Oscillation and asymptotic properties of a class of second-order Emden–Fowler neutral differential equations

In this paper, we study a second-order delay differential equationwhere \(z(t)=x(t)+ p(t)x(\tau (t))\) and \(\alpha\) is a positive constant. Throughout this paper, we assume that

It is recognized that Emden–Fowler equations have a number of applications in physics and engineering; see, e.g., Berkovich (1997). As a result, there has been a great deal of interest in investigating the oscillation or nonoscillation of differential equations; see, e.g., Hale (1977), Džurina and Stavroulakis (2003), Li (2004), Li et al. (2011, 2013, 2015), Li and Rogovchenko (2015), Erbe et al. (2009), Manojlović (1999), Wang and Yang (2004), Wang (2001), Liu et al. (2012), Shi et al. (2016), Baculíková and Džurina (2011), Bohner and Li (2014), Yang et al. (2006), Xu and Meng (2006, 2007). As we known, many results of half-linear or nonlinear equations with positive neutral coefficients were established, see, e.g., Baculíková and Džurina (2011), Erbe et al. (2009), Li et al. (2011, 2013), Li and Rogovchenko (2015), Liu et al. (2012), Shi et al. (2016), Yang et al. (2006), Xu and Meng (2006, 2007). The equations with nonpositive neutral coefficients have been applied to practical life; see, for instance, Brayton (1966) and Kuang (1993, sec. 1.1.7) provided a model about the system with lossless transmission lines. And, there have been a few oscillation and asymptotic results of the equations with nonpositive neutral coefficients, see, e.g., Bohner and Li (2014), Erbe et al. (2009), Li et al. (2015), Yang et al. (2006). In the following, we provide some background details which motivated our research. Manojlović (1999), Wang (2001), and Wang and Yang (2004) considered the half-linear differential equationand gave some different oscillation results by using an inequality due to Hardy, Littlewood and Ploya and averaging functions. Motivated by these ideas, many scholars extended the results to delay differential equations or neutral delay differential equations. Džurina and Stavroulakis (2003) and Li (2004) expanded the Eq. (2) to the delay differential equation Xu and Meng (2006, 2007) extended (2) to the neutral delay differential equationprovided that \(0\le p(t)\le 1\). Li et al. (2015) established some oscillation and asymptotic results to (1) in the case where \(-1<p(t)\le 0\). By using Riccati transformation, Erbe et al. (2009) proposed some oscillation and asymptotic results for (1), under the assumptions that \(0\le p(t)<1\) and \(-1< p(t)< 0\). Liu et al. (2012) considered the the following equationin the case where \(0\le p(t)\le 1\) and \(\alpha \ge \beta >0\). They established some oscillation and asymptotic criteria by employing averaging technique and Riccati transformation. Shi et al. (2016) extended the results of Liu et al. (2012) to dynamic equations on time scales provided that \(0\le p(t)\le 1\) and \(p(t)> 1\).

In what follows, let \(D=\{(t, s):t_0\le s \le t\}\; \mathrm {and}\;D_0=\{(t, s):t_0\le s < t\}.\) We say a function \(H=H(t,s)\) belongs to a function class P, if it satisfies

1. (i)

   \(H(t, t)=0, t\ge t_0; H(t,s)>0, (t, s)\in D_0;\)

    
2. (ii)
   H has partial derivatives \(\partial H/\partial t\) and \(\partial H{/}\partial s\) on \(D_0\), such that

   $$\begin{aligned} \frac{\partial H(t, s)}{\partial t}=h_1(t, s)\sqrt{H(t, s)} \end{aligned}$$

   and

   $$\begin{aligned} \frac{\partial H(t, s)}{\partial s}=-h_2(t, s)\sqrt{H(t, s)}, \end{aligned}$$

   where \(h_1\) and \(h_2\) are nonnegative continuous functions on \(D_0\).
    

We see thatIt follows from Corollary 1 that all solutions of (29) are oscillatory or converge to zero. Letting \(\alpha =\beta =2\), we can certify that \(x(t)=e^{-t}\) is an asymptotic solution of (29).

Letting \(\rho (t)=\frac{\gamma }{2} \frac{\beta \sigma _0\left( t-\frac{t_1}{\sigma _0}\right) ^{\beta -1}}{K^{1-\frac{\beta }{\alpha }}}\) and \(H(t,s)=(t-s)^2\), thenandThat is, (17) holds. It follows from Corollary 2 that all solutions of (30) are oscillatory or converge to zero.

In this paper, we consider the oscillation of a class of second-order differential equations with positive and nonpositive neutral coefficients. It is difficult to study the nonpositive neutral coefficients equations because the sign of z is not explicit. Using Riccati transformation, some oscillation and asymptotic criteria are obtained under the assumptions that \((H_1)-(H_5)\). In Liu et al. (2012), Shi et al. (2016), the results were established for (1) in the case when \(0\le p(t)\le 1\) or \(p(t)>1\). This paper states some oscillation and asymptotic criteria for (1) in the case where \(-1<p(t)<0\) and \(0\le p(t)\le 1\). Erbe et al. (2009), Li et al. (2015) assume that \(\alpha =\beta\), however, in this paper \(\alpha \ne \beta\) is allowed. We give some examples to illustrate our results. There are two interesting questions for future study:

1. (Q1)

   Could we obtain oscillation criteria for Eq. (1) when \(-\infty < p(t)\le 1\, \text{or} -\infty<p(t)\le-1\)?

    
2. (Q2)

   Could we obtain some sufficient conditions which ensure that all solutions of (1) are oscillatory?