Response Solutions for a Singularly Perturbed System Involving Reflection of the Argument

In this paper, the existence and uniqueness of response solutions, which has the same frequency ω with the nonlinear terms, are investigated for a quasiperiodic singularly perturbed system involving reflection of the argument. Firstly, we prove that all quasiperiodic functions with the frequency ω form a Banach space. Then, we obtain the existence and uniqueness of quasiperiodic solutions by means of the fixed-point methods and the B-property of quasiperiodic functions.


Introduction
The following singularly perturbed system x′ t ð Þ = F t, x t ð Þ, y t ð Þ, ε ð Þ , in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems involving large damping or resistance [1][2][3][4], where x and y are vectors with multiple components and ε ≥ 0 is a small parameter. The existence of periodic solutions and almost periodic solutions of (1) had been one of the most attracting topics in the qualitative theory of ordinary differential equations. The early contributions on these topics are due to Anosov [5] and Flatto and Levinson [6]. They investigated system (1) in the case that the degenerate system has a periodic solution θðtÞ, χðtÞ. The authors showed sufficient conditions on F, G which assure that the existence of periodic solutions of (1) and these solutions converge to θðtÞ, χðtÞ as ε ⟶ 0 uniformly. In 1961, Hale and Seifert [7] generalized the results of Flatto and Levinson to the almost periodic case and gave sufficient conditions for the existence of the almost periodic solutions of (1) using the similar method with [6]. Chang [8] obtained the same result of [7] under generalized hypothesis. But, the above papers [5][6][7][8] do not consider the stability properties of the solutions. Smith [9] considered the existence of almost periodic or periodic solutions for system (1). By the construction of manifolds of initial data, the author investigated the stability properties of these solutions, which approach the given solutions as t ⟶ ∞ at an exponential rate, α, independent of ε. He also gave the application in a reaction diffusion system with a traveling wave input.
It is natural to ask whether there is a bounded solution of system (1) for sufficiently small ε and how the stability properties of the solutions for the quasiperiodic case are.
For the Silberstein equation we define yðtÞ = xðe t Þ, then Equation (3) is equivalent to y ′ ðtÞ = e −t yð−tÞ, which is known as the equation involving reflection of argument. This kind of equations has applications in the study of stability of differential-difference equations, see Sharkovskii [10]. One of the earliest contributions to this kind of equations are due to Wiener and Aftabizadeh [11]. They investigated the boundary value problems for the second-order nonlinear differential equation They also considered the boundary value problems for the following equation changing the equation to a higher order one without reflection of the argument, where h, k ≥ 0, h + k > 0. Gupta [12,13] studied more general boundary value problems than Equations (4) and (5) using degree theory arguments. He proved the existence of solutions for the boundary value problems in a simple and straightforward manner. The existence and uniqueness of periodic, almost periodic, pseudo almost periodic, Besicovitch almost periodic, and pseudo almost automorphic solutions of this kind of equations were investigated in [14][15][16][17][18][19]. Cabada et al. [20][21][22] studied the first-order equation with two-point boundary conditions and the nth-order differential equations involving reflection, con-stant coefficients, and initial conditions, adding a new element to the previous studies: the existence of Green's function. However, as far as we know, the quasiperiodic solutions for the equations involving reflection of the argument have not been considered yet. Our present paper is devoted to discuss the existence and uniqueness of response solutions for the following singularly perturbed system where ε ≥ 0 is a small real parameter, and the functions F, G are quasiperiodic in t uniformly on ℝ 2 × ℝ 2 with frequency ω = ðω 1 , ω 2 , ⋯, ω d Þ. A quasiperiodic solution of (6) with the frequency ω is called response solution.
It is assumed that the degenerate system a quasiperiodic "outer" solution which we take to be the trivial solution, that is, we suppose so that ðx, yÞ = ð0, 0Þ satisfies (7). Expanding (6) about the trivial solution gives One can think of, e.g., a 1 ðt, εÞ = ∂F/∂xðt, 0, 0, 0, 0, εÞ. In the following discussion, we mainly consider (9). This paper is organized as follows: in Section 2, we present the Bohr's notion of B-property for quasiperiodic functions and then prove that all ω-frequency continuous quasiperiodic functions form a Banach space under the supremum norm. We prove an existence and uniqueness result for a linear scalar equation with reflection of the argument. In Section 3, the main results on the local existence and uniqueness of response solutions will be stated and proved by means of fixed-point methods in the spirit of Sacker and Sell [23]. We give conclusions of this paper in Section 4.

Preliminary
Firstly, we will give some lemmas which are important in proving our main results. Definition 1. Assume that ω 1 , ω 2 , ⋯, ω d ∈ ℝ are rationally independent. A continuous function uðtÞ on ℝ is said to be quasiperiodic with frequencies ðω 1 , Remark 2. This definition for quasiperiodic function can be found in many references, for example [24]. It is not difficult to prove that this definition is equivalent to the definition of quasiperiodic function in [25].
Proof. The proof of the lemma can be found in [26].

Advances in Mathematical Physics
According to the Lemma 2 of [14], we see that uðtÞ is of the form for some constant C. If C ≠ 0, then uðtÞ will be unbounded. This is a contradiction to the boundedness of quasiperiodic function. So, the operator L 1 ε : h ⟶ zðh, εÞ is well defined. From (16), we see the operator L 1 ε is linear. On the other hand, where To prove the continuity of L 1 ε in ε, we write vðtÞ = zðh, ε 1 ÞðtÞ − zðh, ε 2 ÞðtÞ for any 0 < ε 1 , ε 2 < ε 0 , then vðtÞ satisfies This implies that the map ε ⟶ L 1 ε is continuous for 0 < ε ≤ ε 0 .
Similar to the proof of Lemma 8, one can prove the following Lemma.
For the sake of convenience, we state the following conditions.