An Asymptotic Result for neutral differential equations

We obtain asymptotic result for the solutions of neutral differential equations. Our technique depends on characteristic equations.


Introduction
Neutral differential equations (NDEs) describe a certain form of delay differential equations. Recently, these equations have received much interest, because of they play important part in the mathematical modeling of natural phenomena. (NDEs) emerge in many fields of engineering and mathematical science. Most of solutions of (NDEs) can not be obtained in closed form. For this reason, researching the qualitative behaviour of solutions is an effective option. Until today, many authors have investigated qualitative behaviour of solutions of (NDEs). Particularly, for more results on the qualitative behaviour of solutions of (NDEs) see [1]- [14] and references therein.
There are several methods to investigate asymptotic behaviour of solutions, such as characteristic equations, fixed point methods, and Lyapunov functionals. While studying (NDEs), each of these methods has its advantages and disadvantages. In the present paper, we use characteristic equations to investigate asymptotic properties of solutions of a (NDE).
In [1], Ardjouni and Djoudi deal with the first order (NDE), Applying the fixed point methods, they obtained asymptotic results to solutions of above (NDE).
In [8], Dix et al. obtained asmptotic results of solutions to first order linear (NDEs). Motivated by the results of references therein, we investigate asymptotic properties of solutions to first order (NDE) with initial condition (IC) where a, b i , c j ∈ C(R + , R) and g i , h j ∈ C(R + , R + ).

Main results
We denote By the considered (NDE) (1), we combine the following equation: Lemma. For each (IC) (2), there exists a solution of (NDE) (1). Proof. Firstly, we will show that the characteristic equation has a unique solution. Let l 1 = inf g i (z), h j (z), for i = 1, 2, ..., m and j = 1, 2, ..., n . Let So, from the characteristic equation (3), we get for z 0 ≤ z ≤ z 0 + l 1 . Then, we obtain the solution of above equation as follows: From this, we can define λ (z) as follows: Similarly, from the solution of above equation, we can define ζ (z) on [z 0 + l 1 , z 0 + 2l 1 ] . So, we define λ (z) for all z ≥ z 0 − l.
For existing of solution of (NDE) (1)-(2), we will consider two case: That is So, the characteristic equation has solution such that is a solution of (1)- (2). Here y c is a solution of (1) with θ (z) = c. Theorem 2. Suppose that Then for each solution y of (NDE) (1)-(2), there exists a K (constant), such that Proof. For solutions y of (1)-(2) and λ of (3), we set , z z 0 − l.