 3.1 in ). Consequently, when , the spaces and are Banach spaces which are invariant by translation.
denotes one the following spaces: , , , , , or endowed with the norm . We have , and when , is a Banach space which is invariant by translation. denotes the space of functions such that . Endowed with the norm , is a Banach space. denotes the space of the functions such that . Endowed with the norm , is a Banach space.
3 Preliminary results
For the proof of the main result, we need the following lemmas.
Remark 3.2 Lemma 3.1 is well know for , . In the scalar case, for , this result is proved in , Corollary 5.6, p.59.
Proof Consider the function defined by for . Since is a translation invariant vectorial space, then . The equality shows that the uniform continuity of implies that the sequence with values in the Banach space converges uniformly to on ℝ. Then , and from the definition of , we obtain . □
When (respectively ), this result is a straightforward consequence of Theorem 3.8 in  (respectively Theorem 3.9 in ) and when , this result is proved in , Proposition 3. Then we deduce the result for (respectively ).
Note that the case of the pseudo almost periodic (respectively pseudo almost automorphic) functions is a special case of the weighted pseudo almost periodic (respectively weighted pseudo almost automorphic) functions by taking for all ; remark that the associated measure is exactly the Lebesgue measure.
And so it suffices to prove the cases of weighted pseudo almost periodic functions and of weighted pseudo almost automorphic functions.
When (respectively ), this result is a straightforward consequence of Theorem 3.8 in  (respectively Theorem 3.9 in ). Consequently, , and from the definition of , we deduce that . Since u satisfies Equation (3.1), then , and from the definition of , we obtain . □
Remark 3.5 Contrary to the asymptotically almost periodic case and, in particular, for the almost periodic case, when the dimension of the Banach spaces X and Y is infinite, Lemma 3.4 does not hold for the pseudo almost periodic case, and thus for the weighted pseudo almost periodic case, without additional assumptions. This is due to the fact that the range of a pseudo almost periodic function is only bounded, but not relatively compact, contrary to the asymptotically almost periodic case. This last observation still holds when the word almost periodic is replaced by almost automorphic.
Proof When and , replacing by ℝ, this result is a variation of Theorem 8.4 in .
When and (respectively and ), this result is a straightforward consequence of Theorem 4.1 (respectively Theorem 4.2) in .
When and (respectively and ), using Corollary 4.12 in  (respectively Corollary 5.10 in ), we know that (respectively ). Moreover, using Lemma 1 in , we know that is continuous, and so its restriction to (respectively ) is also continuous. □
Lemma 3.6LetXandYbe two finite-dimensional Banach spaces, and letbe a continuously Fréchet-differentiable mapping. Then the Nemytskii operatoris continuously Fréchet-differentiable on, and we havefor all.
Proof When and , replacing by ℝ, this result is a variation of Theorem 8.5 in .
When and , using Lemma 1 in , we know that is of class and that we have when . Now, using Theorem 2.15 in , we know that and that when . And so and the announced formula for its Fréchet-differential is proven.
As in the proof of Lemma 3.3, the case of (respectively ) is a corollary of the case (respectively ). And so it suffices to prove the cases of the weighted pseudo almost periodic functions and of the weighted pseudo almost automorphic functions.
To prove the result in the case where and (respectively and ), note that, using Lemma 1 in , we know that is of class and that we have when . Now, using Corollary 4.12 in  (respectively Corollary 5.10 in ), we know that (respectively ) and that (respectively ) when (respectively ).
Proof Lemma 3.2 in  asserts that if x is a solution of () in , then , therefore the derivative is uniformly continuous, and by help of Lemma 3.1, we obtain . By using Lemma 3.4, the functions and are in . Applying again Lemma 3.4 to and using the continuous function defined by , we obtain that . Since , then , and from the definition of , we obtain that . □
4 The main result
First we announce the main result of the paper.
Proof First we introduce linear operators: defined by , defined by , defined by and defined by . Since , and for all , and for all , these linear operators are continuous; and consequently, the following assertion holds:
Now we define the Nemytskii operators build on the functions f and g: defined by and defined by . By using Lemma 3.6, and are of class on assimilated to , and Lemma 3.6 provides formulas for the differentials of these Nemytskii operators:
We can assimilate a point to the constant function that belongs to , which permits us to look at P as a closed vector subspace of . Then we can consider the following restrictions of the operators and : defined by and defined by when and .
Since the restriction of a -mapping to a Banach subspace is also a -mapping, a straightforward consequence of the continuous differentiability of and of is that and are of class , and the consequences of (4.3) and (4.4) are the following formulas:
Since and are linear continuous, they are of class . Since B is bilinear continuous, it is of class ; and consequently, using Lemma 3.6, is of class on . Denoting by the mapping , ε is of class after (A3), and is of class as a composition of -mappings. And so we can assert that , , and C are of class .
Now we note that the following equality holds:
which is the announced formula. □
Proof Let . We want to prove that there exists a unique such that . Using the formula provided by Lemma 4.2, this equation is equivalent to saying that y is a solution in of the following second-order linear differential equation (which is a Duffing equation):
Rewriting this second-order equation in the form of a first-order system, we obtain the following equivalent differential system:
For and with condition (A4), the assumptions of Lemma 3.3 are fulfilled, and we can assert that there exists a unique , which is a solution of (4.7). Therefore the first coordinate of X, denoted by y, is the unique solution of (4.6) in since y and , and then y is the unique element of which satisfies . □
Proof of Theorem 4.1 By using (4.2), Lemma 4.2 and Lemma 4.3, we can use the implicit function theorem (, p.61) that permits us to say that there exist a neighborhood of 0 in , a neighborhood of 0 in P and a -mapping from into , which satisfies the following conditions:
And so Theorem 4.1 is proven. □
5 Special cases
We consider the equation
The authors declare that they have no competing interests.
The authors are very pleased to contribute to this special issue in honor of Jean Mawhin, an international expert in the field of nonlinear analysis and differential equations, whose opinion is ever very important and useful to us.
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