Asymptotic integration of higher order nonlinear delay differential equations via principal solutions

Abstract

We study the asymptotic integration problem for a general class of n-th order nonlinear delay differential equations of the form $L_{n} x (t) = f (t, x (g (t)))$ , where $L_{n} x = x^{(n)} + a_{n - 1} (t) x^{(n - 1)} + \dots + a_{0} (t) x$ . It is shown that if ${u_{1}, u_{2}, \dots, u_{n}}$ is a set of principal solutions for $L_{n} x = 0$ , then the n-th order nonlinear delay differential equation above has solutions $x_{k}$ with the property that $x_{k} (t) = b_{1} u_{1} (t) + \dots + b_{k} u_{k} (t) + o (u_{k} (t))$ , $t \to \infty$ for $k = 1, 2, \dots, n$ .

Keywords

Higher order differential equation delay differential equation asymptotic integration fixed point theory principal solutions

1. Introduction

We consider an n-th order nonlinear delay differential equation of the form $\begin{matrix} (1.1) & L_{n} x (t) = f (t, x (g (t))), t ⩾ t_{0}, \end{matrix}$ where $t_{0} ⩾ 1$ is fixed, $L_{n}$ is a linear operator defined by $\begin{matrix} (1.2) & L_{n} x = x^{(n)} + a_{n - 1} (t) x^{(n - 1)} + \dots + a_{0} (t) x, \end{matrix}$ $a_{k} \in C ([t_{0}, \infty), R)$ for $k = 0, \dots, n - 1$ , $f \in C ([t_{0}, \infty) \times R, R)$ , and $g \in C ([t_{0}, \infty), R)$ such that ${lim}_{t \to \infty} g (t) = \infty$ and $- τ ⩽ g (t) ⩽ t$ for some $τ > 0$ .

We make a standing hypothesis that the associated linear homogeneous equation $\begin{matrix} (1.3) & L_{n} x = 0 \end{matrix}$ is disconjugate on an open interval I, i.e, no nontrivial solution has n zeros counting multiplicities on I [4]. It is well-known [3,6] that if Eq. (1.3) is disconjugate, then certain Wronskians of solutions are positive on I. By making use of this, Polya [7] showed that there are positive functions $γ_{k} \in C^{n - k} (I)$ such that for any $x \in C^{n} (I)$ , $\begin{matrix} (1.4) & L_{n} x (t) = γ_{n} (t) {γ_{n - 1} (t) {\dots {γ_{0} (t) x (t)}^{'} \dots}^{'}}^{'}, \end{matrix}$ called the Polya factorization. Indeed, if ${x_{1} (t), x_{2} (t), \dots, x_{n} (t)}$ is a linearly independent set of solutions of (1.3) and $w_{k} (t)$ denotes the Wronskian of the first k solutions, then $\begin{matrix} (1.5) & γ_{n} (t) = \frac{w_{n} (t)}{w_{n - 1} (t)}, γ_{k} (t) = \frac{w_{k}^{2} (t)}{w_{k - 1} (t) w_{k + 1} (t)}, γ_{0} (t) = \frac{1}{w_{1} (t)}, \end{matrix}$ where $1 ⩽ k ⩽ n - 1$ with $w_{0} (t) = 1$ . The validity of this result on an arbitrary interval was shown by Hartman [5]. Later, Trench [9] proved that there is a special Polya factorization, called the Trench Factorization, in which the functions $γ_{i}$ satisfy $\begin{matrix} \int_{t_{0}}^{\infty} \frac{1}{γ_{k} (s)} d s = \infty, 1 ⩽ k ⩽ n - 1 . \end{matrix}$ Indeed, if the Eq. (1.3) is disconjugate on $[t_{0}, \infty)$ , then there are linearly independent solutions $u_{k}$ ’s, $k = 1, \dots, n$ such that $\begin{matrix} (1.6) & lim_{t \to \infty} \frac{u_{k} (t)}{u_{k + 1} (t)} = 0 . \end{matrix}$ Each solution $u_{k}$ is called a k-th principal solution of Eq. (1.3). These functions can be obtained from the functions $γ_{i}$ appearing in Trench factorization as follows: $\begin{matrix} (1.7) & u_{1} (t) : = \frac{1}{γ_{0} (t)} \end{matrix}$ and $\begin{matrix} (1.8) & u_{k} (t) : = \frac{1}{γ_{0} (t)} \int_{t_{0}}^{t} \frac{1}{γ_{1} (s_{1})} \int_{t_{0}}^{s_{1}} \frac{1}{γ_{2} (s_{2})} \dots \int_{t_{0}}^{s_{k - 2}} \frac{1}{γ_{k - 1} (s_{k - 1})} d s_{k - 1} \dots d s_{2} d s_{1}, \end{matrix}$ for all $2 ⩽ k ⩽ n$ . Each $u_{k}$ becomes the k-th principal solution of $L_{n} x = 0$ , see [6, Theorem 6.38].

In the present work, we will use these principal solutions to construct a special operator $F_{k}$ in such a way that a fixed point of which being a solution of Eq. (1.1) has a prescribed asymptotic behavior. Namely, it will be shown that for any given real numbers $b_{1}, b_{2}, \dots, b_{n}$ there exist solutions $x_{1} (t), x_{2} (t), \dots, x_{n} (t)$ of Eq. (1.1) such that as $t \to \infty$ , $\begin{array}{l} x_{1} (t) = b_{1} u_{1} (t) + o (u_{1} (t)), \\ x_{2} (t) = b_{1} u_{1} (t) + b_{2} u_{2} (t) + o (u_{2} (t)), \\ x_{3} (t) = b_{1} u_{1} (t) + b_{2} u_{2} (t) + b_{3} u_{3} (t) + o (u_{3} (t)), \\ ⋮ \\ x_{n} (t) = b_{1} u_{1} (t) + b_{2} u_{2} (t) + \dots + b_{n} u_{n} (t) + o (u_{n} (t)) . \end{array}$ A related work concerning second-order equations by the present authors can be found in [1]. We note that extension to higher order equations is not straightforward, rather requires deeper understanding.

2. The main result

Let $\begin{array}{rcl} (2.1) & z_{k} (t) : = b_{1} u_{1} (t) + b_{2} u_{2} (t) + \dots + b_{k} u_{k} (t), k = 1, 2, \dots, n \end{array}$ and $\begin{array}{rcl} (2.2) & F_{n} (t) : = {(- 1)}^{n} \int_{t}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{f (s_{n}, x (g (s_{n})))}{γ_{n} (s_{n})} d s_{n} \dots d s_{1}, \end{array}$ where the functions $u_{k}$ and $γ_{k}$ are as in the previous section.

We first observe that $L_{n} z_{k} = 0$ for $k = 1, 2, \dots, n$ . Next, by using the Polya factorization (1.4) we also see that $\begin{matrix} L_{n} (u_{1} F_{n}) = f (t, x (g (t))) . \end{matrix}$ Therefore, it follows that a solution $x_{k} (t)$ of the integral equation $\begin{matrix} x_{k} (t) = z_{k} (t) + u_{1} (t) F_{n} (t), k = 1, 2, \dots, n \end{matrix}$ is a solution of Eq. (2.7). This observation will be crucial in proving our main result. We will rely on the following lemma, see [2,8,10], as well.

Lemma 2.1.
Let $Ω \subset R^{n}$ . A set $M \subset L_{p} (Ω)$ is compact if
there exists a number $B > 0$ such that $‖ f ‖_{L_{p} (Ω)} ⩽ B$ for all $f \in M$ ;

$‖ τ_{h} f - f ‖_{L_{p} (Ω)} \to 0$ as $h \to 0$ , where $(τ_{h} f) (x) : = f (x_{1} + h, x_{2} + h, \dots, x_{n} + h)$ , $x \in Ω$ .

Our main result is as follows. Theorem 2.2.
Let $u_{k}$ be the k-th principal solution of ( 1.3 ), where $1 ⩽ k ⩽ n$ . Suppose that $\begin{matrix} (2.3) & | f (t, x) | ⩽ h_{1} (g (t)) ϕ (\frac{| x |}{u_{1} (g (t))}) + h_{2} (t), t ⩾ T \end{matrix}$ for some functions $ϕ \in C ([0, \infty), [0, \infty))$ , $h_{1}, h_{2} \in C ([t_{0}, \infty), [0, \infty))$ and $T > t_{0}$ . If $\begin{matrix} (2.4) & \int_{T}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{1} (g (s_{n})) d s_{n} \dots d s_{2} d s_{1} < \infty, \end{matrix}$ and $\begin{matrix} (2.5) & \int_{T}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{2} (s_{n}) d s_{n} \dots d s_{2} d s_{1} < \infty, \end{matrix}$ then for any given $b_{1}, b_{2}, \dots, b_{n} \in R$ there are solutions $x_{k} (t)$ for $k = 1, 2, \dots, n$ of Eq. ( 1.1 ) satisfying $\begin{matrix} (2.6) & x_{k} (t) = b_{1} u_{1} (t) + b_{2} u_{2} (t) + \dots + b_{k} u_{k} (t) + o (u_{k} (t)), t \to \infty . \end{matrix}$
Proof.
The proof is an application of Schäuder’s fixed point theorem. Let $b_{1}, b_{2}, \dots, b_{n} \in R$ be given. Define $\begin{matrix} (2.7) & M : = max {| ϕ (η) | | 0 ⩽ η ⩽ 1 + | b_{1} | + \dots + | b_{n} |} . \end{matrix}$ Fix a number $T_{1} > T$ such that $u_{1} (t) ⩽ u_{2} (t) ⩽ \dots ⩽ u_{n} (t)$ , $\begin{matrix} (2.8) & \int_{T_{1}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{1} (g (s_{n})) d s_{n} \dots d s_{2} d s_{1} < \frac{1}{2 M} \end{matrix}$ and $\begin{matrix} (2.9) & \int_{T_{1}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{2} (s_{n}) d s_{n} \dots d s_{2} d s_{1} < \frac{1}{2} . \end{matrix}$ Such a number $T_{1}$ exists in view of (1.6), (2.4) and (2.5). Consider the linear spaces $\begin{matrix} X_{k} = {x \in C ([T_{1}, \infty), R) | \frac{| x (t) |}{u_{k} (t)} ⩽ M_{x}, t ⩾ T_{1}} . \end{matrix}$ It is easy to check that $X_{k}$ ’s are Banach spaces with the respective norms $\begin{matrix} ‖ x ‖_{k} = sup_{T_{1} ⩽ t < \infty} \frac{| x (t) |}{u_{k} (t)} . \end{matrix}$ Clearly $X_{k} \subset \dots \subset X_{1}$ . Further $‖ x ‖_{k} ⩽ \dots ⩽ ‖ x ‖_{1}$ , for any given $x \in X_{k}$ and $k = 1, \dots, n$ . The sets $\begin{matrix} Ω_{k} : = {x \in X | ‖ x - b_{1} u_{1} - b_{2} u_{2} - \dots - b_{k} u_{k} ‖ ⩽ 1} \end{matrix}$ are nonempty. Further, $Ω_{k}$ ’s are closed, bounded and convex subsets of $X_{k}$ ’s. Note that there exists a $T_{2} > T_{1}$ such that $g (t) ⩾ T_{1}$ , for all $t ⩾ T_{2}$ . Define the operators $F_{k} : Ω_{k} \to X_{k}$ by $\begin{matrix} (F_{k} x) (t) = \{\begin{matrix} z_{k} (t) + u_{1} (t) F_{n} (t), & T_{2} ⩽ t, \\ z_{k} (t) + u_{1} (t) F_{n} (T_{2}), & t < T_{2}, \end{matrix} \end{matrix}$ where $z_{k}$ and $F_{n}$ are given by (2.1) and (2.2), respectively.

$F_{k}$ map $Ω_{k}$ into $Ω_{k}$ : For each $x \in Ω_{k}$ and for each $t ⩾ T_{1}$ we have $\begin{array}{rcl} \frac{| (F_{k} x) (t) - z_{k} (t) |}{u_{k} (t)} & ⩽ & \frac{| (F_{k} x) (t) - z_{k} (t) |}{u_{1} (t)} \\ ⩽ & \int_{t}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} | f (s_{n}, x (g (s_{n}))) | d s_{n} \dots d s_{1} \\ ⩽ & M \int_{T_{2}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{1} (g (s_{n})) d s_{n} \dots d s_{1} \\ + \int_{T_{2}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{2} (s_{n}) d s_{n} \dots d s_{1} \\ ⩽ & 1, \end{array}$ where (2.3), (2.8), and (2.9) have been used. By taking the supremum we obtain $F_{k} Ω_{k} \subset Ω_{k}$ .

$F_{k}$ ’s are continuous: Let ${x_{i}}_{i = 1}^{\infty} \subset Ω_{k}$ be an arbitrary sequence converging to $x \in Ω_{k}$ . In view of (2.3) for each $n \in N$ we have $\begin{array}{l} \frac{| (F_{k} x_{n}) (t) - (F_{k} x) (t) |}{u_{k} (t)} \\ ⩽ \frac{| (F_{k} x_{n}) (t) - (F_{k} x) (t) |}{u_{1} (t)} \\ ⩽ \int_{t}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} f_{i} (s_{n}) d s_{n} \dots d s_{1} \\ ⩽ 2 M \int_{T_{2}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{1} (g (s_{n})) d s_{n} \dots d s_{1} \\ (2.10) & + 2 \int_{T_{2}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \frac{1}{γ_{2} (s_{2})} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} h_{2} (s_{n}) d s_{n} \dots d s_{1}, \end{array}$ where $\begin{matrix} f_{i} (s_{n}) = | f (s_{n}, x_{i} (g (s_{n}))) - f (s_{n}, x (g (s_{n}))) | . \end{matrix}$ By applying the Lebesgue’s dominated convergence theorem we obtain from (2.10) that $F_{k} x_{i} \to F_{k} x$ as $i \to \infty$ .

The $F_{k}$ ’s are completely continuous: Let ${x_{i}}_{i = 1}^{\infty} \subset Ω_{k}$ be an arbitrary sequence. We need to show that there exist $w \in Ω_{k}$ and a subsequence ${x_{i_{j}}}_{j = 1}^{\infty}$ so that $F_{k} x_{i_{j}} \to w$ as $j \to \infty$ . We will use Lemma 2.1 to show the existence of w. Define $\begin{matrix} f_{i} (s_{1}) : = \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} | f (s_{n}, x_{i} (g (s_{n}))) | d s_{n} \dots d s_{2} . \end{matrix}$ It follows from (2.3) and (2.7) that $\begin{matrix} \int_{T_{2}}^{\infty} | f_{i} (s_{1}) | d s_{1} ⩽ \int_{T_{2}}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} (M h_{1} (g (s_{n})) + h_{2} (s_{n})) d s_{n} \dots d s_{1}, \end{matrix}$ i.e., the condition (a) of Lemma 2.1 holds. To see that (b) is also satisfied we find $\begin{array}{rcl} \int_{T_{2}}^{\infty} | (τ_{h} f_{i}) (s_{1}) - f_{i} (s_{1}) | d s_{1} & ⩽ & \int_{T_{2}}^{\infty} | f_{i} (s_{1} + h) | d s_{1} + \int_{T_{2}}^{\infty} | f_{i} (s_{1}) | d s_{1} \\ ⩽ & \int_{T_{2} + h}^{\infty} | f_{i} (s_{1}) | d s_{1} + \int_{T_{2}}^{\infty} | f_{i} (s_{1}) | d s_{1} \\ (2.11) & ⩽ & 2 \int_{T_{2}}^{\infty} | f_{i} (s_{1}) | d s_{1}, \end{array}$ where $(τ_{h} f) (s_{1}) = f (s_{1} + h)$ . Now, by using the Lebesgue’s dominated convergence theorem, we obtain from (2.11) that $\begin{matrix} ‖ τ_{h} f_{i} - f_{i} ‖_{L_{1} ([T_{2}, \infty))} \to 0 as h \to 0 . \end{matrix}$ In view of Lemma 2.1 we may deduce that there exists a subsequence ${f_{i_{j}}}_{j = 1}^{\infty}$ such that $\begin{matrix} ‖ f_{i_{j}} - z ‖_{L_{1} ([T_{2}, \infty))} \to 0 as k \to \infty \end{matrix}$ for some $z \in L_{1} ([T_{2}, \infty))$ . Define $\begin{matrix} w_{k} (t) : = \{\begin{matrix} z_{k} (t) + \frac{{(- 1)}^{n}}{γ_{0} (t)} \int_{t}^{\infty} z (s_{1}) d s_{1}, & t ⩾ T_{2}, \\ z_{k} (t) + \frac{{(- 1)}^{n}}{γ_{0} (t)} \int_{T_{2}}^{\infty} z (s_{1}) d s_{1}, & t < T_{2} . \end{matrix} \end{matrix}$ Then $\begin{matrix} \frac{| (F_{k} x_{i_{j}}) (t) - w_{k} (t) |}{u_{k} (t)} ⩽ \frac{| (F_{k} x_{i_{j}}) (t) - w_{k} (t) |}{u_{1} (t)} ⩽ ‖ f_{i_{j}} - z ‖_{L_{1} ([T_{2}, \infty))}, \end{matrix}$ for all $T_{1} ⩽ t$ . Applying again the Lebesgue’s dominated convergence theorem, we may conclude that $F$ is completely continuous. Now it follows from Schäuder’s fixed point theorem that the operator $F_{k}$ has a fixed point $x_{k} \in Ω_{k}$ , that is, $\begin{matrix} x_{k} (t) = \{\begin{matrix} z_{k} (t) + u_{1} (t) F_{n} (t), & t ⩾ T_{2}, \\ z_{k} (t) + u_{1} (t) F_{n} (T_{2}), & t < T_{2} . \end{matrix} \end{matrix}$

It is easy to check that $x_{k} (t)$ is solution of (1.1) over the interval $T_{2} ⩽ t < \infty$ . To complete the proof we only need to show the asymptotic representation (2.6) holds. To this end we consider the following estimate: $\begin{array}{rcl} \frac{| x_{k} (t) - z_{k} (t) |}{u_{k} (t)} & ⩽ & \frac{| x_{k} (t) - z_{k} (t) |}{u_{1} (t)} \\ ⩽ & \int_{t}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} | f (s_{n}, x (g (s_{n}))) | d s_{n} \dots d s_{1} \\ ⩽ & \int_{t}^{\infty} \frac{1}{γ_{1} (s_{1})} \int_{s_{1}}^{\infty} \dots \int_{s_{n - 1}}^{\infty} \frac{1}{γ_{n} (s_{n})} (M h_{1} (s_{n}) + h_{2} (s_{n})) d s_{n} \dots d s_{1}, \end{array}$ where (2.3) is used. Taking the limit on both sides as $t \to \infty$ we get $\begin{matrix} x_{k} (t) = b_{1} u_{1} (t) + b_{2} u_{2} (t) + \dots + b_{k} u_{k} (t) + o (u_{k} (t)), t \to \infty . \end{matrix}$ □

3. An example

In this section we apply our main result to a third-order nonlinear delay differential equation to illustrate its usefulness. The example shows how the delay term affects the asymptotic behavior of the solutions. We consider the third-order nonlinear delay differential equation $\begin{matrix} (3.1) & x^{‴} (t) + 3 x^{″} (t) + 2 x^{'} (t) = 6 x^{β} (α t) + (1 - α) e^{ν t}, t ⩾ 1, \end{matrix}$ where $ν < - 2$ , $1 < β < 1 / α^{2}$ and $0 < α < 1$ such that β does not have an even denominator. We observe that $\begin{matrix} L_{3} x = x^{‴} + 3 x^{″} + 2 x^{'} . \end{matrix}$

The following calculations are straightforward. Obviously, ${e^{- 2 t}, e^{- t}, 1}$ is a fundamental set of solutions for $L_{3} x = 0$ . Then it is easy to calculate that $\begin{matrix} w_{0} (t) = 1, w_{1} (t) = e^{- 2 t}, w_{2} (t) = |\begin{matrix} e^{- 2 t} & e^{- t} \\ - 2 e^{- 2 t} & - e^{- t} \end{matrix}| = e^{- 3 t}, \end{matrix}$ and $\begin{matrix} w_{3} (t) = |\begin{matrix} e^{- 2 t} & e^{- t} & 1 \\ - 2 e^{- 2 t} & - e^{- t} & 0 \\ 4 e^{- 2 t} & e^{- t} & 0 \end{matrix}| = 2 e^{- 3 t} . \end{matrix}$ Thus we have $\begin{array}{l} γ_{0} (t) = \frac{1}{w_{1} (t)} = e^{2 t}, γ_{1} (t) = \frac{w_{1}^{2} (t)}{w_{0} (t) w_{2} (t)} = e^{- t}, \\ γ_{2} (t) = \frac{w_{2}^{2} (t)}{w_{1} (t) w_{3} (t)} = \frac{1}{2} e^{- t}, γ_{3} (t) = \frac{w_{3} (t)}{w_{2} (t)} = 2 . \end{array}$ Clearly, $u_{1} (t) = e^{- 2 t}$ , and $\begin{matrix} \int_{1}^{\infty} \frac{1}{γ_{1} (s)} d s = \infty, \int_{1}^{\infty} \frac{1}{γ_{2} (s)} d s = \infty . \end{matrix}$ Hence the Trench Factorization takes the form $\begin{matrix} L_{3} x = 2 {\frac{1}{2} e^{- t} {e^{- t} {e^{2 t} x}^{'}}^{'}}^{'} . \end{matrix}$ Further, we may take $u_{2} (t) = e^{- t}$ and $u_{3} (t) = 1$ . Setting $\begin{matrix} h_{1} (t) = 6 e^{- 2 t / α}, h_{2} (t) = (1 - α) e^{ν t}, ϕ (x) = x^{β}, \end{matrix}$ we easily verify that $\begin{matrix} | f (t, x (g (t))) | ⩽ 6 {| x (α t) |}^{β} + (1 - α) e^{ν t} = h_{1} (g (t)) ϕ (\frac{| x (g (t)) |}{u_{1} (g (t))}) + h_{2} (t) . \end{matrix}$ Finally, $\begin{array}{rcl} \int_{1}^{\infty} \frac{1}{γ_{1} (t)} \int_{t}^{\infty} \frac{1}{γ_{2} (s)} \int_{s}^{\infty} \frac{h_{1} (g (r))}{γ_{3} (r)} d r d s d t & = & \int_{1}^{\infty} e^{t} \int_{t}^{\infty} 2 e^{s} \int_{s}^{\infty} 3 e^{- 2 r / α} d r d s d t \\ = & \frac{3 α^{3}}{2 α^{2} - 6 α + 4} e^{(\frac{2 α - 2}{α})} < \infty \end{array}$ and $\begin{array}{rcl} \int_{1}^{\infty} \frac{1}{γ_{1} (t)} \int_{t}^{\infty} \frac{1}{γ_{2} (s)} \int_{s}^{\infty} \frac{h_{2} (r)}{γ_{3} (r)} d r d s d t & = & \int_{1}^{\infty} e^{t} \int_{t}^{\infty} 2 e^{s} \int_{s}^{\infty} \frac{1}{2} (1 - α) e^{ν r} d r d s d t \\ = & \frac{(1 - α)}{ν^{3}} e^{ν} < \infty . \end{array}$

Since all the conditions of Theorem 2.2 are satisfied, we may conclude that, for any given real numbers $b_{1}$ , $b_{2}$ , $b_{3}$ , there are solutions $x_{1}$ , $x_{2}$ , $x_{3}$ of (3.1) such that $\begin{array}{l} x_{1} (t) = b_{1} e^{- 2 t} + o (e^{- 2 t}), t \to \infty, \\ x_{2} (t) = b_{1} e^{- 2 t} + b_{2} e^{- t} + o (e^{- t}), t \to \infty, \\ x_{3} (t) = b_{1} e^{- 2 t} + b_{2} e^{- t} + b_{3} + o (1), t \to \infty . \end{array}$

We end our paper with the following remark.

Remark 3.1.

If we let $α \to 1$ , then Eq. (3.1) reduces to the linear third-order equation $\begin{matrix} x^{‴} + 3 x^{″} + 2 x^{'} - 6 x = 0 . \end{matrix}$ This equation has the general solution $\begin{matrix} x (t) = c_{1} e^{- 2 t} cos \sqrt{t} + c_{2} e^{- 2 t} sin \sqrt{t} + c_{3} e^{t} . \end{matrix}$ We see that there is no solution having an asymptotic representation similar to the above ones, which means that the delay term plays an important role on the asymptotic integration of the solutions.

Footnotes

Acknowledgement

The second author is supported by The Scientific and Technological Research Council of Turkey (TUBITAK).

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