Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396]

Abstract

In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations

$$(r(t)\psi (x(t))|Z^{\prime }(t){|}^{\alpha -1}{Z}^{\prime }(t){)}^{\prime }+q(t)f(x(\sigma (t)))=0\text{,}t⩾t_0>0\text{,}$$

where $Z(t)=x(t)+p(t)x(t-\tau )$ and $\alpha >0\text{,}0⩽p(t)<1$. On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations

$$[r(t)[x(t)+p(t)x(\tau (t)){]}^{\prime }{]}^{\prime }+q(t)f(x(\sigma (t)))=0\text{,}t⩾t_0>0\text{,}$$

where ${\int }_{t_0}^{\infty }\frac{\mathrm{d}t}{r(t)}<\infty \text{,}0⩽p(t)⩽p_0<+\infty$. The results obtained here complement and correct some known results in [1]. Some examples are given to illustrate the main results.

Introduction

In [1], the authors studied the oscillation of the second-order quasilinear neutral delay differential equations

$$(r(t)\psi (x(t))|Z^{\prime }(t){|}^{\alpha -1}{Z}^{\prime }(t){)}^{\prime }+q(t)f(x(\sigma (t)))=0\text{,}t⩾t_0>0\text{,}$$

where $Z(t)=x(t)+p(t)x(\tau (t))$ for $t⩾t_0$, and $\alpha >0$. Throughout [1], the authors assume that the following conditions hold:

• $r\text{,}p\text{,}q\in C(I\text{,}\mathbb{R})\text{,}r(t)>0\text{,}0⩽p(t)⩽1$, and $q(t)⩾0$ for $t\in I\text{,}I=[t_0\text{,}\infty )$;

• $\psi \text{,}f\in C^1(\mathbb{R}\text{,}\mathbb{R})\text{,}\psi (x)>0$ and $\mathit{xf}(x)>0$ for all $x\ne 0$, and there exist two positive constants k and L such that

  $$\frac{f(x)}{|x{|}^{\alpha -1}x}⩾k\text{,}\psi (x)⩽L^{-1}\text{,}x\ne 0\text{;}$$

• $\tau \in C(I\text{,}\mathbb{R})\text{,}\tau (t)⩽t$ for $t⩾t_0$, and ${\lim }_{t\to \infty }\tau (t)=\infty$;

• $\sigma \in C(I\text{,}\mathbb{R})\text{,}{\sigma }^{\prime }(t)>0\text{,}\sigma (t)⩽t$ for $t⩾t_0$, and ${\lim }_{t\to \infty }\sigma (t)=\infty$.

In [1], denote $\epsilon =(\alpha /(\alpha +1){)}^{\alpha +1}\text{,}Q(t)=q(t)(1-p(\sigma (t)){)}^{\alpha }$ and $\pi (t)={\int }_t^{\infty }\frac{\mathrm{d}s}{r^{\frac{1}{\alpha }}(s)}$.

The method in [1] is very useful for studying the oscillation of second-order half-linear differential equations

$$(r(t)|x^{\prime }(t){|}^{\alpha -1}{x}^{\prime }(t){)}^{\prime }+q(t)|x(t){|}^{\alpha -1}x(t)=0\text{,}t⩾t_0>0\text{.}$$

However, the statements of their theorems and representations include several mistakes as follows:

Theorem 1.1 [1, Theorem 2.3]

Suppose that $\pi (t_0)<\infty$. If

$${\int }^{\infty }\left[Q(t){\pi }^{\alpha }(\sigma (t))-\frac{\epsilon }{\mathit{Lk}}\frac{{\sigma }^{\prime }(t)}{\pi (\sigma (t))r^{\frac{1}{\alpha }}(\sigma (t))}\right]\mathrm{d}t=\infty$$

and

$${\int }^{\infty }\left[Q(t){\pi }^{\alpha }(t)-\frac{\epsilon }{\mathit{Lk}}\frac{r(\sigma (t))}{\pi (t)({\sigma }^{\prime }(t){)}^{\alpha }{r}^{\frac{\alpha +1}{\alpha }}(t)}\right]\mathrm{d}t=\infty \text{,}$$

then (1.1) is oscillatory.

Theorem 1.2 [1, Theorem 2.4]

Suppose that (1.2) holds and $\pi (t_0)<\infty$. If

$${\int }^{\infty }Q(t){\pi }^{\alpha +1}(t)\mathrm{d}t=\infty \text{,}$$

then (1.1) is oscillatory.

Theorem 1.3 [1, Theorem 2.5]

Suppose that (1.3) holds and $\pi (t_0)<\infty$. If

$${\int }^{\infty }\left[Q(t)\pi (\sigma (t))-\frac{1}{\mathit{Lk}(\alpha +1{)}^{\alpha +1}}\frac{{\sigma }^{\prime }(t)}{r^{\frac{1}{\alpha }}(\sigma (t)){\pi }^{\alpha }(\sigma (t))}\right]\mathrm{d}t=\infty \text{,}$$

then (1.1) is oscillatory.

In the proof of Theorem 2.3 [1], the authors said “On the other hand, as in the proof of Theorem 2.1, we get (2.4) and (2.7) hold.” ([p. 394, line 13]) Note that (2.7) in [1] is

$$(r(t)\psi (x(t))(Z^{\prime }(t){)}^{\alpha }{)}^{\prime }+\mathit{kQ}(t)Z^{\alpha }(\sigma (t))⩽0\text{,}t⩾T_0\text{,}$$

where $T_0⩾t_0$ is sufficiently large. But (1.6) does not hold under the case $Z^{\prime }(t)<0$. We shall give an example to illustrate it.

Consider the differential equations

$${\left(e^{2t}{\left(x(t)+\frac{1}{2}x(t-2)\right)}^{\prime }\right)}^{\prime }+\left(e^{2t}+\frac{1}{2}{e}^{2t+2}\right)x(t)=0\text{,}t⩾t_0>0\text{.}$$

Let $r(t)=e^{2t}\text{,}\psi (x(t))=1\text{,}p(t)=1/2\text{,}q(t)=e^{2t}+e^{2t+2}/2\text{,}\tau (t)=t-2\text{,}\sigma (t)=t\text{,}\alpha =1\text{,}L=1\text{,}k=1$. Then $\pi (t)=e^{-2t}/2\text{,}\pi (t_0)<\infty \text{,}Q(t)=q(t)/2=(e^{2t}+e^{2t+2}/2)/2$. It is easy to see that $x(t)=e^{-t}$ is a positive solution of (1.7).

Let $x(t)=e^{-t}\text{,}Z(t)=x(t)+x(t-2)/2$. Then $Z^{\prime }(t)=-e^{-t}-e^{-t+2}/2<0$, and

$$(r(t)\psi (x(t))(Z^{\prime }(t){)}^{\alpha }{)}^{\prime }+\mathit{kQ}(t)Z^{\alpha }(\sigma (t))=\frac{e^4-4}{8}{e}^t>0\text{.}$$

Hence, (1.6) does not hold under the case $Z^{\prime }(t)<0$.

Furthermore, we have

$${\int }^{\infty }\left[Q(t){\pi }^{\alpha }(\sigma (t))-\frac{\epsilon }{\mathit{Lk}}\frac{{\sigma }^{\prime }(t)}{\pi (\sigma (t))r^{\frac{1}{\alpha }}(\sigma (t))}\right]\mathrm{d}t={\int }^{\infty }\left[Q(t){\pi }^{\alpha }(t)-\frac{\epsilon }{\mathit{Lk}}\frac{r(\sigma (t))}{\pi (t)({\sigma }^{\prime }(t){)}^{\alpha }{r}^{\frac{\alpha +1}{\alpha }}(t)}\right]\mathrm{d}t={\int }^{\infty }\left[Q(t)\pi (\sigma (t))-\frac{1}{\mathit{Lk}(\alpha +1{)}^{\alpha +1}}\frac{{\sigma }^{\prime }(t)}{r^{\frac{1}{\alpha }}(\sigma (t)){\pi }^{\alpha }(\sigma (t))}\right]\mathrm{d}t={\int }^{\infty }\frac{e^2-2}{8}\mathrm{d}t=\infty \text{,}$$

which satisfy all conditions of Theorems 2.3 and 2.5 in [1]. However, $x(t)=e^{-t}$ is a nonoscillatory solution of (1.7).

As a result, we cannot obtain that (1.1) is oscillatory from Theorems 2.3 and 2.5 in [1].

In the following, we show that the proof of Theorem 2.4 in [1] has a mistake, that is, “Then proceeding the proof of case 2 of Theorem 2.3, we get that (2.23) holds.” ([1, p. 394, line 28]). Note that (2.23) in [1] is

$$u^{\prime }(t)+\mathit{kQ}(t)+\frac{\alpha L^{\frac{1}{\alpha }}{\sigma }^{\prime }(t)}{r^{\frac{1}{\alpha }}(\sigma (t))}|u(t){|}^{\frac{\alpha +1}{\alpha }}⩽0\text{,}t⩾T_1\text{,}$$

where $T_1⩾t_0$ is sufficiently large. But (1.8) does not hold under the case $Z^{\prime }(t)<0$. We consider (1.7) to illustrate it.

Note that $\psi (x(t))=1\text{,}\sigma (t)=t$, then

$$u(t)=\frac{r(t)Z^{\prime }(t)}{Z(t)}\text{.}$$

Let $Z(t)=e^{-t}+e^{-t+2}/2$. Thus

$$u^{\prime }(t)+\mathit{kQ}(t)+\frac{\alpha L^{\frac{1}{\alpha }}{\sigma }^{\prime }(t)}{r^{\frac{1}{\alpha }}(\sigma (t))}|u(t){|}^{\frac{\alpha +1}{\alpha }}=\frac{e^2-2}{4}{e}^{2t}>0\text{.}$$

Hence, the process of the proof of Theorem 2.4 in [1] does not hold.

On the other hand, consider the second-order delay differential equations

$$(e^{2t}{x}^{\prime }(t){)}^{\prime }+\frac{3}{4}{e}^{2t-1}x(t-2)=0\text{,}t⩾t_0>0\text{.}$$

Let $r(t)=e^{2t}\text{,}\psi (x(t))=1\text{,}p(t)=0\text{,}q(t)=3e^{2t-1}/4\text{,}\sigma (t)=t-2\text{,}\alpha =1\text{,}L=1\text{,}k=1$. Then $\pi (t)=e^{-2t}/2\text{,}\pi (t_0)<\infty \text{,}Q(t)=q(t)=3e^{2t-1}/4$. It is easy to verify

$${\int }^{\infty }\left[Q(t){\pi }^{\alpha }(\sigma (t))-\frac{\epsilon }{\mathit{Lk}}\frac{{\sigma }^{\prime }(t)}{\pi (\sigma (t))r^{\frac{1}{\alpha }}(\sigma (t))}\right]\mathrm{d}t={\int }^{\infty }\left(\frac{3}{8}{e}^3-\frac{1}{2}\right)\mathrm{d}t=\infty$$

and

$${\int }^{\infty }\left[Q(t){\pi }^{\alpha }(t)-\frac{\epsilon }{\mathit{Lk}}\frac{r(\sigma (t))}{\pi (t)({\sigma }^{\prime }(t){)}^{\alpha }{r}^{\frac{\alpha +1}{\alpha }}(t)}\right]\mathrm{d}t={\int }^{\infty }\left(\frac{3}{8}{e}^{-1}-\frac{1}{2}{e}^{-4}\right)\mathrm{d}t=\infty \text{,}$$

so by [1, Theorem 2.3], Eq. (1.9) is oscillatory. However, $x(t)=e^{-\frac{t}{2}}$ is a nonoscillatory solution of (1.9).

In order to correct the mistakes, in Section 2, we will continue to investigate the oscillation of Eq. (1.1), where $\tau (t)=t-\tau \text{,}\tau ⩾0$ is a constant. In Section 3, we shall examine the oscillation of the following equations:

$${\left[r(t)[x(t)+p(t)x(\tau (t)){]}^{\prime }\right]}^{\prime }+q(t)f(x(\sigma (t)))=0\text{,}t⩾t_0>0\text{.}$$

In the last section we give some examples and remarks to illustrate the main results.