Besicovitch almost periodic solutions for a class of second order differential equations involving reflection of the argument

In this paper, using the Fourier series expansion and fixed point methods, we investigate the existence and uniqueness of Besicovitch almost periodic solutions for a class of second order differential equations involving reflection of the argument. Lipschitz nonlinear case is considered.


Introduction
The differential equations involving reflection of the argument have applications in the study of stability of differential-difference equations, see Sharkovskii [14], and such equations have very interesting properties, so many authors worked on them.First-order equations with constant coefficients and reflection have been studied in detail in [1,11,13,15].There is also an indication ([13, p.169] and [15, p.241]) that "The problem is much more difficult in the case of differential equations with reflection of order greater than one".Wiener and Aftabizadeh [16] initiated the study of boundary value problems for the second order differential equations involving reflection of the argument.Gupta [6,7] investigated two point boundary value problems for this kind of equations under the Carathéodory conditions.In [11,12], one of the present authors investigated existence and uniqueness of periodic, almost periodic and pseudo almost periodic solutions of the equations ẋ(t) + ax(t) + bx(−t) = g(t), b = 0, t ∈ R and ẋ(t) Recently Cabada et al. [2] studied the first order operator x (t) + mx(−t) coupled with periodic boundary value conditions, and described the eigenvalues of the operator and obtained the expression of its related Green's function in the non resonant case.Also Cabada et al. [3], using the theory of fixed point index, established new results for the existence of nonzero solutions of Hammerstein integral equations with reflections.They applied their results to a first-order periodic boundary value problem with reflections.On the other hand, Layton [8] studied the existence and uniqueness of Besicovitch almost periodic solutions for the delay equation ẋ(t) under any Besicovitch almost periodic forcing term e(t).But as far as we know, there are no works on the almost periodic solutions for such second-order equations.Motivated by the above references, our present paper is devoted to investigate the existence of a unique Besicovitch almost periodic solution of the second order nonlinear differential equation with reflection of the argument Remark 1.1.Ma żbic-Kulma [10] investigated firstly the equation The left hand side of equation (1.1) is a special case of this.
In order to develop our results, we review some facts about Bohr almost periodic and Besicovitch almost periodic functions.For further knowledge on almost periodic functions we refer the readers to the books [5,4,9].
We denote by AP (R) the set of all almost periodic functions in the sense of Bohr on R. The Besicovitch space of almost periodic functions, B 2 (R) is the closure of trigonometric polynomials of the form under the norm Here • on B 2 (R) is induced by the inner product Alternatively, B 2 (R) could be defined as the set of all The space B 2,1 (R) is defined to be the closure of the trigonometric polynomials (1.2) in the norm For details on some notations see Layton [8].

The linear problem
For e(t) ∈ B 2 , e n e iλ n t , consider the problem of finding a solution x(t) ∈ B 2 to the linear equation Let Λ be the Fourier exponents of e(t), then formally, the solution x(t) ∈ B 2 Λ to (2.1) is given by x n e iλ n t . (2.2) Putting it into equation (2.1), we obtain (2.4) We denote the coefficient determinant of the system (2.4) by d(λ n ), then .
It is easy to see that (ii) From the proof of Lemma 2.1, we see that (a where P 1 (λ) and P 2 (λ) are polynomials of λ with degree 4. Since d 2 (λ) is a polynomial of λ with degree 8, lim λ→∞ P k (λ)/d 2 (λ) = 0, k = 1, 2. On the other hand, by the proof of Lemma Hence, the infinite series ∑ ∞ n=−∞ (α n + iβ n )e iλ n t is absolutely convergent, so we have next theorem.
a,b exists on B 2 .The solution x(t) to (2.1) exists, is an element of B 2 Λ , is unique (up to a function with B 2 -norm zero) and is given by (2.2).Furthermore, there exists a constant M > 0, such that ) 0 is a sufficient condition for the existence of a unique B 2 -solution of the equation (2.1), but not a necessary one for just the existence of B 2 -solutions.As an example, let a 1 be any real number, and set λ 0 = 1/(a 2 1 + 2) 1/2 .Then the function x(t) defined by Utilizing the fact d 2 (λ)| ≥ |∆/(4(a 2 0 − b 2 0 ))| 2 > 0 we conclude that there exists a constant C > 0 such that

The nonlinear equation
Now let us consider the nonlinear equation The next lemma is an extension of Lemma 4.
It is easy to see that the following lemma holds.
for some constant L and 2ML < 1, then the equation (3.1) has a unique Besicovitch almost periodic solution x(t), and x(t) ∈ B Since 2ML < 1, T is a contraction mapping.So T has a unique fixed point in B 2 .Since f (t, x(t), x(−t)) ∈ B 2 and x = L −1 ] 2 < 0 is not satisfied, then d(λ n ) may become arbitrarily close to zero.That means we will meet the socalled small denominator problem.We will consider this case in future by means of KAM theory.