On the Asymptotic Behavior of a Class of Second-Order Non-Linear Neutral Differential Equations with Multiple Delays

Abstract

In this work, we present some new sufficient conditions for the oscillation of a class of second-order neutral delay differential equation. Our oscillation results, complement, simplify and improve recent results on oscillation theory of this type of non-linear neutral differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

1. Introduction

Consider the second-order neutral differential equation with several delays of the form

$$\begin{array}{c}\hfill {\left(a\left(y\right)\left(w^{\prime }\left(y\right)\right){}^{\mu }\right)}^{{}^{\prime }}+\sum _{i=1}^m{c}_i\left(y\right)F_i\left(u(y-q_i)\right)=0,w\left(y\right)=u\left(y\right)+b\left(y\right)u(y-p)\end{array}$$

(1)

where $p,q_i\in {\mathbb{R}}_{+}=(0,+\infty )$; $b\in PC({\mathbb{R}}_{+},\mathbb{R})$; $F_i\in C(\mathbb{R},\mathbb{R})$ is nondecreasing such that $u{F}_i\left(u\right)>0$ for $u\ne 0$; $c_i,a\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})$ for $i=1,2,\cdots ,m$ and $\mu$ is a ratio of two odd positive integers.

Recently there has been an increasing interest in dynamical systems both neutral and involving time delays with applications ranging from Biology and Population Dynamics to Physics and Engineering, and from Economics to Medicine. In particular, it is natural to ask why time-delayed systems are so important. Time delays are intrinsic in many real systems, and therefore must be properly accounted for evolution models. For further details we refer the reader to [1,2,3,4,5,6,7]. As a matter of fact, Equation (1) (i.e., half-linear/Emden-Fowler differential equation) arises in a variety of real world problems such as in the study of p-Laplace equations non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so forth; see [8,9,10] for more details.

Now we recall some basic definitions.

A function $u\left(y\right):[y_u,\infty )\to \mathbb{R}$, $y_u\ge y_0$ is said to be a solution of (1) if $u\left(y\right)$ and $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ are once continuously differentiable and $w\left(y\right)$ twice continuously differentiable for all $y\in [y_u,\infty )$ and it satisfies Equation (1) for all $y\in [y_u,\infty )$

We assume that (1) admits a solution in the sense of the above definition.

A solution $w\left(y\right)$ of (1) is said to be non-oscillatory if it is eventually positive or eventually negative; otherwise, it is said to be oscillatory.

The Equation (1) is said to be oscillatory if all its solutions are oscillatory.

In this paper, we restrict our attention to study the oscillation and non-oscillation of (1). First of all, it is interesting to make a review in the context of functional differential equations.

In 1978, Brands [11] has proved that the equation

$$u^{″}\left(y\right)+c\left(y\right)u(y-q\left(y\right))=0$$

is oscillatory if and only if

$$u^{″}\left(y\right)+c\left(y\right)u\left(y\right)=0$$

is oscillatory. In [8] Bohner et al. have studied oscillatory behavior of solutions to a class of second-order half-linear dynamic equations with deviating arguments under the assumptions that allow applications to dynamic equations with delayed and advanced arguments. They obtained several Fite–Hille–Wintner-type criteria that do not need some restrictive assumptions required in related results. In [12,13] Chatzarakis et al. have considered a more general equation

$$\begin{array}{c}\hfill {\left(a{\left(u^{\prime }\right)}^{\mu }\right)}^{\prime }\left(y\right)+c\left(y\right)u^{\mu }\left(q\left(y\right)\right)=0,\end{array}$$

(2)

and established new oscillation criteria for (2) when $A\left(y\right)={\int }^{y}a{\left(s\right)}^{-\frac{1}{\mu }}\mathrm{d}s$, ${lim}_{y\to \infty }A\left(y\right)=\infty$ and ${lim}_{y\to \infty }A\left(y\right)<\infty$.

Wong [14] has obtained the oscillation conditions of

$${\left(u\left(y\right)+bu(y-p)\right)}^{″}+c\left(y\right)F\left(u(y-q)\right)=0,-1<b<0,$$

in which the neutral coefficient and delays are constants. However, we have seen in [15,16] that the authors Baculǐkovǎ and Džurina have studied the equation

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+c\left(y\right)u^{\beta }\left(q\left(y\right)\right)=0,w\left(y\right)=u\left(y\right)+b\left(y\right)u\left(p\left(y\right)\right),y\ge y_0,\end{array}$$

(3)

and established the oscillation of solutions of (3) using comparison techniques when $\mu =\beta =1$, 0 $\le$ $b\left(y\right)$ and ${lim}_{y\to \infty }A\left(y\right)=\infty$. With the same technique, Baculikova and Džurina [17] have considered (3) and obtained oscillation conditions of (3) considering the assumptions $0\le b\left(y\right)$ and ${lim}_{y\to \infty }A\left(y\right)=\infty$. In [18], Tripathy et al. have studied (3) and established several conditions of the solutions of (3) considering the assumptions ${lim}_{y\to \infty }A\left(y\right)=\infty$ and ${lim}_{y\to \infty }A\left(y\right)<\infty$ for different ranges of the neutral coefficient b. In [19], Bohner et al. have obtained sufficient conditions for oscillation of solutions of (3) when $\mu =\beta$, ${lim}_{y\to \infty }A\left(y\right)<\infty$ and $0\le b\left(y\right)<1$. Grace et al. [20] studied the oscillation of (3) when $\mu =\beta$ and considering the assumptions ${lim}_{y\to \infty }A\left(y\right)<\infty$, ${lim}_{y\to \infty }A\left(y\right)=\infty$ and $0\le b\left(y\right)<1$. In [21], Li et al. have established sufficient conditions for the oscillation of the solutions of (3), under the assumptions ${lim}_{y\to \infty }A\left(y\right)<\infty$ and $b\left(y\right)\ge 0$. Karpuz and Santra [22] considered the equation

$${\left(a\left(y\right){(u\left(y\right)+b\left(y\right)u\left(p\left(y\right)\right))}^{\prime }\right)}^{\prime }+c\left(y\right)F\left(u\left(q\right(y\left)\right)\right)=0,$$

by considering the assumptions ${lim}_{y\to \infty }A\left(y\right)<\infty$ and ${lim}_{y\to \infty }A\left(y\right)=\infty$ for different ranges of b.

For further details regarding oscillatory properties of second-order differential equations, we refer the interested reader to [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51].

2. Necessary and Sufficient Conditions for Oscillation

This section deals with the necessary and sufficient conditions for oscillation of solutions of (1). We introduce the following assumptions for our use in the sequel:

$\left(C1\right)$

$-1<-b_0\le b\left(y\right)\le 0$, $b_0>0$ for $y\in {\mathbb{R}}_{+}$;

$\left(C2\right)$

$0\le b\left(y\right)\le b_0<\infty$ for $y\in {\mathbb{R}}_{+}$;

$\left(C3\right)$

$-\infty <-b_1\le b\left(y\right)\le -b_2<-1$, $b_1,b_2>0$ for $y\in {\mathbb{R}}_{+}$;

$\left(A1\right)$

$F_i(-x)=-F_i\left(x\right)$, $x\in \mathbb{R}$;

Theorem 1.

Assume that (C1) and (A1) hold. Furthermore, assume that

$\left(A2\right)$

${\int }_Y^{\infty }{\sum }_{i=1}^m{c}_i\left(y\right)F_i\left(\delta A(y-q_i)\right)\mathrm{d}y<+\infty$for every constant$Y,\delta >0$where$A\left(y\right)={\int }_Y^{y}a{\left(s\right)}^{-\frac{1}{\mu }}\mathrm{d}s$holds. Then every unbounded solution of (1) is oscillatory if and only if

$\left(A3\right)$

${lim}_{y\to \infty }A\left(y\right)<+\infty$.

Proof. 

We prove sufficiency by contradiction. Let $u\left(y\right)$ be an eventually positive unbounded solution of (1). Then there exists some $y_0>0$ such that $u\left(y\right)>0,u(y-p)>0$ and $u(y-q_i)>0$ for $i=1,2,\cdots ,m$ and $y\ge y_1>y_0+r$ where $r=max\{p,q_i\}$. We define

$$\begin{array}{c}\hfill w\left(y\right)=u\left(y\right)+b\left(y\right)u(y-p).\end{array}$$

(4)

From (1), it follows that

$$\begin{array}{cc}\hfill {\left(a\left(y\right)(w^{\prime }\left(y\right)\right)}^{\mu }{)}^{\prime }& =-\sum _{i=1}^m{c}_i\left(y\right)F_i\left(u(y-q_i)\right)<0\hfill \end{array}$$

(5)

for $y\ge y_1$. Hence, there exists $y_2>y_1$ such that $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ is nonincreasing on $[y_2,\infty ).$ Since $w\left(y\right)$ is monotonic, then for $y\ge y_3>y_2$ we have following two possibilities:

Case 1. Let $w\left(y\right)<0$ for $y\ge y_3$. Then $u\left(y\right)<u(y-p),$ and hence

$$\begin{array}{c}\hfill u\left(y\right)<u(y-p)<u(y-2p)<\cdots <u\left(y_3\right),\end{array}$$

that is, $u\left(y\right)$ is bounded, which contradicts u being unbounded.

Case 2. Let $w\left(y\right)>0$ for $y\ge y_3$.

Sub-case $2_1$. Let $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }>0$ for $y\ge y_3$. Since $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ is nonincreasing on $[y_3,\infty )$, hence there exist a constant $\kappa >0$ and $y_4>y_3$ such that $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\le \kappa$ for $y\ge y_4$. Consequently,

$$\begin{array}{c}\hfill w\left(y\right)\le w\left(y_4\right)+{\kappa }^{\frac{1}{\mu }}{\int }_{y_4}^y{\left(a\left(s\right)\right)}^{-\frac{1}{\mu }}\mathrm{d}s<\infty \end{array}$$

(6)

as $t\to \infty$ due to $\left(A3\right)$. On the other hand, $u\left(y\right)$ is unbounded, thus there exists $\left\{{\varsigma }_n\right\}$ such that ${\varsigma }_n\to \infty$ as $n\to \infty ,u\left({\varsigma }_n\right)\to \infty$ as $n\to \infty$ and

$$u\left({\varsigma }_n\right)=max\{u\left(\vartheta \right):y_3\le \vartheta \le {\varsigma }_n\}.$$

Therefore,

$$\begin{array}{cc}\hfill w\left({\varsigma }_n\right)& =u\left({\varsigma }_n\right)+b\left({\varsigma }_n\right)u({\varsigma }_n-p)\ge (1-b_0)u\left({\varsigma }_n\right)\to +\infty \mathrm{as}{\varsigma }_n\to \infty \hfill \end{array}$$

implies that $w\left(y\right)$ is unbounded, which leads a contradiction to (6).

Sub-case $2_2$. Let $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }<0$ for $y\ge y_3$. Since, a is positive and $\mu$ is a ratio of two odd positive integers. Therefore, $w^{\prime }\left(t\right)<0$ and hence w is bounded, which contradicts u being unbounded.

Hence, every positive unbounded solution of (1) oscillates.

If $u\left(y\right)<0$ for $y\ge y_0,$ then we set $v\left(y\right)=-u\left(y\right)$ for $y\ge y_0$ in (1) and we find

$$\begin{array}{c}\hfill {\left(a\left(y\right)\left(w^{\prime }\left(y\right)\right){}^{\mu }\right)}^{{}^{\prime }}+\sum _{i=1}^m{c}_i\left(y\right)F_i\left(v(y-q_i)\right)=0,w\left(y\right)=v\left(y\right)+b\left(y\right)v(y-p)\end{array}$$

due to $\left(A1\right)$. Hence proceeding as above, we find the same contradiction.

To proof necessary by contradiction, we suppose that $\left(A3\right)$ does not hold. Assume that

$$\begin{array}{c}\hfill {\int }_{y_0}^{\infty }{\left(a\left(s\right)\right)}^{-\frac{1}{\mu }}\mathrm{d}s=+\infty \end{array}$$

and due to our assumption $\left(A2\right)$, let

$$\begin{array}{c}\hfill {\int }_Y^{\infty }\sum _{i=1}^m{c}_i\left(y\right)F_i\left(\delta A(y-q_i)\right)\mathrm{d}y\le \delta ,\mathrm{for}\mathrm{all}\delta >0.\end{array}$$

In particular, we use a positive $ϵ$ such that ${\left(2\delta \right)}^{\frac{1}{\mu }}=(1-b_0)ϵ$ and $0<{\delta }^{\frac{1}{\mu }}\le \frac{(1-b_0)ϵ}{2^{\frac{1}{\mu }}}<ϵ$. We define a set of continuous function

$$\begin{array}{cc}\hfill S=\{u:u\in C([Y-r,+\infty ),\mathbb{R}),u(y)=0\mathrm{for}y\in [Y-r,Y)& \mathrm{and}{\delta }^{\frac{1}{\mu }}A\left(y\right)\le u\left(y\right)\le ϵA\left(y\right)\}\hfill \end{array}$$

and a mapping $\mathrm{\Phi }:S\to C\left(\right[Y-r,+\infty ),\mathbb{R})$ such that

$$\left(\mathrm{\Phi }u\right)\left(y\right)=\left\{\begin{array}{c}0,y\in [Y-r,Y)\hfill \\ -b\left(y\right)u(y-p)+{\int }_Y^y\frac{1}{{\left(a\left(v\right)\right)}^{\frac{1}{\mu }}}[\delta +{\int }_v^{\infty }\sum _{i=1}^m{c}_i\left(s\right)F_i{\left(u(s-q_i))\mathrm{d}s\right]}^{\frac{1}{\mu }}\mathrm{d}v,y\ge Y.\hfill \end{array}\right.$$

For every $u\in S$,

$$\begin{array}{cc}\hfill \left(\mathrm{\Phi }u\right)\left(y\right)& \ge {\int }_Y^y\frac{1}{{\left(a\left(v\right)\right)}^{\frac{1}{\mu }}}[\delta +{\int }_v^{\infty }\sum _{i=1}^m{c}_i\left(s\right)F_i{\left(u(s-q_i))\mathrm{d}s\right]}^{\frac{1}{\mu }}\mathrm{d}v\hfill \\ \hfill & \ge {\delta }^{\frac{1}{\mu }}{\int }_Y^y\frac{\mathrm{d}v}{{\left(a\left(v\right)\right)}^{\frac{1}{\mu }}}={\delta }^{\frac{1}{\mu }}A\left(y\right)\hfill \end{array}$$

and $u\left(y\right)\le ϵA\left(y\right)$ implies that

$$\begin{array}{cc}\hfill \left(\mathrm{\Phi }u\right)\left(y\right)& \le -b\left(y\right)u(y-p)+{\left(2\delta \right)}^{\frac{1}{\mu }}{\int }_Y^y\frac{\mathrm{d}v}{{\left(a\left(v\right)\right)}^{\frac{1}{\mu }}}\hfill \\ & \le b_0ϵA\left(y\right)+{\left(2\delta \right)}^{\frac{1}{\mu }}A\left(y\right)\hfill \\ & =b_0ϵA\left(y\right)+(1-b_0)ϵA\left(y\right)=ϵA\left(y\right)\hfill \end{array}$$

implies that $\left(\mathrm{\Phi }u\right)\left(y\right)\in S$. Define $v_n:[Y-r,+\infty )\to \mathbb{R}$ by the recurrence relation

$$\begin{array}{c}\hfill v_n\left(y\right)=\left(\mathrm{\Phi }{v}_{n-1}\right)\left(y\right),n\ge 1,\end{array}$$

with the initial condition

$$v_0\left(y\right)=\left\{\begin{array}{c}0,y\in [Y-r,Y)\hfill \\ {\delta }^{\frac{1}{\mu }}A\left(y\right),y\ge Y.\hfill \end{array}\right.$$

By the mathematical induction, it is easy to prove that

$$\begin{array}{c}\hfill {\delta }^{\frac{1}{\mu }}A\left(y\right)\le v_{n-1}\left(y\right)\le v_n\left(y\right)\le ϵA\left(y\right).\end{array}$$

for $y\ge Y$. Therefore for $y\ge Y-r$, ${lim}_{n\to \infty }{v}_n\left(y\right)$ exists. Let ${lim}_{n\to \infty }{v}_n\left(y\right)=v\left(y\right)$ for $y\ge Y-r$. By the Lebesgue’s dominated convergence Theorem $v\in S$ and $\left(\mathrm{\Phi }v\right)\left(y\right)=v\left(y\right)$, where $v\left(y\right)$ is a solution of Equation (1) such that $v\left(y\right)>0$. Hence, $\left(A3\right)$ is necessary. This completes the proof of the Theorem. □

Remark 1.

In the above theorem, the function $F_i$ could be sublinear, superlinear or linear.

Theorem 2.

Assume that (C1), (A1) and (A3) hold. Furthermore, assume that

$\left(A4\right)$

${\int }_Y^{\infty }\frac{1}{{\left(a\left(y\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_Y^y{\sum }_{i=1}^m{c}_i\left(s\right)F_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\right]}^{\frac{1}{\mu }}\mathrm{d}y=+\infty$for$Y,ϵ>0$where$A_1\left(y\right)={\int }_y^s\frac{\mathrm{d}\theta }{{\left(a\left(\theta \right)\right)}^{\frac{1}{\mu }}}$

and

$\left(A5\right)$

${\int }_Y^{\infty }{\sum }_{i=1}^m{c}_i\left(s\right)F_i\left(\delta \right)\mathrm{d}s=+\infty$for$\delta >0$

hold. Then every solution of Equation (1) either oscillates or converges to zero.

Proof. 

On the contrary, let u be an eventually positive solution of (1) that does not converge to zero. By Theorem 1, we have (5) for $y\ge y_1$. Thus, there exists $y_2>y_1$ such that $w^{\prime }\left(y\right)$ and $w\left(t\right)$ are of constant sign on $[y_2,\infty )$. Therefore, we have following two possibilities:

Case 1. Let $w\left(y\right)<0$ for $y\ge y_2$. By Case 1 of Theorem 1, we have that $u\left(y\right)$ is bounded. Consequently, w is bounded, and since w is monotonic, ${lim}_{y\to \infty }w\left(y\right)$ exists. As a result,

$$\begin{array}{cc}\hfill 0& \ge \underset{y\to \infty }{lim}w\left(y\right)=\underset{y\to \infty }{lim\; sup}w\left(y\right)\hfill \\ \hfill & \ge \underset{y\to \infty }{lim\; sup}\left(u\left(y\right)-b_{0}u(y-p)\right)\hfill \\ \hfill & \ge \underset{y\to \infty }{lim\; sup}u\left(y\right)+\underset{y\to \infty }{lim\; inf}\left(-b_{0}u(y-p)\right)\hfill \\ \hfill & =(1-b_0)\underset{y\to \infty }{lim\; sup}u\left(y\right)\hfill \end{array}$$

implies that ${lim\; sup}_{y\to \infty }u\left(y\right)=0$, because $1-b_0>0$ and thus ${lim}_{y\to \infty }u\left(y\right)=0$.

Case 2. Let $w\left(y\right)>0$ for $y\ge y_2$.

Sub-case $2_1$. Let $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }<0$ for $y\ge y_2$. Therefore, $w\left(y\right)$ is bounded and monotonic, hence ${lim}_{y\to \infty }w\left(y\right)$ exists. Therefore, for $s\ge y>y_2$, $a\left(s\right){\left(w^{\prime }\left(s\right)\right)}^{\mu }\le a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ implies that

$$\begin{array}{c}\hfill w^{\prime }\left(s\right)\le \frac{a{\left(y\right)}^{\frac{1}{\mu }}{w}^{\prime }\left(y\right)}{{\left(a\left(s\right)\right)}^{\frac{1}{\mu }}}.\end{array}$$

Since $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ is decreasing and $\mu$ is the quotient of positive odd integers, $a{\left(y\right)}^{1/\mu }\left(w^{\prime }\left(y\right)\right)$ is also decreasing. Then

$$\begin{array}{c}\hfill w\left(s\right)\le w\left(y\right)+{\left(a\left(y\right)\right)}^{\frac{1}{\mu }}{w}^{\prime }\left(y\right){\int }_y^s\frac{\mathrm{d}\theta }{{\left(a\left(\theta \right)\right)}^{\frac{1}{\mu }}}.\end{array}$$

Since, $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ is nonincreasing, we can find a constant $\delta >0$ such that $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\le -\delta$ for $y\ge y_2$. As a result, $w\left(s\right)\le w\left(y\right)-{\delta }^{\frac{1}{\mu }}{\int }_y^s\frac{d\theta }{{\left(a\left(\theta \right)\right)}^{\frac{1}{\mu }}}$ and hence $0\le w\left(y\right)-ϵA_1\left(y\right)$ for $y\ge y_2$, where $ϵ={\delta }^{\frac{1}{\mu }}$, $ϵ>0$. From (1), it is not difficult to see that

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{c}_i\left(y\right)F_i\left(ϵA_1(y-q_1)\right)\le 0.\end{array}$$

Integrating the last inequality from $y_2$ to $y(>y_2)$, we obtain

$$\begin{array}{c}\hfill {\left[a\left(s\right){\left(w^{\prime }\left(s\right)\right)}^{\mu }\right]}_{y_2}^y+{\int }_{y_2}^y\sum _{i=1}^m{c}_i\left(s\right)F_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\le 0,\end{array}$$

that is,

$$\begin{array}{c}\hfill {\int }_{y_2}^y\sum _{i=1}^m{c}_i\left(s\right)F_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\le -{\left[a\left(s\right){\left(w^{\prime }\left(s\right)\right)}^{\mu }\right]}_{y_2}^y\le -a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\end{array}$$

implies that

$$\begin{array}{c}\hfill \frac{1}{{\left(a\left(y\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_{y_2}^y\sum _{i=1}^m{c}_i\left(s\right)F_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\right]}^{\frac{1}{\mu }}\le -w^{\prime }\left(y\right)\end{array}$$

and further integration of the preceeding inequality, we have

$$\begin{array}{c}\hfill {\int }_Y^v\frac{1}{{\left(a\left(y\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_Y^y\sum _{i=1}^m{c}_i\left(s\right)F_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\right]}^{\frac{1}{\mu }}\mathrm{d}y\le -{\left[w\left(y\right)\right]}_Y^v\le w\left(Y\right)<+\infty \mathrm{as}v\to \infty .\end{array}$$

gives a contradiction to (A4).

Sub-case $2_2$. Let $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }>0$ for $y\ge y_2$. Since $w\left(y\right)$ is nondecreasing on $[y_2,\infty )$, there exist a constant $\delta >0$ and $Y>y_2$ such that $w\left(y\right)\ge \delta$ for $y\ge Y$. Therefore, (5) becomes

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{c}_i\left(y\right)F_i\left(\delta \right)\le 0.\end{array}$$

We integrate the inequality from Y to $+\infty$ and obtain

$$\begin{array}{c}\hfill {\int }_Y^{\infty }\sum _{i=1}^m{c}_i\left(s\right)F_i\left(\delta \right)\mathrm{d}s<+\infty ,\end{array}$$

which is a contradiction to (A5).

The case where u is eventually negative is very similar and we omit it here. Thus, the theorem is proved. □

Theorem 3.

Assume that (C2), (A1) and (A3) hold. Furthermore, assume that

$\left(A6\right)$

there exists$\lambda >0$such that${\sum }_{i=1}^m{F}_i\left(u\right)+{\sum }_{i=1}^m{F}_i\left(v\right)\ge \lambda {\sum }_{i=1}^m{F}_i(u+v)$for$u,v\in {\mathbb{R}}_{+}$,

$\left(A7\right)$

${\sum }_{i=1}^m{F}_i\left(uv\right)\le {\sum }_{i=1}^m{F}_i\left(u\right){\sum }_{i=1}^m{F}_i\left(v\right)$for$u,v\in {\mathbb{R}}_{+}$,

$\left(A8\right)$

${\int }_Y^{\infty }\frac{1}{{\left(a\left(\eta \right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_{y_3}^{\eta }{c}_n\left(\zeta \right){\sum }_{i=1}^m{F}_i\left(ϵA_1(\zeta -q_i)\right)\mathrm{d}\zeta \right]}^{\frac{1}{\mu }}\mathrm{d}\eta =+\infty$for$Y,y_3,\delta ,ϵ>0$

and

$\left(A9\right)$

${\int }_Y^{\infty }{c}_n\left(y\right)\mathrm{d}y=+\infty$, where$c_n\left(y\right)=min\{c_i\left(y\right),c_i(y-p)\}$

hold. Then every solution of (1) oscillates.

Proof. 

On the contrary, let $u\left(y\right)$ be a eventually positive solution of (1). Proceeding as in Theorem 1, we have that $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ is nonincreasing and constant sign on $[y_2,\infty )$. Since $w\left(y\right)$ is positive. Therefore, we have following two possible cases.

Case 1. Let $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }<0$ for $y\in [y_2,\infty )$. Ultimately, $u\left(y\right)$ is bounded. From the system (1) it is not difficult to see that

$$\begin{array}{cc}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }& +\sum _{i=1}^m{c}_i\left(y\right)F_i\left(u(y-q_i)\right)\hfill \\ & +\sum _{i=1}^m{F}_i\left(b_0\right)\left[{\left(a(y-p){\left(w^{\prime }(y-p)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{c}_i(y-p)F_i\left(u(y-p-q_i)\right)\right]=0.\hfill \end{array}$$

Using $\left(A6\right)$ and $\left(A7\right)$ in the above system, it follows that

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{F}_i\left(b_0\right){\left(a(y-p){\left(w^{\prime }(y-p)\right)}^{\mu }\right)}^{\prime }+\lambda c_n\left(y\right)\sum _{i=1}^m{F}_i\left(w(y-q_i)\right)\le 0.\end{array}$$

(7)

Since, $u\left(y\right)$ is bounded. By Theorem 2 Case $2_1$, we have that $w\left(y\right)\ge ϵA_1\left(y\right)$ for $y\ge y_2$. Therefore, (7) becomes

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{F}_i\left(b_0\right){\left(a(y-p){\left(w^{\prime }(y-p)\right)}^{\mu }\right)}^{\prime }+\lambda c_n\left(y\right)\sum _{i=1}^m{F}_i\left(ϵA_1(y-q_i)\right)\le 0\end{array}$$

for $y\ge y_3>y_2$. Integrating the last inequality from $y_3$ to $y(>y_3)$, we get

$$\begin{array}{c}\hfill {\left[a\left(s\right){\left(w^{\prime }\left(s\right)\right)}^{\mu }\right]}_{y_3}^y+\sum _{i=1}^m{F}_i\left(b_0\right){\left[a(s-p){\left(w^{\prime }(s-p)\right)}^{\mu }\right]}_{y_3}^y+\lambda {\int }_{y_3}^y{c}_n\left(s\right)\sum _{i=1}^m{F}_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\le 0,\end{array}$$

that is,

$$\begin{array}{cc}\hfill \lambda {\int }_{y_3}^y{c}_n\left(s\right)\sum _{i=1}^m{F}_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s& \le -{\left[a\left(s\right){\left(w^{\prime }\left(s\right)\right)}^{\mu }+\sum _{i=1}^m{F}_i\left(b_0\right)a(s-p){\left(w^{\prime }(s-p)\right)}^{\mu }\right]}_{y_3}^y\hfill \\ \hfill & \le -\left[a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }+\sum _{i=1}^m{F}_i\left(b_0\right)a(y-p){\left(w^{\prime }(y-p)\right)}^{\mu }\right]\hfill \\ \hfill & \le -\left(1+\sum _{i=1}^m{F}_i\left(b_0\right)\right)a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\left(\frac{\lambda }{(1+{\sum }_{i=1}^m{F}_i\left(b_0\right))}\right)}^{\frac{1}{\mu }}\frac{1}{{\left(a\left(y\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_{y_3}^y{c}_n\left(s\right)\sum _{i=1}^m{F}_i\left(ϵA_1(s-q_i)\right)\mathrm{d}s\right]}^{\frac{1}{\mu }}\le -w^{\prime }\left(y\right).\end{array}$$

Integrating the above inequality, we obtain

$$\begin{array}{c}\hfill {\left(\frac{\lambda }{(1+{\sum }_{i=1}^m{F}_i\left(b_0\right))}\right)}^{\frac{1}{\mu }}{\int }_Y^{\infty }\frac{1}{{\left(a\left(\eta \right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_{y_3}^{\eta }{c}_n\left(\zeta \right)\sum _{i=1}^m{F}_i\left(ϵA_1(\zeta -q_i)\right)\mathrm{d}\zeta \right]}^{\frac{1}{\mu }}\mathrm{d}\eta <\infty \end{array}$$

which is a contradiction to $\left(A8\right)$.

Case 2. Let $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }>0$ for $y\in [y_2,\infty )$. Then there exist a constant $\delta >0$ and $y_3>y_2$ such that $w\left(y\right)\ge \delta$ for $y\ge y_3$. From (7), it follows that

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{F}_i\left(b_0\right){\left(a(y-p){\left(w^{\prime }(y-p)\right)}^{\mu }\right)}^{\prime }+\lambda c_n\left(y\right)\sum _{i=1}^m{F}_i\left(\delta \right)\le 0.\end{array}$$

Integrating the last inequality from Y to $+\infty$, we get a contradiction to $\left(A9\right)$.

The case where u is an eventually negative solution is omitted since it can be dealt similarly. □

Theorem 4.

Assume that (C3), (A1) and (A3)–(A5) hold. Furthermore, assume that

$\left(A10\right)$

${\int }_Y^{\infty }{\sum }_{i=1}^m{c}_i\left(s\right)F_i\left(-b_1^{-1}\beta \right)\mathrm{d}s=\infty$for$\beta <0$

and

$\left(A11\right)$

${\int }_Y^{\infty }\frac{1}{{\left(a\left(s\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_Y^s{\sum }_{i=1}^m{c}_i\left(\zeta \right)F_i\left(-b_1^{-1}\beta \right)\mathrm{d}\zeta \right]}^{\frac{1}{\mu }}\mathrm{d}s=+\infty$for$\beta <0$

hold. Then every bounded solution of the system (1) either oscillates or converges to zero.

Proof. 

Let $u\left(y\right)$ be a bounded solution of the system (1). Proceeding as in Theorem 1, it follows that $w\left(y\right)$ and $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }$ are monotonic on $[y_2,\infty )$. Since $u\left(y\right)$ is bounded, then $w\left(y\right)$ is also bounded. Hence, ${lim}_{y\to \infty }w\left(y\right)$ exists. By Theorem 2, we get contradictions to (A4) and (A5) for cases $w\left(y\right)>0,a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }<0$ and $w\left(y\right)>0,a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }>0$ respectively. Therefore, we have the following two possible cases.

Case 1. Let $w\left(y\right)<0$, $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }>0$ for $y\ge y_2.$ We claim that ${lim}_{y\to \infty }w\left(y\right)=0$. If not, there exist $\beta <0$ and $y_3>y_2$ such that $w(y+p-q_i)<\beta$ for $y\ge y_3$. Hence, $w\left(y\right)\ge -b_{1}u(y-p)$ implies that $u(y-q_i)\ge -b_1^{-1}\beta$ for $y\ge y_3$. Consequently, Equation (5) reduces to

$$\begin{array}{c}\hfill {\left(a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }\right)}^{\prime }+\sum _{i=1}^m{F}_i\left(-b_1^{-1}\beta \right)c_i\left(y\right)\le 0\end{array}$$

(8)

for $y\ge y_3$. Integrating (8) from Y to $+\infty$, we get

$$\begin{array}{c}\hfill {\int }_Y^{\infty }\sum _{i=1}^m{c}_i\left(s\right)F_i\left(-b_1^{-1}\beta \right)\mathrm{d}s<\infty \end{array}$$

which contradicts to (A10). Therefore, our claim holds and

$$\begin{array}{cc}\hfill 0& =\underset{y\to \infty }{lim}w\left(y\right)=\underset{y\to \infty }{lim\; inf}\left(u\left(y\right)+b\left(y\right)u(y-p)\right)\hfill \\ \hfill & \le \underset{y\to \infty }{lim\; inf}(u\left(y\right)-b_{2}u(y-p))\hfill \\ \hfill & \le \underset{y\to \infty }{lim\; sup}u\left(y\right)+\underset{y\to \infty }{lim\; inf}\left(-b_{2}u(y-p)\right)\hfill \\ \hfill & =(1-b_2)\underset{y\to \infty }{lim\; sup}u\left(y\right)\hfill \end{array}$$

implies that ${lim\; sup}_{y\to \infty }u\left(y\right)=0$, because $1-b_2<0$. Thus, ${lim}_{y\to \infty }u\left(y\right)=0$.

Case 2. Let $w\left(y\right)<0$, $a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu }<0$ for $y\ge y_2$. Proceeding as in the previous case, we get (8). Integrating (8) from Y to y, we get

$$\begin{array}{c}\hfill {\int }_Y^y\sum _{i=1}^m{c}_i\left(s\right)F_i\left(-b_1^{-1}\beta \right)\mathrm{d}s\le -a\left(y\right){\left(w^{\prime }\left(y\right)\right)}^{\mu },\end{array}$$

that is,

$$\begin{array}{c}\hfill \frac{1}{{\left(a\left(y\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_Y^y\sum _{i=1}^m{c}_i\left(s\right)F_i\left(-b_1^{-1}\beta \right)\mathrm{d}s\right]}^{\frac{1}{\mu }}\le -w^{\prime }\left(y\right)\end{array}$$

for $y\ge Y$. Further integration the above inequality from Y to $+\infty$, we get

$$\begin{array}{c}\hfill {\int }_Y^{\infty }\frac{1}{{\left(a\left(s\right)\right)}^{\frac{1}{\mu }}}{\left[{\int }_Y^s\sum _{i=1}^m{c}_i\left(\zeta \right)F_i\left(-b_1^{-1}\beta \right)\mathrm{d}\zeta \right]}^{\frac{1}{\mu }}\mathrm{d}s<\infty \end{array}$$

which is a contradiction (A11). Thus, ${lim}_{y\to \infty }w\left(y\right)=0$. Rest of the case follows from Case 1. Hence, the proof of the Theorem is complete. □

We now present the following example:

Example

Consider the differential equation

$$\begin{array}{c}\hfill {\left(y^6{\left({\left(u\left(y\right)-\frac{1}{y}u(y-2)\right)}^{\prime }\right)}^3\right)}^{\prime }+\frac{1}{y^3}{F}_1\left(u\left(y\right)\right)+\frac{1}{{(y-1)}^5}{F}_2\left(u(y-1)\right)=0,y\ge 2,\end{array}$$

(9)

where $a\left(y\right)=y^6$, $b\left(y\right)=-\frac{1}{y}$, $p=2$, $c_i\left(y\right)={(y-i+1)}^{-(2i+1)}$, $F_i\left(y\right)=y^{2i+1}$, $\mu =3$, $q_i=i-1$, with the index $i=1,2.$ Here $F_i\left(y\right)$ is an odd function. Since, ${\int }_{y_0}^{\infty }\frac{1}{{\left(a\left(s\right)\right)}^{\frac{1}{\mu }}}=\frac{1}{y_0}<\infty$ and in order to verify (A3), we have

$$\begin{array}{cc}\hfill & {\int }_{y_0}^{\infty }\left(c_1\left(y\right)F_1\left(\delta A(y-q_1)\right)+c_2\left(y\right)F_2\left(\delta A(y-q_2)\right)\right)dy\hfill \\ & ={\int }_{y_0}^{\infty }\frac{1}{y^3}{\left(\frac{1}{Y}-\frac{1}{y}\right)}^{3}dy+{\int }_{y_0}^{\infty }\frac{1}{{(y-1)}^5}{\left(\frac{1}{(Y-1)}-\frac{1}{(y-1)}\right)}^{5}dy<\infty .\hfill \end{array}$$

Therefore, all the assumptions of Theorem 1 are verified. Hence, due to Theorem 1 every solution of (9) is oscillatory.

3. Conclusions

In this work, we obtained necessary and sufficient conditions for the oscillation of a second-order non-linear neutral differential equation with multiple delays (1) under the assumption (A3) when $-1<b\left(y\right)<0$. However, we failed to obtain the necessary and sufficient conditions for the oscillation of the solution of (1) for the other ranges of the neutral coefficient $b\left(y\right)$.