Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments

We study oscillatory properties of a class of second-order nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results.

MSC:34C10, 34K11.

This paper is concerned with oscillation of the second-order nonlinear functional differential equation

where $t\ge t_0>0$, $\alpha \ge 1$ is a constant, and $z:=x+p\cdot x\circ \tau$. Throughout, we assume that the following hypotheses hold:

(H₁) $\mathbb{I}:=[t_0,\mathrm{\infty })$, $r,p\in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$, $r(t)>0$, and $p(t)\ge 0$;

(H₂) $q\in \mathrm{C}(\mathbb{I}\times [a,b],[0,\mathrm{\infty }))$ and $q(t,\xi )$ is not eventually zero on any $[t_{\mu },\mathrm{\infty })\times [a,b]$, $t_{\mu }\in \mathbb{I}$;

(H₃) $g\in \mathrm{C}(\mathbb{I}\times [a,b],[0,\mathrm{\infty }))$, ${lim\hspace{0.17em}inf}_{t\to \mathrm{\infty }}g(t,\xi )=\mathrm{\infty }$, and $g(t,a)\le g(t,\xi )$ for $\xi \in [a,b]$;

(H₄) $\tau \in {\mathrm{C}}^2(\mathbb{I},\mathbb{R})$, ${\tau }^{\prime }(t)>0$, ${lim}_{t\to \mathrm{\infty }}\tau (t)=\mathrm{\infty }$, and $g(\tau (t),\xi )=\tau [g(t,\xi )]$;

(H₅) $\sigma \in \mathrm{C}([a,b],\mathbb{R})$ is nondecreasing and the integral of (1.1) is taken in the sense of Riemann-Stieltijes.

By a solution of (1.1), we mean a function $x\in \mathrm{C}([t_x,\mathrm{\infty }),\mathbb{R})$ for some $t_x\ge t_0$, which has the properties that $z\in {\mathrm{C}}^1([t_x,\mathrm{\infty }),\mathbb{R})$, $r{|z^{\prime }|}^{\alpha -1}{z}^{\prime }\in {\mathrm{C}}^1([t_x,\mathrm{\infty }),\mathbb{R})$, and satisfies (1.1) on $[t_x,\mathrm{\infty })$. We restrict our attention to those solutions x of (1.1) which exist on $[t_x,\mathrm{\infty })$ and satisfy $sup\{|x(t)|:t\ge T\}>0$ for any $T\ge t_x$. A solution x of (1.1) is termed oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions oscillate.

As is well known, neutral differential equations have a great number of applications in electric networks. For instance, they are frequently used in the study of distributed networks containing lossless transmission lines, which rise in high speed computers, where the lossless transmission lines are used to interconnect switching circuits; see [1]. Hence, there has been much research activity concerning oscillatory and nonoscillatory behavior of solutions to different classes of neutral differential equations, we refer the reader to [2–30] and the references cited therein.

In the following, we present some background details that motivate our research. Recently, Baculíková and Lacková [6], Džurina and Hudáková [12], Li et al. [15, 18], and Sun et al. [22] established some oscillation criteria for the second-order half-linear neutral differential equation

where $z:=x+p\cdot x\circ \tau$,

Baculíková and Džurina [4, 5] and Li et al. [17] investigated oscillatory behavior of a second-order neutral differential equation

where

Ye and Xu [26] and Yu and Fu [27] considered oscillation of the second-order differential equation

Assuming $0\le p(t)<1$, Thandapani and Piramanantham [23], Wang [24], Xu and Weng [25], and Zhao and Meng [30] studied oscillation of an equation

As yet, there are few results regarding the study of oscillatory properties of (1.1) under the conditions $p(t)\ge 1$ or ${lim}_{t\to \mathrm{\infty }}p(t)=\mathrm{\infty }$. Thereinto, Li and Thandapani [19] obtained several oscillation results for (1.1) in the case where (1.2) holds, $\sigma (\xi )=\xi$, and

In the subsequent sections, we shall utilize the Riccati substitution technique and some inequalities to establish several new oscillation criteria for (1.1) assuming that (1.3) holds or

All functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.

In what follows, we use the following notation for the convenience of the reader:

where h, ρ, and η will be specified later.

Theorem 2.1 Assume (H₁)-(H₅), (1.3), and let $g(t,a)\in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$, $g^{\prime }(t,a)>0$, $g(t,a)\le t$, and $g(t,a)\le \tau (t)$ for $t\in \mathbb{I}$. Suppose further that there exists a real-valued function $h\in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$ such that $p[g(t,\xi )]\le p[h(t)]$ for $t\in \mathbb{I}$ and $\xi \in [a,b]$. If there exists a real-valued function $\rho \in {\mathrm{C}}^1(\mathbb{I},(0,\mathrm{\infty }))$ such that

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a $t_1\in \mathbb{I}$ such that $x(t)>0$, $x[\tau (t)]>0$, and $x[g(t,\xi )]>0$ for all $t\ge t_1$ and $\xi \in [a,b]$. Then $z(t)>0$. Applying (1.1), one has, for all sufficiently large t,

Using the inequality (see [[5], Lemma 1])

the definition of z, $g(\tau (t),\xi )=\tau [g(t,\xi )]$, and $p[g(t,\xi )]\le p[h(t)]$, we conclude that

By virtue of (1.1), we get

Thus, $r{|z^{\prime }|}^{\alpha -1}{z}^{\prime }$ is nonincreasing. Now we have two possible cases for the sign of $z^{\prime }$: (i) $z^{\prime }<0$ eventually, or (ii) $z^{\prime }>0$ eventually.

1. (i)

   Assume that $z^{\prime }(t)<0$ for $t\ge t_2\ge t_1$. Then we have by (2.3)

   $$r(t){|z^{\prime }(t)|}^{\alpha -1}{z}^{\prime }(t)\le r(t_2){|z^{\prime }(t_2)|}^{\alpha -1}{z}^{\prime }(t_2)<0,t\ge t_2,$$

   which yields

   $$z(t)\le z(t_2)-r^{1/\alpha }(t_2)|z^{\prime }(t_2)|{\int }_{t_2}^t{r}^{-1/\alpha }(s)\mathrm{d}s.$$

   Then we obtain ${lim}_{t\to \mathrm{\infty }}z(t)=-\mathrm{\infty }$ due to (1.3), which is a contradiction.

1. (ii)

   Assume that $z^{\prime }(t)>0$ for $t\ge t_2\ge t_1$. It follows from (2.2) and $g(t,\xi )\ge g(t,a)$ that

   $$\begin{array}{r}{(r(t){(z^{\prime }(t))}^{\alpha })}^{\prime }+\frac{p^{\alpha }[h(t)]}{{\tau }^{\prime }(t)}{\left(r[\tau (t)]{\left(z^{\prime }[\tau (t)]\right)}^{\alpha }\right)}^{\prime }\\ +\frac{1}{2^{\alpha -1}}{z}^{\alpha }[g(t,a)]{\int }_a^{b}Q(t,\xi )\mathrm{d}\sigma (\xi )\le 0.\end{array}$$

   (2.4)

We define a Riccati substitution

Then $\omega (t)>0$. From (2.3) and $g(t,a)\le t$, we have

Differentiating (2.5), we get

Therefore, by (2.5), (2.6), and (2.7), we see that

Similarly, we introduce another Riccati transformation:

Then $\upsilon (t)>0$. From (2.3) and $g(t,a)\le \tau (t)$, we obtain

Differentiating (2.9), we have

Therefore, by (2.9), (2.10), and (2.11), we find

Combining (2.8) and (2.12), we get

It follows from (2.4) that

Integrating the latter inequality from $t_2$ to t, we obtain

Define

Using the inequality

we get

On the other hand, define

Then we have by (2.14)

Thus, from (2.13), we get

which contradicts (2.1). This completes the proof. □

Assuming (1.2), where $p_0$ and ${\tau }_0$ are constants, we obtain the following result.

Theorem 2.2 Suppose (H₁)-(H₅), (1.2), (1.3), and let $g(t,a)\in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$, $g^{\prime }(t,a)>0$, $g(t,a)\le t$, and $g(t,a)\le \tau (t)$ for $t\in \mathbb{I}$. If there exists a real-valued function $\rho \in {\mathrm{C}}^1(\mathbb{I},(0,\mathrm{\infty }))$ such that

then (1.1) is oscillatory.

Proof As above, let x be an eventually positive solution of (1.1). Proceeding as in the proof of Theorem 2.1, we have $z^{\prime }(t)>0$, (2.3), and (2.4) for all sufficiently large t. Using (1.2), (2.3), and (2.4), we obtain

The remainder of the proof is similar to that of Theorem 2.1, and hence it is omitted. □

Theorem 2.3 Suppose we have (H₁)-(H₅), (1.3), and let $\tau (t)\le t$ and $g(t,a)\ge \tau (t)$ for $t\in \mathbb{I}$. Assume also that there exists a real-valued function $h\in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$ such that $p[g(t,\xi )]\le p[h(t)]$ for $t\in \mathbb{I}$ and $\xi \in [a,b]$. If there exists a real-valued function $\rho \in {\mathrm{C}}^1(\mathbb{I},(0,\mathrm{\infty }))$ such that

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a $t_1\in \mathbb{I}$ such that $x(t)>0$, $x[\tau (t)]>0$, and $x[g(t,\xi )]>0$ for all $t\ge t_1$ and $\xi \in [a,b]$. As in the proof of Theorem 2.1, we obtain (2.3) and (2.4). In view of (2.3), $r{|z^{\prime }|}^{\alpha -1}{z}^{\prime }$ is nonincreasing. Now we have two possible cases for the sign of $z^{\prime }$: (i) $z^{\prime }<0$ eventually, or (ii) $z^{\prime }>0$ eventually.

1. (i)

   Suppose that $z^{\prime }(t)<0$ for $t\ge t_2\ge t_1$. Then, with a proof similar to the proof of case (i) in Theorem 2.1, we obtain a contradiction.

2. (ii)

   Suppose that $z^{\prime }(t)>0$ for $t\ge t_2\ge t_1$. We define a Riccati substitution

   $$\omega (t):=\rho (t)\frac{r(t){(z^{\prime }(t))}^{\alpha }}{{(z[\tau (t)])}^{\alpha }},t\ge t_2.$$

   (2.18)

Then $\omega (t)>0$. From (2.3) and $\tau (t)\le t$, we have

Differentiating (2.18), we obtain

Therefore, by (2.18), (2.19), and (2.20), we see that

Similarly, we introduce another Riccati substitution:

Then $\upsilon (t)>0$. Differentiating (2.22), we have

Therefore, by (2.22) and (2.23), we get

Combining (2.21) and (2.24), we have

It follows from (2.4) and $g(t,a)\ge \tau (t)$ that

Integrating the latter inequality from $t_2$ to t, we obtain

Define

Using inequality (2.14), we have

On the other hand, define

Then, by (2.14), we obtain

Thus, from (2.25), we get

which contradicts (2.17). This completes the proof. □

Assuming we have (1.2), where $p_0$ and ${\tau }_0$ are constants, we get the following result.

Theorem 2.4 Suppose we have (H₁)-(H₅), (1.2), (1.3), and let $\tau (t)\le t$ and $g(t,a)\ge \tau (t)$ for $t\in \mathbb{I}$. If there exists a real-valued function $\rho \in {\mathrm{C}}^1(\mathbb{I},(0,\mathrm{\infty }))$ such that

then (1.1) is oscillatory.

Proof Assume again that x is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, we have $z^{\prime }(t)>0$, (2.3), and (2.4) for all sufficiently large t. By virtue of (1.2), (2.3), and (2.4), we have (2.16) for all sufficiently large t. The rest of the proof is similar to that of Theorem 2.3, and so it is omitted. □

In the following, we present some oscillation criteria for (1.1) in the case where (1.4) holds.

Theorem 2.5 Suppose we have (H₁)-(H₅), (1.2), (1.4), and let $g(t,a)\in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$, $g^{\prime }(t,a)>0$, $g(t,a)\le \tau (t)\le t$ for $t\in \mathbb{I}$, and $g(t,\xi )\le g(t,b)$ for $\xi \in [a,b]$. Assume further that there exists a real-valued function $\rho \in {\mathrm{C}}^1(\mathbb{I},(0,\mathrm{\infty }))$ such that (2.15) is satisfied. If there exists a real-valued function $\eta \in {\mathrm{C}}^1(\mathbb{I},\mathbb{R})$ such that $\eta (t)\ge t$, $\eta (t)\ge g(t,b)$, ${\eta }^{\prime }(t)>0$ for $t\in \mathbb{I}$, and