An Asymptotic Result for neutral differential equations

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), $x^{'} (t) = - \sum_{j = 1}^{N} b_{j} (t) x (t - τ_{j} (t)) + \sum_{j = 1}^{N} c_{j} (t) x^{'} (t - τ_{j} (t)) .$ 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) (1) $y^{'} (z) - a (z) y (z) = \sum_{i = 1}^{m} b_{i} (z) y (z - g_{i} (z)) + \sum_{j = 1}^{n} c_{j} (z) y^{'} (z - h_{j} (z)) = 0, z \geq z_{0}$ 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) (2) $y (z) = θ (z), for inf {z - g_{i} (z), z - h_{j} (z) : i = 1, 2, \dots, m, j = 1, 2, \dots, n},$ 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,b_i,c _j ∈ C(ℝ ⁺ ,ℝ ) and g_i,h_j ∈ C(ℝ ⁺ ,ℝ ⁺ ).

Main results

We denote $\begin{array}{l} h = sup {h_{j} (z) : j = 1, 2, \dots n} \\ l = sup {g_{i} (z), h_{j} (z) : i = 1, 2, \dots, m, j = 1, 2, \dots, n} . \end{array}$ \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: (3) $\begin{matrix} λ (z) - a (z) = \sum_{i = 1}^{m} b_{i} (z) e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} + \sum_{j = 1}^{n} c_{j} (z) λ (z - h_{j} (z)) e^{\int_{z}^{z - h_{j} (z)} λ (t) dt}, for z \geq z_{0} \\ λ (z) = λ (z_{0}), for z \in [z_{0} - l, z_{0}] . \end{matrix}$ \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 = {infg_i(z),h_j(z), for i = 1,2,...,m and j = 1,2,...,n. Let $ζ (z) = e^{\int_{z_{0}}^{z} λ (t) dt}, for z \geq z_{0} - l$ \zeta \left( z \right) = {e^{\int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }},{\rm{for}}\,\,z \ge {z_0} - l and $β (z) = e^{\int_{z_{0}}^{z} λ (t) dt}, for z_{0} - l \leq z \leq z_{0} .$ \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 $ζ^{'} (z) = λ (z) ζ (z) = a (z) ζ (z) + \sum_{i = 1}^{m} b_{i} (z) β (z - g_{i} (z)) + \sum_{j = 1}^{n} c_{j} (z) λ (z - h_{j} (z)) β (z - h_{j} (z))$ \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: $\begin{array}{l} ζ (z) & = {ζ (z_{0}) + \int_{z_{0}}^{z} (\sum_{i = 1}^{m} b_{i} (t) e^{\int_{z_{0}}^{t - g_{i} (t)} λ_{z_{0}} (s) ds} \\ + \sum_{j = 1}^{n} c_{j} (t) λ_{z_{0}} (t - h_{j} (t)) e^{\int_{z_{0}}^{t - h_{j} (t)} λ_{z_{0}} (s) ds}) e^{- \int_{z_{0}}^{t} a (s) ds} dt} e^{\int_{z_{0}}^{z} a (s) ds} . \end{array}$ 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: $λ (z) = \frac{ζ^{'} (z)}{ζ (z)}$ \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 $ζ^{'} (z) = λ (z) ζ (z) = a (z) ζ (z) + \sum_{i = 1}^{m} b_{i} (z) β (z - g_{i} (z)) + \sum_{j = 1}^{n} c_{j} (z) λ (z - h_{j} (z)) β (z - h_{j} (z)) .$ \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 $θ (z) = θ (z_{0}) e^{\int_{z_{0}}^{z} λ_{z_{0}} (t) dt} .$ \theta \left( z \right) = \theta \left( {{z_0}} \right){e^{\int\limits_{{z_0}}^z {{\lambda _{{z_0}}}\left( t \right)dt} }}.

That is $λ_{z_{0}} (z) = \frac{θ^{'} (z)}{θ (z)} .$ {\lambda _{{z_0}}}\left( z \right) = \frac{{\theta \prime\left( z \right)}}{{\theta \left( z \right)}}.

So, the characteristic equation has solution such that $y (z) = θ (z_{0}) e^{\int_{z_{0}}^{z} λ_{z_{0}} (t) dt}$ 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 $ω (z) = θ (z) + c > 0$ \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, $y (z) = u (z) - y c (z)$ y\left( z \right) = u\left( z \right) - yc\left( z \right) is a solution of (1)–(2). Here y_c is a solution of (1) with θ (z) = c.

Theorem 2. Suppose that (4) $sup_{z \geq z_{0} + l - h} {\sum_{i = 1}^{m} | b_{i} (z) | | g_{i} (z) | e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} + \sum_{j = 1}^{n} | c_{j} (z) | | 1 - h_{j} (z) | | λ (z - h_{j} (z)) | e^{\int_{z}^{z - h_{j} (z)} λ (t) dt}} < 1$ \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 $y (z) e^{- \int_{z_{0}}^{z} λ (t) dt} \to K, for z \to \infty$ y\left( z \right){e^{ - \int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }} \to K,\,\,\,\,{\rm{for}}\,\,z \to \infty and ${y (z) e^{- \int_{z_{0}}^{z} λ (t) dt}}^{'} \to 0, for z \to \infty .$ {\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 $γ (z) = y (z) e^{- \int_{z_{0}}^{z} λ (t) dt}, z \geq z_{0} - l .$ \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 $\begin{array}{l} γ^{'} (z) & = (y^{'} (z) - y (z) λ (z)) e^{- \int_{z_{0}}^{z} λ (t) dt} \\ = (\sum_{i = 1}^{m} b_{i} (z) y (z - g_{i} (z)) - y (z) \sum_{i = 1}^{m} b_{i} (z) e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} \\ + \sum_{j = 1}^{n} c_{j} (z) y^{'} (z - h_{j} (z)) - y (z) \sum_{j = 1}^{n} c_{j} (z) λ (z - h_{j} (z)) e^{\int_{z}^{z - h_{j} (z)} λ (t) dt}) e^{- \int_{z_{0}}^{z} λ (t) dt} . \end{array}$ \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 $y (z) = γ (z) e^{\int_{z_{0}}^{z} λ (t) dt}$ y\left( z \right) = \gamma \left( z \right){e^{\int\limits_{{z_0}}^z {\lambda \left( t \right)dt} }}, and $y^{'} (z) = (γ^{'} (z) + γ (z) λ (z)) e^{\int_{z_{0}}^{z} λ (t) dt}$ 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 $\begin{array}{l} γ^{'} (z) & = \sum_{i = 1}^{m} b_{i} (z) [γ (z - g_{i} (z)) - γ (z)] e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} \\ + \sum_{j = 1}^{n} c_{j} (z) [γ^{'} (z - h_{j} (z)) + (γ (z - h_{j} (z)) - γ (z)) λ (z - h_{j} (z))] e^{\int_{z}^{z - h_{j} (z)} λ (t) dt}, \end{array}$ \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 (5) $\begin{array}{l} γ^{'} (z) & = \sum_{i = 1}^{m} b_{i} (z) \int_{z}^{z - g_{i} (z)} e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} γ^{'} (s) ds \\ + \sum_{j = 1}^{n} c_{j} (z) [γ^{'} (z - h_{j} (z)) + λ (z - h_{j} (z)) \int_{z}^{z - h_{j} (z)} γ^{'} (s) ds] e^{\int_{z}^{z - h_{j} (z)} λ (t) dt} \end{array}$ \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 b_i’s and c_j’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 b_i ≠ 0 or cj ≠ 0 on [z₀ + l − h,∞). Let $\begin{array}{l} σ & = sup_{z \geq z_{0} + l - h} {\sum_{i = 1}^{m} | b_{i} (z) | | g_{i} (z) | e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} \\ + \sum_{j = 1}^{n} | c_{j} (z) | | 1 - h_{j} (z) | | λ (z - h_{j} (z)) | e^{\int_{z}^{z - h_{j} (z)} λ (t) dt}} . \end{array}$ \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}

From (4), (6) $0 < σ < 1 .$ 0 < \sigma < 1.

Let $M = max {| γ^{'} (z) | : z \in [z_{0} - h, z_{0} + l - h]} .$ 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 (7) $| γ^{'} (z) | \leq M for all z \geq z_{0} - h,$ \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 $| γ^{'} (z) | < M + ɛ, for z \in [z_{0} - h, z *)$ \left| {\gamma \prime\left( z \right)} \right| < M + \varepsilon ,\,\,{\rm{for}}\,z \in [{z_0} - h,z*) and $| γ^{'} (z) | = M + ɛ,$ \left| {\gamma \prime\left( z \right)} \right| = M + \varepsilon , for z₀ − h ≤ z ≤ z₀ + l − h.

From the definition of σ,(5) and (6), we get $\begin{array}{l} M + ɛ & = | γ^{'} (z *) | \\ \leq \sum_{i = 1}^{m} | b_{i} (z *) | \int_{z *}^{z * - g_{i} (z *)} e^{\int_{z *}^{z * - g_{i} (z *)} λ (t) dt} | γ^{'} (s) | ds \\ + \sum_{j = 1}^{n} | c_{j} (z *) | [| γ^{'} (z * - h_{j} (z *)) | + | λ (z * - h_{j} (z *)) | \int_{z *}^{z * - h_{j} (z *)} | γ^{'} (s) | ds] e^{\int_{z *}^{z * - h_{j} (z *)} λ (t) dt} \\ \leq (M + ɛ) {\sum_{i = 1}^{m} | b_{i} (z *) | | g_{i} (z *) | e^{\int_{z *}^{z * - g_{i} (z *)} λ (t) dt}} \\ + \sum_{j = 1}^{n} | c_{j} (z *) | | 1 - h_{j} (z *) | | λ (z * - h_{j} (z *)) | e^{\int_{z *}^{z * - h_{j} (z *)} λ (t) dt}} \\ \leq (M + ɛ) σ < M + ɛ . \end{array}$ \begin{array}{*{20}{l}}{M + \varepsilon }&{ = \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)} {{e^{\int\limits_{z*}^{z* - {g_i}\left( {z*} \right)} {\lambda \left( t \right)dt} }}} \left| {\gamma \prime\left( s \right)} \right|ds} }\\{}&{\,\,\,\, + \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 \left( {M + \varepsilon } \right)\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} }}} } \right\}}\\{}&{\,\,\, + \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 \left( {M + \varepsilon } \right)\sigma < M + \varepsilon .}\end{array}

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), $\begin{array}{l} | γ^{'} (z) | & \leq \sum_{i = 1}^{m} | b_{i} (z) | \int_{z}^{z - g_{i} (z)} | γ^{'} (s) | ds e^{\int_{z}^{z - g_{i} (z)} λ (t) dt} \\ + \sum_{j = 1}^{n} | c_{j} (z) | [| γ^{'} (z - h_{j} (z)) | + | λ (z - h_{j} (z)) | \int_{z}^{z - h_{j} (z)} | γ^{'} (s) | ds] e^{\int_{z}^{z - h_{j} (z)} λ (t) dt} \\ \leq M {{\sum_{i = 1}^{m} | b_{i} (z) | g_{i} (z) e}^{\int_{z}^{z - g_{i} (z)} λ (t) dt} \\ + \sum_{j = 1}^{n} | c_{j} (z) | | 1 - h_{j} (z) | | λ (z - h_{j} (z)) | e^{\int_{z}^{z - h_{j} (z)} λ (t) dt}} \\ \leq M σ, for z \geq z_{0} + l - h . \end{array}$ \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, (8) $| γ^{'} (z) | \leq M σ^{n}, for z \geq z_{0} + nl - h (n = 0, 1, \dots) .$ \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 $n = \frac{z - z_{0} + h}{l}$ n = \frac{{z - {z_0} + h}}{l}.

So, z ⩾ z₀ + nl − h and $\frac{z - z_{0} + h}{l} - 1 < n$ \frac{{z - {z_0} + h}}{l} - 1 < n . Therefore, from (6) and (8), (9) $| γ^{'} (z) | \leq M σ^{n} \leq M σ^{\frac{z - z_{0} + h}{l} - 1} .$ \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), $lim_{z \to \infty} γ^{'} (z) = 0 .$ \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 $\begin{array}{l} | γ (z) - γ (Z) | & \leq \int_{Z}^{z} | γ^{'} (t) | dt = \int_{Z}^{z} M σ^{\frac{s - z_{0} + h}{l} - 1} ds \\ = M \frac{l}{ln σ} {[μ^{\frac{s - z_{0} + h}{l}} - 1]}_{s = Z}^{s = z} \\ = M \frac{l}{ln σ} [μ^{\frac{z - z_{0} + h}{l} - 1} - μ^{\frac{z - z_{0} + h}{l} - 1}] . \end{array}$ \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, $lim | γ (z) - γ (Z) | = 0$ {\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.

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

Journal RSS Feed

An Asymptotic Result for neutral differential equations

Published Online: Mar 31, 2020

Page range: 189 - 194

Received: May 07, 2019

Accepted: May 31, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00017

Keywordsneutral differential equation, characteristic equation, asymptotic behaviour

© 2020 Emel Bicer, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
neutral differential equation, characteristic equation, asymptotic behaviour