Nonoscillation and Oscillation Criteria for a Class of Second-Order Nonlinear Neutral Delay Differential Equations with Positive and Negative Coefficients

Abstract

In this paper, we investigate some nonoscillatory and oscillatory solutions for a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. By means of the method of contraction mapping principle and some integral inequality techniques, we extend the recent results provided in the literature.

1. Introduction

Only recently, some scholars ([1,2]) studied the oscillation of the following mixed-type second-order equation:

$${\left(r\left(x\right){\left(y^{\prime }\left(x\right)\right)}^{\alpha }\right)}^{\prime }=p\left(x\right)y^{\alpha }\left(\tau \left(x\right)\right),x\ge 0,$$

where $p,r\in C([x_0,\infty ),(0,\infty ))$, $\alpha$ is the ratio of two positive odd integers, $\underset{x\to \infty }{lim}\tau \left(x\right)=\infty$, $\tau \left(x\right)\in C^1([x_0,\infty ),\Re )$ and ${\tau }^{\prime }\left(x\right)>0$.

The researchers ([3,4]) studied the oscillation of the following second-order half-linear neutral delay differential equation:

$${\left(r\left(x\right){(y\left(x\right)+p\left(x\right)y\left(\tau \left(x\right)\right))}^{\alpha }\right)}^{\prime }{)}^{\prime }+q\left(x\right)y^{\alpha }\left(\sigma \left(x\right)\right)=0,x\ge x_0>0,$$

where $\alpha$ is the ratio of two positive odd integers, $r,p\in C^1([x_0,\infty ),(0,\infty ))$, $q\in C([x_0,\infty ),\Re )$, $\tau ,\sigma \in C([x_0,\infty ),\Re )$, $\tau \left(x\right)\le x$, $\sigma \left(x\right)\le x$ and $\underset{x\to \infty }{lim}\tau \left(x\right)=\underset{x\to \infty }{lim}\sigma \left(x\right)=\infty$.

Baculíková et al. [5] considered the oscillation of the following second-order delay differential equation:

$${\left(a\left(x\right){(y\left(x\right)-p\left(x\right)y^{\alpha }\left(\tau \left(x\right)\right))}^{\prime }\right)}^{\prime }+q\left(x\right)y^{\beta }\left(\sigma \left(x\right)\right)=0,x\ge x_0>0,$$

where $0<\alpha \le 1$, $\alpha$ and $\beta$ are the ratio of two positive odd integers, $a\in C^1([x_0,\infty ),(0,\infty ))$, $p,q\in C([x_0,\infty ),(0,\infty ))$, $0<p\left(x\right)\le p<1$, $\tau ,\sigma \in C^1([x_0,\infty ),(0,\infty ))$, $\tau \left(x\right)\le x$, $\sigma \left(x\right)\le x$, ${\tau }^{\prime }\left(x\right)>0$, ${\sigma }^{\prime }\left(x\right)>0$ and $\underset{x\to \infty }{lim}\tau \left(x\right)=\underset{x\to \infty }{lim}\sigma \left(x\right)=\infty$.

Oscillation phenomena take part in delay differential equations from real world applications. We refer the reader to [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] (where oscillation and/or delay situations take part in models from mathematical biology and physics when their formulation includes cross-diffusion terms) and the references cited therein.

Thus, many scholars were concerned about the second-order equation with positive and negative coefficients. In [24], Lin et al. studied the following equation:

$${\left[a\left(x\right){(y\left(x\right)+p\left(x\right)y(x-\tau ))}^{\prime }\right]}^{\prime }+q\left(x\right)G\left(y(x-\delta )\right)-r\left(x\right)H\left(y(x-\sigma )\right)=0,$$

(1)

where $x\ge x_0$, $\tau \in (0,\infty )$, $\delta ,\sigma \in [0,\infty )$, $p,q,r\in C([x_0,\infty ),\Re )$ and $G,H\in C(\Re ,\Re )$, and $a\left(x\right),p\left(x\right),q\left(x\right),r\left(x\right),G\left(x\right)$ and $H\left(x\right)$ satisfy some of the following assumptions.

Assumption 1 (c1). 

G and H satisfy local Lipschitz condition, and $uG\left(u\right)>0$, $uH\left(u\right)>0$ for $u\ne 0$.

Assumption 2 (c2). 

$a\left(x\right)>0$, $q\left(x\right),r\left(x\right)\ge 0$, ${\int }^{\infty }A\left(x\right)q\left(x\right)dx<\infty$, ${\int }^{\infty }A\left(x\right)r\left(x\right)dx<\infty$, where $A\left(x\right)={\int }_{x_0}^x\frac{1}{a\left(s\right)}ds$.

Assumption 3 (c3). 

$mq\left(x\right)-r\left(x\right)$ is eventually non-negative for every $m>0$.

Assumption 4 (c4). 

$\left|p\left(x\right)\right|\le p_0<\frac{1}{2}$ eventually.

Assumption 5 (c5). 

$p\left(x\right)\ge 0$ eventually, and $0<p_1<1$; $p\left(x\right)\le 0$ eventually, and $-1<p_2<0$, where $p_1=\underset{x\to \infty }{lim\; sup}p\left(x\right)$, $p_2=\underset{x\to \infty }{lim\; inf}p\left(x\right)$.

Assumption 6 (c6). 

$p\left(x\right)>1$ eventually, and $1<p_2<p_1<p_2^2<+\infty$; $p\left(x\right)<-1$ eventually, and $-\infty <p_2<p_1<-1$, where $p_1$ and $p_2$ are the same as that in (c5).

By using the contraction mapping principle, they obtained the existence of nonoscillatory solutions of (1) when (c1)–(c3), (c4) (or (c1)–(c3), (c5) or (c1)–(c3) and (c6)) hold.

In [25], Zhang et al. extended the results of [24] to the case $p\left(x\right)=1$ and indicated that the condition (c3) is redundant.

When $a\left(x\right)\equiv 1$, we extend the number of neutral terms and positive and negative coefficient terms from single to multiple, and then we obtain the following equation:

$${\left[y\left(x\right)+\eta \sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)\right]}^{″}+\sum _{j=1}^m{q}_j\left(x\right)G\left(y(x-{\delta }_j)\right)-\sum _{k=1}^n{r}_k\left(x\right)H\left(y(x-{\sigma }_k)\right)=0,$$

(2)

where $x\ge x_0$, $\eta =\pm 1$, $l,m,n\in N$, $p_i\left(x\right)(i=1,\cdots ,l)\in C^2([x_0,\infty ),\Re )$, $q_j\left(x\right)$$(j=1,\cdots ,m)$ and $r_k\left(x\right)(k=1,\cdots ,n)\in C([x_0,\infty ),\Re )$, $G,H\in C([x_0,\infty ),\Re )$ and $G\left(v\right)=H\left(v\right)=0$ for $v=0$.

When we consider (2), some of the following five assumptions are satisfied.

Assumption 7 (H1). 

$0<h_1\le q_j\left(x\right)\le h_2$, $j=1,2,\cdots ,m$;

Assumption 8 (H2). 

Set the following values:

$$A:=\{y\in X:M_2\le y\left(x\right)\le M_1,x\ge x_0\},$$

where X denotes the set which includes all continuous and bounded functions on $[x_0,\infty )$ with the sup norm, $M_1>1$ and $M_2>0$. Let G and H satisfy Lipschitz conditions in A; that is, for any $y_1,y_2\in A$, there exist $L_1,L_2>0$ such that the following is the case.

$$|G\left(y_1\right)-G\left(y_2\right)|\le L_1|y_1-y_2|,$$

$$|H\left(y_1\right)-H\left(y_2\right)|\le L_2|y_1-y_2|;$$

Assumption 9 (H3). 

$0<m_1\le \frac{G\left(u\right)}{u}\le m_2$ and $0<N_1\le \frac{H\left(u\right)}{u}\le N_2$ for $u\ne 0$.

Assumption 10 (H4). 

${\sum }_{k=1}^n{\int }_{x_0}^{+\infty }x{r}_k\left(x\right)dx<\infty$.

Assumption 11 (H5). 

${\sum }_{j=1}^m{\int }_{x_0}^{+\infty }x{q}_j\left(x\right)dx<\infty .$

Let $\kappa :=max\{{\tau }_1,{\tau }_2,\cdots ,{\tau }_l,{\delta }_1,{\delta }_2,\cdots ,{\delta }_m$, ${\sigma }_1,{\sigma }_2,\cdots ,{\sigma }_n\}$.

Definition 1.

A function y is called a solution of (2) on the interval $I=[x_0,\infty )$, if y is continuous, $y\left(x\right)+\eta {\sum }_{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)$ is continuously differentiable and y satisfies (2) on $x\in I$.

We only consider the nontrivial solution of (2), which satisfies $sup\left\{\right|y\left(x\right)|:x\ge X\}>0$ for all $X\ge x_0$.

Definition 2.

A nontrivial solution of (2) is nonoscillatory if it is eventually positive or eventually negative. Otherwise, it is oscillatory.

Motivated by the useful work of Lin et al. and Zhang et al., in this paper, we obtain some new conditions of the existence of nonoscillatory solutions of the Equation (2).

Recently, the scholars ([26,27,28,29,30,31,32,33]) investigated the oscillatory properties of Equations (1) and (2). When $a\left(x\right)\equiv 1$, in [31], Thandapani et al. obtained that every solution of (1) is oscillatory if the following assumptions are satisfied.

Assumption 12 (B0). 

$\tau ,\delta$ and σ are nonnegative constants with $\delta \ge \sigma \ge \tau$;

Assumption 13 (B1). 

There exist $\alpha \ge 1$ and a positive constant $M_1$ such that $\frac{G\left(v\right)}{v^{\alpha }}\le M_1$ for $v\ne 0$.

Assumption 14 (B2). 

There exist $M_2,M>0$ such that $0\le \frac{H\left(v\right)}{v}\le M_2$ and $0\le \frac{G\left(v\right)}{H\left(v\right)}\le M$ for $v\ne 0$.

Assumption 15 (B3). 

$p\left(x\right)$ is bounded.

Assumption 16 (B4). 

${\int }_{x_0}^{\infty }{\int }_{u-\delta +\sigma }^{u}r\left(v\right)dvdu<\infty$.

Assumption 17 (B5). 

There exists a constant k such that $q\left(x\right)-Mr(x-\delta +\sigma )\ge k>0$ for all $x\ge x_0$.

In [28,29,33], the authors established some criteria that ensured that every solution of (1) with $G\equiv H$ is oscillatory. In particular, some authors ([26,27,32]) considered the oscillatory and asymptotic behavior of Equation (2) with $G\left(v\right)=H\left(v\right)=v$.

The above research has greatly stimulated our interest. Thus, in this article, we investigate the oscillatory behavior of the Equation (2) under some assumptions that are different from the previous ones.

Under some new assumptions (i.e., (c3) is not needed and we replace assumption (c2) with (H4) and (H5); (B0), (B1) and (B5) are not required and we provide assumption (H4) instead of (B4)), we study the second-order nonlinear delay differential equation with multiple neutral terms and positive and negative coefficients terms. Motivated by the above research, we obtain some new conditions of the existence of nonoscillatory solution of (2) by using the contraction mapping principle, and we obtain some criteria that ensure the oscillation of bounded solutions of Equation (2) by utilizing the integral inequality technique. Our results extend the research work in this field.

2. Nonoscillatory Solution

When $\eta =1$, (2) becomes the following.

$${\left[y\left(x\right)+\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)\right]}^{″}+\sum _{j=1}^m{q}_j\left(x\right)G\left(y(x-{\delta }_j)\right)-\sum _{k=1}^n{r}_k\left(x\right)H\left(y(x-{\sigma }_k)\right)=0,x\ge x_0.$$

(3)

When $\eta =-1$, (2) becomes the following.

$${\left[y\left(x\right)-\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)\right]}^{″}+\sum _{j=1}^m{q}_j\left(x\right)G\left(y(x-{\delta }_j)\right)-\sum _{k=1}^n{r}_k\left(x\right)H\left(y(x-{\sigma }_k)\right)=0,x\ge x_0.$$

(4)

In this section, we investigate the existence of the nonoscillatory solution of Equations (3) and (4).

Lemma 1.

Suppose $\left(H2\right)$, $\left(H4\right)$ and $\left(H5\right)$ hold. If $p_i\left(x\right)$ satisfies the following:

$$0<\sum _{i=1}^l{p}_i\left(x\right)\le p<1,$$

(5)

then (3) has a nonoscillatory solution.

Proof. 

It is easy to verify that if (H2) holds, then (H3) holds. Let $L=max\{L_1,L_2\}$, ${\alpha }_1=max\{G\left(y\right):y\in A\}$ and ${\alpha }_2=max\{H\left(y\right):y\in A\}$. According to (H5) and (H4), we have the following:

$$0<\sum _{j=1}^m{\int }_{x_0}^{+\infty }(s-x_0)q_j\left(s\right)ds<\sum _{j=1}^m{\int }_{x_0}^{+\infty }s{q}_j\left(s\right)ds<\infty$$

(6)

and the following is obtained.

$$0<\sum _{k=1}^n{\int }_{x_0}^{+\infty }(s-x_0)r_k\left(s\right)ds<\sum _{k=1}^n{\int }_{x_0}^{+\infty }s{r}_k\left(s\right)ds<\infty .$$

(7)

By (6) and (7), we obtain the following.

$$\underset{x_0\to \infty }{lim}\sum _{j=1}^m{\int }_{x_0}^{+\infty }(s-x_0)q_j\left(s\right)ds=0$$

$$\underset{x_0\to \infty }{lim}\sum _{k=1}^n{\int }_{x_0}^{+\infty }(s-x_0)r_k\left(s\right)ds=0.$$

Thus, we can choose a sufficiently large x such that the following is the case.

$$\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)ds+\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)ds<\frac{1-p}{2L},$$

(8)

$$\sum _{j=1}^m{\int }_x^{+\infty }{\alpha }_1(s-x)q_j\left(s\right)ds<1-p{M}_1-M_2,$$

(9)

$$\sum _{k=1}^n{\int }_x^{+\infty }{\alpha }_2(s-x)r_k\left(s\right)ds<M_1-1.$$

(10)

Set the following.

$$z\left(x\right):=y\left(x\right)+\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i),$$

(11)

$$w\left(x\right):=-\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)G\left(y(s-{\delta }_j)\right)ds$$

$$u\left(x\right):=\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)H\left(y(s-{\sigma }_k)\right)ds.$$

(12)

Define a mapping $T:A\to X$ with the following.

$$\begin{array}{cc}\hfill \left(Ty\right)\left(x\right):=& 1-\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)-\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)G\left(y(s-{\delta }_j)\right)ds\hfill \\ \hfill & +\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)H\left(y(s-{\sigma }_k)\right)ds.\hfill \end{array}$$

Clearly, $Ty$ is continuous. For every $y\in A$ and $x\ge x_1$, from (10), we obtain the following.

$$\begin{array}{cc}\hfill \left(Ty\right)\left(x\right)& \le 1+\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)H\left(y(s-{\sigma }_k)\right)ds\hfill \\ \hfill & \le 1+{\alpha }_2\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)ds\le M_1.\hfill \end{array}$$

From (9), we have the following.

$$\begin{array}{cc}\hfill \left(Ty\right)\left(x\right)& \ge 1-p{M}_1-\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)G\left(y(s-{\delta }_j)\right)ds\hfill \\ \hfill & \ge 1-p{M}_1-{\alpha }_1\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)ds\ge M_2.\hfill \end{array}$$

Thus, $TA\subset A$. We claim that T is a contraction mapping on A. Indeed, for any $y_1,y_2\in A$ and $x\ge x_1$, by (8), we have the following.

$$\begin{array}{cc}\hfill |\left(T{y}_1\right)\left(x\right)-& \left(T{y}_2\right)\left(x\right)|\le |p(y_1(x-{\tau }_i)-y_2(x-{\tau }_i))|\hfill \\ \hfill & +L_1\sum _{j=1}^m{\int }_x^{\infty }(s-x)q_j\left(s\right)\left|(y_1(s-{\delta }_j)-y_2(s-{\delta }_j))\right|ds\hfill \\ \hfill & +L_2\sum _{k=1}^n{\int }_x^{\infty }(s-x)r_k\left(s\right)\left|(y_1(s-{\sigma }_k)-y_2(s-{\sigma }_k))\right|ds\hfill \\ \hfill & \le \parallel y_1-y_2\parallel \left\{p+L\left(\sum _{j=1}^m{\int }_x^{\infty }(s-x)q_j\left(s\right)ds+\sum _{k=1}^n{\int }_x^{\infty }(s-x)r_k\left(s\right)ds\right)\right\}\hfill \\ \hfill & <\frac{p+1}{2}\parallel y_1-y_2\parallel .\hfill \end{array}$$

By taking the sup norm of the above inequality, we have the following.

$$\parallel T{y}_1-T{y}_2\parallel <\frac{p+1}{2}\parallel y_1-y_2\parallel .$$

Because of (5), we obtain $\frac{p+1}{2}<1$. Refer to a similar proof of ([24], Theorem 2), and we know that T has a fixed point $y*$. □

Lemma 2.

If the assumptions of Lemma 1 are satisfied, then Equation (4) has a nonoscillatory solution.

Proof. 

Similarly to the proof of Lemma 1, according to (H4) and (H5), we have (6), and (7) holds. Thus, we choose a sufficiently large x such that (8) the following is the case:

$$\sum _{j=1}^m{\int }_x^{+\infty }{\alpha }_1(s-x)q_j\left(s\right)ds<1-M_2,$$

(13)

and

$$\sum _{k=1}^n{\int }_x^{+\infty }{\alpha }_2(s-x)r_k\left(s\right)ds<(1-p)M_1-1,$$

(14)

hold. Similarly, we define the mapping $T_1:A\to X$ as follows.

$$\begin{array}{cc}\hfill \left(T_{1}y\right)\left(x\right):=& 1+\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)-\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)G\left(y(s-{\delta }_j)\right)ds\hfill \\ \hfill & +\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)H\left(y(s-{\sigma }_k)\right)ds.\hfill \end{array}$$

Obviously, $T_{1}y$ is continuous. For any $y\in A$ and $x\ge x_2$, by (14), we have the following.

$$\begin{array}{cc}\hfill \left(T_{1}y\right)\left(x\right)& \le 1+\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i)+\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)H\left(y(s-{\sigma }_k)\right)ds\hfill \\ \hfill & \le 1+p{M}_1+{\alpha }_2\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)ds\le M_1.\hfill \end{array}$$

By (13), we have the following.

$$\begin{array}{cc}\hfill \left(T_{1}y\right)\left(x\right)& \ge 1-\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)G\left(y(s-{\delta }_j)\right)ds\hfill \\ \hfill & \ge 1-{\alpha }_1\sum _{j=1}^m{\int }_x^{+\infty }(s-x)q_j\left(s\right)ds\ge M_2.\hfill \end{array}$$

Thus, $T_{1}A\subset A$. Next, we prove that $T_1$ is a contraction mapping on A. For any $y_1,y_2\in A$ and $x\ge x_2$, by (8), we have the following.

$$\begin{array}{cc}\hfill |\left(T_1{y}_1\right)\left(x\right)-& \left(T_1{y}_2\right)\left(x\right)|\le |p(y_1(x-{\tau }_i)-y_2(x-{\tau }_i))|\hfill \\ \hfill & +L_1\sum _{j=1}^m{\int }_x^{\infty }(s-x)q_j\left(s\right)\left|(y_1(s-{\delta }_j)-y_2(s-{\delta }_j))\right|ds\hfill \\ \hfill & +L_2\sum _{k=1}^n{\int }_x^{\infty }(s-x)r_k\left(s\right)\left|(y_1(s-{\sigma }_k)-y_2(s-{\sigma }_k))\right|ds\hfill \\ \hfill & \le \parallel y_1-y_2\parallel \left\{p+L\left(\sum _{j=1}^m{\int }_x^{\infty }(s-x)q_j\left(s\right)ds+\sum _{k=1}^n{\int }_x^{\infty }(s-x)r_k\left(s\right)ds\right)\right\}\hfill \\ \hfill & <\frac{p+1}{2}\parallel y_1-y_2\parallel .\hfill \end{array}$$

Refer to a similar proof of ([24], Theorem 2), we obtain that $T_1$ has a fixed point $y*$. □

Theorem 1.

Suppose $\left(H2\right)$, $\left(H4\right)$ and $\left(H5\right)$ hold. If $p_i\left(x\right)$ satisfies (5), then (2) has a nonoscillatory solution.

Proof. 

According to Lemmas 1 and 2, we obtain (2), which has a nonoscillatory solution. □

Example 1.

Consider the following equation.

$$\begin{array}{cc}\hfill {\left[y\left(x\right)+\frac{1}{8}y(x-1)\right]}^{″}& -e^{-\frac{x}{2}}\left(1-\frac{1}{e^{-x}+2}\right)\frac{y\left(\frac{x}{2}\right)[y^2\left(\frac{x}{2}\right)+2]}{y^2\left(\frac{x}{2}\right)+1}\hfill \\ \hfill & -\frac{1}{8}{e}^{-\frac{x-1}{2}}\left(1-\frac{2}{e^{-x+1}+3}\right)\frac{y\left(\frac{x-1}{2}\right)[y^2\left(\frac{x-1}{2}\right)+3]}{y^2\left(\frac{x-1}{2}\right)+1}=0.\hfill \end{array}$$

(15)

Here, we have the following.

$$l=1,p_1\left(x\right)=\frac{1}{8},$$

$$q\left(x\right)=0,$$

$$n=2,r_1\left(x\right)=e^{-\frac{x}{2}}\left(1-\frac{1}{e^{-x}+2}\right),r_2\left(x\right)=\frac{1}{8}{e}^{-\frac{x-1}{2}}\left(1-\frac{2}{e^{-x+1}+3}\right).$$

It is easy to verify that $p_1\left(x\right)$ and $q\left(x\right)$ satisfy (5) and (H5), respectively, and $r_1\left(x\right)$ and $r_2\left(x\right)$ satisfy (H4). Therefore, by Theorem 1, $y\left(x\right)=e^{-x}$ is a nonoscillatory solution of (15).

Example 2.

Consider the following equation:

$$\begin{array}{cc}\hfill {\left[y\left(x\right)-\frac{3}{4}y(x-2)\right]}^{″}& +\frac{3}{4}{e}^{-\frac{x-2}{2}}\left(1-\frac{2}{e^{-x+2}+3}\right)\frac{y\left(\frac{x-2}{2}\right)[y^2\left(\frac{x-2}{2}\right)+3]}{y^2\left(\frac{x-2}{2}\right)+1}\hfill \\ \hfill & -e^{-\frac{x}{2}}\left(1-\frac{1}{e^{-x}+2}\right)\frac{y\left(\frac{x}{2}\right)[y^2\left(\frac{x}{2}\right)+2]}{y^2\left(\frac{x}{2}\right)+1}=0,\hfill \end{array}$$

(16)

and we have the following.

$$l=1,p_1\left(x\right)=\frac{3}{4},$$

$$m=1,q_1\left(x\right)=\frac{3}{4}{e}^{-\frac{x-2}{2}}\left(1-\frac{2}{e^{-x+2}+3}\right),$$

$$n=1,r_1\left(x\right)=e^{-\frac{x}{2}}\left(1-\frac{1}{e^{-x}+2}\right).$$

It is easily verified that $p_1\left(x\right)$, $q_1\left(x\right)$ and $r_1\left(x\right)$ satisfy (5), (H5) and (H4), respectively. By Theorem 1, (16) has a nonoscillatory $y\left(x\right)=e^{-x}$.

3. Oscillatory Criteria

In this section, the oscillation criteria of (2) will be given, and some examples will be illustrated to demonstrate the results.

Lemma 3.

Suppose that (H1), (H3) and (H4) hold and $p_i\left(x\right)\in C([x_0,\infty ),{\Re }^{+})$ is bounded. If the bounded solution $y\left(x\right)$ of (3) satisfies $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$, then $y\left(x\right)$ is oscillatory.

Proof. 

Suppose toward a contradiction, there is no loss of generality in assuming that y is an eventually a positive-bounded solution of (3). Thus, there exists $x_1\ge x_0+\kappa$ such that $y(x-\kappa )>0$ for $x\ge x_1$. Furthermore, there exists $K>0$ such that $y\left(x\right)\le K$ for $x\ge x_1$. From (7), we may choose a sufficiently large $x>x_2\ge x_1+\kappa$, such that the following is the case.

$$\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)ds<\frac{1}{2N_2}.$$

(17)

Let the following be the case:

$$w\left(x\right)=z\left(x\right)-u\left(x\right),$$

(18)

where $z\left(x\right)$ and $u\left(x\right)$ are defined in (11) and (12), respectively. Then, we have the following.

$$w^{″}\left(x\right)=-\sum _{j=1}^m{q}_j\left(x\right)G\left(y(x-{\delta }_j)\right).$$

(19)

From (H1) and (H3), we obtain $w^{″}\left(x\right)<0$. Hence, $w^{\prime }\left(x\right)>0$ or $w^{\prime }\left(x\right)<0$ for all $x\ge x_3\ge x_2+\kappa$, and $x_3$ is sufficiently large.

If $w^{\prime }\left(x\right)<0$ for all $x\ge x_3$, then the following is the case.

$$\underset{x\to \infty }{lim}w\left(x\right)=-\infty .$$

(20)

According to (3), (17), (18) and (H3), we obtain the following:

$$w\left(x\right)\ge -N_{2}K\sum _{k=1}^n{\int }_x^{+\infty }(s-x)r_k\left(s\right)ds\ge -\frac{K}{2}>-\infty ,$$

which contradicts (20). Thus, $w\left(x\right)$ is increasing for all $x\ge x_3$. From (H1), $q_j\left(x\right)\ge h_1$, $j=1,\cdots ,m$ for all $x\ge x_3$. Integrating (19) from $x_3$ to $+\infty$, we obtain the following.

$$\infty >w^{\prime }\left(x_3\right)\ge \sum _{j=1}^m{\int }_{x_3}^{+\infty }{q}_j\left(x\right)G\left(y(x-{\delta }_j)\right)dx\ge h_1{m}_1\sum _{j=1}^m{\int }_{x_3}^{+\infty }y(x-{\delta }_j)dx.$$

Therefore, $y\in L^1\left([x_3,\infty )\right)$, which contradicts $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$. The proof is complete. □

Lemma 4.

If the assumptions of Lemma 3 are satisfied and if the bounded solution $y\left(x\right)$ of Equation (4) satisfies $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$, then $y\left(x\right)$ is oscillatory.

Proof. 

Just as in the proof of Lemma 3, assume that y is an eventually positive-bounded solution of (4). Since $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$, $0<y(x-\kappa )<K$ for $x\ge x_1\ge x_0+\kappa$, where $K>0$.

Define the following.

$$Y\left(x\right)=y\left(x\right)-\sum _{i=1}^l{p}_i\left(x\right)y(x-{\tau }_i).$$

$$w_1\left(x\right)=Y\left(x\right)-u\left(x\right).$$

(21)

Then, the following is the case.

$$\begin{array}{cc}\hfill w_1^{″}\left(x\right)& =Y^{″}\left(x\right)-u^{″}\left(x\right)\hfill \\ \hfill & =-\sum _{j=1}^m{q}_j\left(x\right)G\left(y(x-{\delta }_j)\right)<0,x\ge x_2\ge x_1+\kappa .\hfill \end{array}$$

(22)

Hence, $w_1^{\prime }\left(x\right)>0$ or $w_1^{\prime }\left(x\right)<0$ for all $x\ge x_3$, where $x_3\ge x_2+\kappa$ is sufficiently large.

If $w_1^{\prime }\left(x\right)<0$ for all $x\ge x_3$, we have the following.

$$\underset{x\to \infty }{lim}{w}_1\left(x\right)=-\infty .$$

(23)

Because $p_i\left(x\right)$ is bounded, then we have the following:

$$p_i\left(x\right)\le p_i,i=1,\cdots ,l,\sum _{i=1}^l{p}_i<P$$

(24)

where $p_i$ and P are non-negative constants. By means of (17), (21), (24) and (H3), we have the following:

$$w_1\left(x\right)\ge -\left[\sum _{i=1}^l{p}_i\left(x\right)+N_2\sum _{k=1}^n{\int }_{x_2}^{+\infty }(s-x)r_k\left(s\right)ds\right]K\ge -\left(P+\frac{1}{2}\right)K>-\infty ,$$

which contradicts (23). Hence, $w_1\left(x\right)$ is increasing for all $x\ge x_3$. Following the same method as in Lemma 3, from (H1) and inequality (22), we obtain $y\in L^1\left([x_3,\infty )\right)$, which contradicts $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$. The proof is complete. □

Theorem 2.

Suppose that (H1), (H3) and (H4) hold and $p_i\left(x\right)\in C([x_0,\infty ),{\Re }^{+})$ is bounded. If the bounded solution $y\left(x\right)$ of (2) satisfies $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$, then $y\left(x\right)$ is oscillatory.

Proof. 

According to Lemmas 3 and 4, we obtain that if the bounded solution $y\left(x\right)$ of (2) satisfies $\underset{x\to \infty }{lim}y\left(x\right)\ne 0$, then $y\left(x\right)$ is oscillatory. □

Example 3.

Consider the following equation.

$$\begin{array}{cc}\hfill & {\left[y\left(x\right)+y(x-\frac{3\pi }{2})\right]}^{\prime \prime }+(e^{-x}+1)\left(1-\frac{1}{{sin}^{2}x+2}\right)\frac{y(x-2\pi )[y^2(x-2\pi )+2]}{y^2(x-2\pi )+1}\hfill \\ \hfill & +\left(1-\frac{1}{{cos}^{2}x+2}\right)\frac{y(x-\frac{3\pi }{2})[y^2(x-\frac{3\pi }{2})+2]}{y^2(x-\frac{3\pi }{2})+1}\hfill \\ \hfill & -e^{-x}\left(1-\frac{2}{{sin}^{4}x+3}\right)\frac{y(x-2\pi )[y^4(x-2\pi )+3]}{y^4(x-2\pi )+1}=0.\hfill \end{array}$$

(25)

Here, we have the following.

$$l=1,p_1\left(x\right)=1,$$

$$m=2,q_1\left(x\right)=(e^{-x}+1)\left(1-\frac{1}{{sin}^{2}x+2}\right),q_2\left(x\right)=\left(1-\frac{1}{{cos}^{2}x+2}\right),$$

$$n=1,r_1\left(x\right)=e^{-x}\left(1-\frac{2}{{sin}^{4}x+3}\right).$$

It is easy to verify that $\frac{1}{2}\le q_1\left(x\right)\le \frac{4}{3}$, $\frac{1}{2}\le q_2\left(x\right)\le \frac{2}{3}$ and ${\int }_{x_0}^{+\infty }(s-x)r_1\left(s\right)ds<\infty$. Therefore, according to Theorem 2, we know that every bounded solution of (2) that does not tend to zero is oscillatory. Indeed, $y\left(x\right)=sinx$ is a bounded oscillatory solution of (25).

Example 4.

Consider the following equation.

$$\begin{array}{cc}\hfill & {\left[y\left(x\right)-\frac{1}{2^3}y(x-\pi )-\frac{1}{2^3}y(x-2\pi )\right]}^{″}\hfill \\ \hfill & +(e^{-x}+1)\left(1-\frac{1}{{sin}^{2}x+2}\right)\frac{y(x-2\pi )[y^2(x-2\pi )+2]}{y^2(x-2\pi )+1}\hfill \\ \hfill & -e^{-x}\left(1-\frac{2}{si{n}^{4}x+3}\right)\frac{y(x-2\pi )[y^4(x-2\pi )+3]}{y^4(x-2\pi )+1}=0.\hfill \end{array}$$

(26)

We have the following.

$$l=2,p_1\left(x\right)=p_2\left(x\right)=\frac{1}{2^3},$$

$$m=1,q_1\left(x\right)=(e^{-x}+1)\left(1-\frac{1}{{sin}^{2}x+2}\right),$$

$$n=1,r_1\left(x\right)=e^{-x}\left(1-\frac{2}{{sin}^{4}x+3}\right).$$

It is clear that ${\sum }_{i=1}^2{p}_i\left(x\right)\le 1$, $\frac{1}{2}\le q_1\left(x\right)\le \frac{4}{3}$ and ${\int }_{x_0}^{+\infty }(s-x)r_1\left(s\right)ds<\infty$. Therefore, according to Theorem 2, we know that every bounded solution of (2) that does not tend to zero is oscillatory. Indeed, $y\left(x\right)=sinx$ is a bounded oscillatory solution of (26).

4. Remark

Comparing with the results of [24,25,26,27,28,29,31,32,33], we increased the number of the positive and negative coefficient terms and the neutral terms of the second-order delay differential equation with positive and negative coefficients from single to multiple and generalized the equation from a linear case to a nonlinear case.

Motivated by the useful work of Lin et al. and Zhang et al. ([24,25]), we provide some new conditions under which Equation (2) has a nonoscillatory solution. More precisely, (c3) is not needed and we replace assumption (c2) with (H4) and (H5).

For the oscillation of Equation (2), we present some assumptions that are different from those in [31], i.e., (B0), (B1) and (B5) are not necessary and we provide assumption (H4) instead of (B4). Compared with the studies of Malojlović et al. ([27,32]), we generalize their work to the nonlinear situation and provide different assumptions. Firstly, we provide condition (H4) instead of the following condition:

$$\sum _{i=1}^n{\int }_0^{\infty }{\int }_{s-{\delta }_i}^{s-{\sigma }_i}{q}_i\left(\xi \right)d\xi ds<1$$

in [27] or

$$\sum _{i=1}^n{\int }_0^{\infty }{\int }_{s-{\delta }_i}^{s-{\sigma }_i}{q}_i\left(\xi \right)d\xi ds<p_{j_1}\left(x\right)$$

in [32]. Secondly, assumptions (H1) and (H2) in [27] (or (H2) and (H3) in [32]) are not needed, which means that there is no relationship between the positive and negative coefficients.

We obtain not only the oscillation criteria but also the existence of the nonoscillation solution of (2); thus, our results are an extension of theirs.