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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),
x\prime\left( t \right) = - \sum\limits_{j = 1}^N {{b_j}} \left( t \right)x\left( {t - {\tau _j}\left( t \right)} \right) + \sum\limits_{j = 1}^N {{c_j}\left( t \right)x\prime\left( {t - {\tau _j}\left( t \right)} \right).}
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)
y\prime\left( z \right) - a\left( z \right)y\left( z \right) = \sum\limits_{i = 1}^m {{b_i}} \left( z \right)y\left( {z - {g_i}\left( z \right)} \right) + \sum\limits_{j = 1}^n {{c_j}\left( z \right)y\prime\left( {z - {h_j}\left( z \right)} \right) = 0,\,\,z \ge {z_0}}
with initial condition (IC)
y\left( z \right) = \theta \left( z \right),\,\,\,{\rm{for}}\,\,{\rm{inf}}\left\{ {z - {g_i}\left( z \right),z - {h_j}\left( z \right):i = 1,2, \ldots ,m,j = 1,2, \ldots ,n} \right\},
where a,bi,c
j ∈ C(ℝ
+
,ℝ
) and gi,hj ∈ C(ℝ
+
,ℝ
+
).
Main results
We denote
\begin{array}{*{20}{l}}{h = \sup \left\{ {{h_j}\left( z \right):j = 1,2, \ldots n} \right\}}\\{l = \sup \left\{ {{g_i}\left( z \right),{h_j}\left( z \right):i = 1,2, \ldots ,m,j = 1,2, \ldots ,n} \right\}.}\end{array}
Let C([z0
− l,z0]
,ℝ
) represent the set of continuous real-valued functions on [z0
− l,z0].
By the considered (NDE) (1), we combine the following equation:
\begin{array}{*{20}{c}}{\lambda \left( z \right) - a\left( z \right) = \sum\limits_{i = 1}^m {{b_i}\left( z \right){e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} + \sum\limits_{j = 1}^n {{c_j}\left( z \right)\lambda \left( {z - {h_j}\left( z \right)} \right){e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }},{\rm{for}}\,z \ge {z_0}} }\\{\lambda \left( z \right) = \lambda \left( {{z_0}} \right),\,\,\,{\rm{for}}\,\,z \in \left[ {{z_0} - l,{z_0}} \right].}\end{array}
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 l1 = {infgi(z),hj(z), for i = 1,2,...,m and j = 1,2,...,n. Let
\zeta \left( z \right) = {e^{\int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }},{\rm{for}}\,\,z \ge {z_0} - l
and
\beta \left( z \right) = {e^{\int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }},{\rm{for}}\,\,\,{z_0} - l \le z \le {z_0}.
So, from the characteristic equation (3), we get
\zeta \prime\left( z \right) = \lambda \left( z \right)\zeta \left( z \right) = a\left( z \right)\zeta \left( z \right) + \sum\limits_{i = 1}^m {{b_i}} \left( z \right)\beta \left( {z - {g_i}\left( z \right)} \right) + \sum\limits_{j = 1}^n {{c_j}\left( z \right)\lambda \left( {z - {h_j}\left( z \right)} \right)\beta \left( {z - {h_j}\left( z \right)} \right)}
for z0
≤ z ≤ z0 + l1. Then, we obtain the solution of above equation as follows:
begin{array}{*{20}{l}}{\zeta \left( z \right)}&{ = \{ \zeta \left( {{z_0}} \right) + \int\limits_{{z_0}}^z {(\sum\limits_{i = 1}^m {{b_i}\left( t \right)} {e^{\int\limits_{{z_0}}^{t - {g_i}\left( t \right)} {{\lambda _{{z_0}}}} \left( s \right)ds}}} }\\{}&{ + \sum\limits_{j = 1}^n {{c_j}\left( t \right){\lambda _{{z_0}}}\left( {t - {h_j}\left( t \right)} \right){e^{\int\limits_{{z_0}}^{t - {h_j}\left( t \right)} {{\lambda _{{z_0}}}\left( s \right)ds} }}){e^{\int\limits_{{z_0}}^t {a\left( s \right)} ds}}} dt\} {e^{\int\limits_{{z_0}}^z {a\left( s \right)ds} }}.}\end{array}
From this, we can define λ (z) as follows:
\lambda \left( z \right) = \frac{{\zeta \prime\left( z \right)}}{{\zeta \left( z \right)}}
on [z0,z0 + l1]. Now, let β (z) = ζ (z) on [z0
− l,z0 + l1]. Then, for z ∈ [z0 + l1,z0 + 2l1] , we get
\zeta \prime\left( z \right) = \lambda \left( z \right)\zeta \left( z \right) = a\left( z \right)\zeta \left( z \right) + \sum\limits_{i = 1}^m {{b_i}} \left( z \right)\beta \left( {z - {g_i}\left( z \right)} \right) + \sum\limits_{j = 1}^n {{c_j}\left( z \right)\lambda \left( {z - {h_j}\left( z \right)} \right)\beta \left( {z - {h_j}\left( z \right)} \right)} .
Similarly, from the solution of above equation, we can define ζ (z) on [z0 + l1,z0 + 2l1]. So, we define λ (z) for all z ≥ z0
− l.
For existing of solution of (NDE) (1)–(2), we will consider two case:
Case 1. Let, θ (z) does not have zeros on [z0
− l,z0]. Let
\theta \left( z \right) = \theta \left( {{z_0}} \right){e^{\int\limits_{{z_0}}^z {{\lambda _{{z_0}}}\left( t \right)dt} }}.
That is
{\lambda _{{z_0}}}\left( z \right) = \frac{{\theta \prime\left( z \right)}}{{\theta \left( z \right)}}.
So, the characteristic equation has solution such that
y\left( z \right) = \theta \left( {{z_0}} \right){e^{\int\limits_{{z_0}}^z {{\lambda _{{z_0}}}\left( t \right)dt} }}
is a solution of (1)–(2) on [z0
− l,∞). Especially, equation (1) by constant θ (z) = c
≠ 0 has a solution.
Case 2. Let, θ (z) has zeros on [z0
− l,z0]. Since θ is continuous on [z0
− l,z0], there is a constant c
≠ 0. So, we can write
\omega \left( z \right) = \theta \left( z \right) + c > 0
for [z0 − l,z0]. Then, equation (1) has a solution u by initial condition ω. Thus,
y\left( z \right) = u\left( z \right) - yc\left( z \right)
is a solution of (1)–(2). Here yc is a solution of (1) with θ (z) = c.
Theorem 2. Suppose that
\mathop {\sup }\limits_{z \ge {z_0} + l - h} \left\{ {\sum\limits_{i = 1}^m {\left| {{b_i}\left( z \right)} \right|\left| {{g_i}\left( z \right)} \right|{e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} + \sum\limits_{j = 1}^n {\left| {{c_j}\left( z \right)} \right|\left| {1 - {h_j}\left( z \right)} \right|\left| {\lambda \left( {z - {h_j}\left( z \right)} \right)} \right|{e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }}} } \right\} < 1
Then for each solution y of (NDE) (1)–(2), there exists a K (constant), such that
y\left( z \right){e^{ - \int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }} \to K,\,\,\,\,{\rm{for}}\,\,z \to \infty
and
{\left\{ {y\left( z \right){e^{ - \int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }}} \right\}^\prime } \to 0,\,\,{\rm{for}}\,\,z \to \infty .
Proof
For solutions y of (1)–(2) and λ of (3), we set
\gamma \left( z \right) = y\left( z \right){e^{ - \int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }},\,\,\,z \ge {z_0} - l.
Applying (1) and (3), we obtain
\begin{array}{*{20}{l}}{\gamma \prime\left( z \right)}&{ = \left( {y\prime\left( z \right) - y\left( z \right)\lambda \left( z \right)} \right){e^{ - \int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }}}\\{}&{ = \left( {\sum\limits_{i = 1}^m {{b_i}\left( z \right)y\left( {z - {g_i}\left( z \right)} \right) - y\left( z \right)\sum\limits_{i = 1}^m {{b_i}\left( z \right){e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} } } \right.}\\{}&{\,\,\, + \sum\limits_{j = 1}^n {{c_j}} \left( z \right)y\prime\left( {z - {h_j}\left( z \right)} \right) - y\left( z \right)\sum\limits_{j = 1}^n {{c_j}\left( z \right)\lambda \left( {z - {h_j}\left( z \right)} \right){e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }}){e^{ - \int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }}} .}\end{array}
Since
y\left( z \right) = \gamma \left( z \right){e^{\int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }}, and
y\prime\left( z \right) = \left( {\gamma \prime\left( z \right) + \gamma \left( z \right)\lambda \left( z \right)} \right){e^{\int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }}, from above equality, we obtain
\begin{array}{*{20}{l}}{\gamma \prime\left( z \right)}&{ = \sum\limits_{i = 1}^m {{b_i}\left( z \right)\left[ {\gamma \left( {z - {g_i}\left( z \right)} \right) - \gamma \left( z \right)} \right]{e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} }\\{}&{ + \sum\limits_{j = 1}^n {{c_j}\left( z \right)\left[ {\gamma \prime\left( {z - {h_j}\left( z \right)} \right) + \left( {\gamma \left( {z - {h_j}\left( z \right)} \right) - \gamma \left( z \right)} \right)\lambda \left( {z - {h_j}\left( z \right)} \right)} \right]{e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }},} }\end{array}
for z ≥ z0. From this, we get
\begin{array}{*{20}{l}}{\gamma \prime\left( z \right)}&{ = \sum\limits_{i = 1}^m {{b_i}\left( z \right)\int\limits_z^{z - {g_i}\left( z \right)} {{e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} \gamma \prime\left( s \right)ds} }\\{}&{ + \sum\limits_{j = 1}^n {{c_j}} \left( z \right)\left[ {\gamma \prime\left( {z - {h_j}\left( z \right)} \right) + \lambda \left( {z - {h_j}\left( z \right)} \right)\int\limits_z^{z - {h_j}\left( z \right)} {\gamma \prime\left( s \right)ds} } \right]{e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }}}\end{array}
for z ≥ z0 + l − h.
If all bi’s and cj’s are equal zero on [z0 + l − h,∞)
, from (5), γ is constant and γ′ = 0 on [z0 + l − h,∞) which would complete the proof. Thus, we suppose that bi ≠ 0 or cj
≠ 0 on [z0 + l − h,∞). Let
\begin{array}{*{20}{l}}\sigma &{ = \mathop {\sup }\limits_{z \ge {z_0} + l - h} \{ \sum\limits_{i = 1}^m {\left| {{b_i}\left( z \right)} \right|\left| {{g_i}\left( z \right)} \right|{e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} }\\{}&{ + \sum\limits_{j = 1}^n {\left| {{c_j}\left( z \right)} \right|\left| {1 - {h_j}\left( z \right)} \right|\left| {\lambda \left( {z - {h_j}\left( z \right)} \right)} \right|{e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }}} \} .}\end{array}
Let
M = {\rm{max}}\left\{ {\left| {\gamma \prime\left( z \right)} \right|:z \in \left[ {{z_0} - h,{z_0} + l - h} \right]} \right\}.
We shall prove that
\left| {\gamma \prime\left( z \right)} \right| \le M\,{\rm{for}}\,{\rm{all}}\,z \ge {z_0} - h,
on [z0
− h,∞).
Conversely, suppose that there exist ɛ > 0 and z ≥ z0
− h such that |γ′(z)| > M + ɛ. Since |γ′(z)| ≤ M and from the continuity of γ′, there exists z* > z0 + l − h such that
\left| {\gamma \prime\left( z \right)} \right| < M + \varepsilon ,\,\,{\rm{for}}\,z \in [{z_0} - h,z*)
and
\left| {\gamma \prime\left( z \right)} \right| = M + \varepsilon ,
for z0
− h ≤ z ≤ z0 + l − h.
So, we obtain a contradiction. Thus, inequality (7) holds. If M = 0
, from (7), γ is constant and γ′ = 0 on [z0
− h,∞). Thus, we suppose that M > 0.
In view of, (5) and (7),
\begin{array}{*{20}{l}}{\left| {\gamma \prime\left( z \right)} \right|}&{ \le \sum\limits_{i = 1}^m {\left| {{b_i}\left( z \right)} \right|\int\limits_z^{z - {g_i}\left( z \right)} {\left| {\gamma \prime\left( s \right)} \right|ds{e^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}} } }\\{}&{ + \sum\limits_{j = 1}^n {\left| {{c_j}\left( z \right)} \right|\left[ {\left| {\gamma \prime\left( {z - {h_j}\left( z \right)} \right)} \right| + \left| {\lambda \left( {z - {h_j}\left( z \right)} \right)} \right|\int\limits_z^{z - {h_j}\left( z \right)} {\left| {\gamma \prime\left( s \right)} \right|ds} } \right]{e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }}} }\\{}&{ \le M\{ {{\sum\limits_{i = 1}^m {\left| {{b_i}\left( z \right)} \right|{g_i}\left( z \right)e} }^{\int\limits_z^{z - {g_i}\left( z \right)} {\lambda \left( t \right)dt} }}}\\{}&{ + \sum\limits_{j = 1}^n {\left| {{c_j}\left( z \right)} \right|\left| {1 - {h_j}\left( z \right)} \right|\left| {\lambda \left( {z - {h_j}\left( z \right)} \right)} \right|{e^{\int\limits_z^{z - {h_j}\left( z \right)} {\lambda \left( t \right)dt} }}\} } }\\{}&{ \le M\sigma ,\,\,{\rm{for}}\,\,z \ge {z_0} + l - h.}\end{array}
Applying above inequality, we can show from induction,
\left| {\gamma \prime\left( z \right)} \right| \le M{\sigma ^n},{\rm{for}}\,z \ge {z_0} + nl - h\left( {n = 0,1, \ldots } \right).
For an arbitrary z ⩾ z0
− h, we define
n = \frac{{z - {z_0} + h}}{l}.
So, z ⩾ z0 + nl − h and
\frac{{z - {z_0} + h}}{l} - 1 < n
. Therefore, from (6) and (8),
\left| {\gamma \prime\left( z \right)} \right| \le M{\sigma ^n} \le M{\sigma ^{\frac{{z - {z_0} + h}}{l} - 1}}.
We get n → ∞, as z → ∞
, and from (6), σn
→ 0. Thus, from (9),
\mathop {\lim }\limits_{z \to \infty } \gamma \prime\left( z \right) = 0.
To prove that limγ(z) exists, as z → ∞
, we benefit from the Cauchy convergence criterion. For z > Z ⩾ z0−h, by (9)
, we get
\begin{array}{*{20}{l}}{\left| {\gamma \left( z \right) - \gamma \left( Z \right)} \right|}&{ \le \int\limits_Z^z {\left| {\gamma \prime\left( t \right)} \right|dt = \int\limits_Z^z {M{\sigma ^{\frac{{s - {z_0} + h}}{l} - 1}}ds} } }\\{}&{ = M\frac{l}{{{\rm{ln}}\sigma }}\left[ {{\mu ^{\frac{{s - {z_0} + h}}{l}}} - 1} \right]_{s = Z}^{s = z}}\\{}&{ = M\frac{l}{{{\rm{ln}}\sigma }}\left[ {{\mu ^{\frac{{z - {z_0} + h}}{l} - 1}} - {\mu ^{\frac{{z - {z_0} + h}}{l} - 1}}} \right].}\end{array}
We have z → ∞
, as Z → ∞
, and from (6), the right sides of above inequality tends zero. Thus,
{\rm{lim}}\left| {\gamma \left( z \right) - \gamma \left( Z \right)} \right| = 0
which implies that the existence of limγ(z), as z → ∞. The proof is completed.