Oscillation of Bounded Solutions of Delay Differential Equations

Abstract

The purpose of this paper is to give new oscillation criteria for second-order delay differential equations $y^{″}\left(t\right)=p\left(t\right)y\left(\tau \left(t\right)\right).$ We introduce a new technique for the elimination of bounded nonoscillatory solutions.

1. Introduction

We consider the second-order functional differential equation with delayed argument:

$$y^{″}\left(t\right)=p\left(t\right)y\left(\tau \left(t\right)\right).$$

(E)

Throughout this paper, it is assumed that

• (H1) $p\left(t\right)\in C\left([t_0,\infty )\right)$, $p\left(t\right)>0,$
• (H2) $\tau \left(t\right)\in C^1\left([t_0,\infty )\right),$$\tau \left(t\right)<t,$${\tau }^{\prime }\left(t\right)>0$, $\underset{t\to \infty }{lim}\tau \left(t\right)=\infty$.

As usual, by a proper solution of Equation (E), we mean a function $y:[t_0,\infty )\to (-\infty ,\infty )$ which satisfies (E) for all sufficiently large t and $sup\{\left|y\left(t\right)\right|:t\ge T\}>0$ for all $T\ge t_0.$

Definition 1.

A proper solution is called oscillatory if it has arbitrary large zero; otherwise, it is called nonoscillatory.

There are two eventual cases for nonoscillatory solutions of (E), namely if $y\left(t\right)$ is a nonoscillatory solution of (E), then there exists a number $j\in \{0,2\}$ such that

$$\begin{array}{cc}\hfill y\left(t\right)y^{\left(i\right)}\left(t\right)& >0\mathrm{for}0\le i\le j,\hfill \\ \hfill {(-1)}^{i}y\left(t\right)y^{\left(i\right)}\left(t\right)& >0\mathrm{for}j\le i\le 2.\hfill \end{array}$$

(1)

Definition 2.

Such $y\left(t\right)$ is said to be a solution of degree j, and the totality of solutions of degree j is denoted by ${\mathcal{N}}_j$.

Therefore, if we denote the set of all nonoscillatory solutions of (E) by $\mathcal{N}$, then $\mathcal{N}$ has the following decomposition:

$$\mathcal{N}={\mathcal{N}}_0\cup {\mathcal{N}}_2.$$

The problem of determining oscillation criteria for particular functional differential equations has been a very active research area in the past few decades, and some of them are mentioned in references [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The qualitative study of such equations has, besides its theoretical interest, significant practical importance. This is due to fact that differential equations arise in various phenomena, including in problems concerning electric networks containing lossless transmission lines (as seen in high-speed computers, where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, and in the solution of variational problems with time delays. We refer the reader to [22,23] for models from mathematical biology where oscillation and/or delay actions may be formulated by means of cross-diffusion terms.

Our aim is to present a new technique for the investigation of asymptotic properties of (E). For this, we recall the classical result of Koplatadze and Chanturia [16].

Lemma 1.

Assume that ($H_1$) and ($H_2$) holds true. If

$$\underset{t\to \infty }{lim\; sup}{\int }_{\tau \left(t\right)}^t(s-\tau \left(t\right))p\left(s\right)\mathrm{d}s>1,$$

(2)

then ${\mathcal{N}}_0=\varnothing$ for (E).

This result has been extended to more general differential equations (see, e.g., [9] or [19]). But we are about to make meaningful progress exactly for (E). Baculikova (see [3]) suggested two ways for improving Theorem 1. She noticed that the main idea of Koplatadze and Chanturia’s proof consists in the following estimates for possible solutions of degree 0:

$$\begin{array}{cc}\hfill y\left(\tau \right(t\left)\right)& ={\int }_{\tau \left(t\right)}^{\infty }(s-\tau \left(t\right))p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ \hfill & \ge {\int }_{\tau \left(t\right)}^t(s-\tau \left(t\right))p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ \hfill & \ge y\left(\tau \left(t\right)\right){\int }_{\tau \left(t\right)}^t(s-\tau \left(t\right))p\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(3)

Firstly, Koplatadze and Chanturia used the fact that $y\left(t\right)$ is decreasing (see the third line of (3)), and Baculikova provided natural improvement in the sense of establishing a new monotonicity for $y\left(t\right)$ in the form of $\tilde{\alpha }\left(t\right)y\left(t\right)$, decreasing for a suitable function $\tilde{\alpha }\left(t\right)$ such that $\tilde{\alpha }\left(t\right)\to \infty$ as $t\to \infty$.

Secondly, the information about function $(s-\tau (t\left)\right)p\left(s\right)y\left(\tau \right(s\left)\right)$ is lost on the interval $(t,\infty )$ (see the second line of (3)), and Baculikova eliminates this insufficiency by establishing “the opposite” monotonicity of $y\left(t\right)$, in the sense that $\beta \left(t\right)y\left(t\right)$ is increasing for certain function $\beta \left(t\right)$. In this paper, we introduce a new idea for preserving information about $(s-\tau (t\left)\right)p\left(s\right)y\left(\tau \right(s\left)\right)$ over the interval $(t,\infty )$ by establishing the important estimate

$${\int }_t^{\infty }(s-\tau \left(t\right))p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\ge \ell {\int }_{\tau \left(t\right)}^t(s-\tau \left(t\right))p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s$$

for a certain positive constant $\ell >0$. Moreover, we improve Baculikova’s monotonicity.

2. Preliminary Results

In this section, we recall some results from Baculikova [3].

Lemma 2.

If

$${\int }_{t_0}^{\infty }sp\left(s\right)\mathrm{d}s=\infty ,$$

(4)

then every solution $y\left(t\right)$ of degree 0 of (E) satisfies

$$y\left(t\right)\to 0ast\to \infty .$$

(5)

Let us denote that

$$\tilde{P}\left(t\right)={\tau }^{\prime }\left(t\right){\int }_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s,{\widetilde{{\alpha }_0}}^{\prime }\left(t\right)=\tilde{P}\left(t\right)\mathrm{and}\tilde{\alpha }\left(t\right)={\mathrm{e}}^{\widetilde{{\alpha }_0}\left(t\right)}.$$

(6)

Lemma 3.

Let $y\left(t\right)$ be a positive solution of degree 0 for Equation (E). Then,

$$\tilde{\alpha }\left(t\right)y\left(\tau \left(t\right)\right)isadecreasingfunction.$$

(7)

We introduce the auxiliary function $\xi \left(t\right)\in C^1\left([t_0,\infty )\right)$ such that

$$\xi \left(\xi \left(t\right)\right)={\tau }^{-1}\left(t\right).$$

(8)

Moreover, we employ the following supplementary functions:

$$\begin{array}{cc}\hfill P_1\left(t\right)& =\tilde{\alpha }\left(\tau \left(\xi \left(t\right)\right)\right){\int }_t^{\xi \left(t\right)}{\displaystyle \frac{s-t}{\tilde{\alpha }\left(\tau \left(s\right)\right)}}p\left(s\right)\mathrm{d}s,\hfill \\ \hfill P_2\left(t\right)& =\tilde{\alpha }\left(t\right){\int }_{\xi \left(t\right)}^{{\tau }^{-1}\left(t\right)}{\displaystyle \frac{s-t}{\tilde{\alpha }\left(\tau \left(s\right)\right)}}p\left(s\right)\mathrm{d}s,\hfill \\ \hfill P_3\left(t\right)& =\tilde{\alpha }\left(\xi \left(t\right)\right){\int }_{{\tau }^{-1}\left(t\right)}^{{\tau }^{-1}\left(\xi \left(t\right)\right)}{\displaystyle \frac{s-t}{\tilde{\alpha }\left(\tau \left(s\right)\right)}}p\left(s\right)\mathrm{d}s.\hfill \end{array}$$

(9)

It is assumed that there exist positive constants $P_i^{*}$, $i=1,3$ such that

$$\begin{array}{cc}\hfill P_1^{*}\left(t\right)& ={\displaystyle \frac{P_1\left(t\right)}{1-P_2\left(t\right)}}\ge P_1^{*},\hfill \\ \hfill P_3^{*}\left(t\right)& ={\displaystyle \frac{P_3\left(t\right)}{1-P_2\left(t\right)}}\ge P_3^{*}.\hfill \end{array}$$

(10)

Lemma 4.

Assume that there exists a function $\xi \left(t\right)\in C^1\left([t_0,\infty )\right)$ satisfying (8) and that $y\left(t\right)$ is a positive solution of degree 0 of (E); then,

$$y\left(\tau \right(t\left)\right)\le Ky\left(t\right),$$

(11)

where

$$K={\displaystyle \frac{1-2P_3^{*}{P}_1^{*}}{{\left(P_1^{*}\right)}^2}}.$$

(12)

If it happens that

$$1-2P_3^{*}{P}_1^{*}\le 0,$$

then the class ${\mathcal{N}}_0=\varnothing .$

3. Results

In deriving the main results, we will rely on the Leibniz integral rule for differentiation under the integral sign, which claims that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\int }_{a\left(t\right)}^{b\left(t\right)}f(s,t)\mathrm{d}s\right)\hfill \\ \hfill & =f(b\left(t\right),t){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}b\left(t\right)-f(a\left(t\right),t){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}a\left(t\right)+{\int }_{a\left(t\right)}^{b\left(t\right)}{\displaystyle \frac{\partial }{\partial t}}f(s,t)\mathrm{d}s.\hfill \end{array}$$

(13)

Lemma 5.

Assume that $K>0$ is defined by (12) and

$${\tau }^{\prime }\left(t\right){\displaystyle \frac{p\left(\tau \right(t\left)\right)}{p\left(t\right)}}\le A,A>0.$$

(14)

Then, there exists a constant $\ell >0$ such that, for any positive solution $y\left(t\right)\in {\mathcal{N}}_0$ of (E),

$${\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\ge \ell {\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s,$$

(15)

where

(i)

if $KA\le 1$, then ℓ is an optional positive number;

($ii$)

if $KA>1$, then ${\displaystyle \ell =\frac{1}{KA-1}}$.

Proof. 

Assume that $y\left(t\right)\in {\mathcal{N}}_0$ is a positive solution of (E). Then,

$$y^{\prime }\left(t\right)\to 0\mathrm{as}t\to \infty .$$

(16)

An integration of (E) yields

$$-y^{\prime }\left(\tau \left(t\right)\right)={\int }_{\tau \left(t\right)}^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s.$$

(17)

We consider the auxiliary function

$$h\left(t\right)={\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s-\ell {\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s.$$

It follows from (16) and (17) that $h(\infty )=0$. Consequently, to verify (15), it is sufficient to show that $h^{\prime }\left(t\right)\le 0$. The Leibniz integral rule implies that

$$h^{\prime }\left(t\right)=-p\left(t\right)y\left(\tau \left(t\right)\right)-\ell \left[p\left(t\right)y\left(\tau \left(t\right)\right)-p\left(\tau \left(t\right)\right)y\left(\tau \left(\tau \left(t\right)\right)\right){\tau }^{\prime }\left(t\right)\right]$$

which, in view of (11), provides

$$h^{\prime }\left(t\right)\le p\left(t\right)y\left(\tau \left(t\right)\right)\left[-(1+\ell )+\ell K{\displaystyle \frac{p\left(\tau \right(t\left)\right)}{p\left(t\right)}}{\tau }^{\prime }\left(t\right)\right].$$

Taking (14) into account, one obtains

$$h^{\prime }\left(t\right)\le p\left(t\right)y\left(\tau \left(t\right)\right)\left[-(1+\ell )+\ell KA\right]\le 0.$$

The proof is complete now. □

Lemma 5 permits us to improve the monotonicity presented in (7).

We consider $\ell >0$ such as in Lemma 5. We denote that

$$P\left(t\right)=(1+\ell ){\tau }^{\prime }\left(t\right)\tilde{\alpha }\left(t\right){\int }_{\tau \left(t\right)}^t{\displaystyle \frac{p\left(s\right)}{\tilde{\alpha }\left(s\right)}}\mathrm{d}s,{\alpha }_0^{\prime }\left(t\right)=P\left(t\right)\mathrm{and}\alpha \left(t\right)={\mathrm{e}}^{{\alpha }_0\left(t\right)}.$$

(18)

Lemma 6.

Let $y\left(t\right)$ be a positive solution of degree 0 for Equation (E). Then,

$$\alpha \left(t\right)y\left(\tau \right(t\left)\right)isadecreasingfunction.$$

(19)

Proof. 

Assume that $y\left(t\right)\in {\mathcal{N}}_0$ is a positive solution of (E). Setting (15) into (17), we have

$$\begin{array}{cc}\hfill -y^{\prime }\left(\tau \left(t\right)\right)& \ge (1+\ell ){\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ \hfill & \ge (1+\ell )\tilde{\alpha }\left(t\right)y\left(\tau \right(t\left)\right){\int }_{\tau \left(t\right)}^t{\displaystyle \frac{p\left(s\right)}{\tilde{\alpha }\left(s\right)}}\mathrm{d}s,\hfill \end{array}$$

(20)

where we have used (7). The last inequality is equivalent to

$$y^{\prime }\left(\tau \left(t\right)\right){\tau }^{\prime }\left(t\right)+P\left(t\right)y\left(\tau \left(t\right)\right)\le 0$$

which gives

$${\left[y\left(\tau \left(t\right)\right)\alpha \left(t\right)\right]}^{\prime }\le 0,$$

and we can conclude that $y\left(\tau \right(t\left)\right)\alpha \left(t\right)$ is a decreasing function. □

Now, we are prepared to establish new criteria for ${\mathcal{N}}_0=\varnothing$ for (E), or in other words, we are prepared for every bounded solution of (E) to be oscillatory.

Theorem 1.

Let (4) hold and the positive constants K, A, ℓ be the same as in Lemma 5. Assume that there exists a function $\xi \left(t\right)\in C^1\left([t_0,\infty )\right)$ satisfying (8) and that $\alpha \left(t\right)$ is defined by (18). If

$$\underset{t\to \infty }{lim\; sup}\left[\alpha \left(t\right){\int }_{\tau \left(t\right)}^t{\displaystyle \frac{p\left(s\right)(s-\tau (t\left)\right)}{\alpha \left(s\right)}}\mathrm{d}s\right]>{\displaystyle \frac{1}{1+\ell }},$$

(21)

then ${\mathcal{N}}_0=\varnothing$ for (E).

Proof. 

Assume, on the contrary, that (E) possesses a positive solution $y\left(t\right)\in {\mathcal{N}}_0$. Integrating (E) twice from t to ∞ and changing the order of integration, we obtain

$$y\left(t\right)\ge {\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)(s-t)\mathrm{d}s$$

Hence,

$$y\left(\tau \left(t\right)\right)\ge {\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)(s-\tau \left(t\right))\mathrm{d}s+{\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)(s-\tau \left(t\right))\mathrm{d}s.$$

(22)

To simplify our notation, we employ the following functions:

$$\begin{array}{cc}\hfill F\left(t\right)& ={\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)(s-\tau \left(t\right))\mathrm{d}s,\hfill \\ \hfill G\left(t\right)& ={\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)(s-\tau \left(t\right))\mathrm{d}s,\hfill \end{array}$$

(23)

We claim that $G\left(t\right)\ge \ell F\left(t\right).$ To verify this, we employ a supplementary function:

$$r\left(t\right)=G\left(t\right)-\ell F\left(t\right)$$

(24)

And we shall show that $r\left(t\right)\ge 0$. It follows from (5) and (22) that $r(\infty )=0$, and therefore, it is sufficient to confirm that $r^{\prime }\left(t\right)\le 0.$ By the Leibniz integral rule,

$$\begin{array}{cc}\hfill r^{\prime }\left(t\right)& =-p\left(t\right)y\left(\tau \left(t\right)\right)(t-\tau \left(t\right))-{\tau }^{\prime }\left(t\right){\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ \hfill & -\ell p\left(t\right)y\left(\tau \left(t\right)\right)(t-\tau \left(t\right))+\ell {\tau }^{\prime }\left(t\right){\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\hfill \\ \hfill & =-(\ell +1)p\left(t\right)y\left(\tau \right(t\left)\right)(t-\tau (t\left)\right)\hfill \\ \hfill & +{\tau }^{\prime }\left(t\right)\left[\ell {\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s-{\int }_t^{\infty }p\left(s\right)y\left(\tau \left(s\right)\right)\mathrm{d}s\right]\le 0,\hfill \end{array}$$

(25)

where we have used (15). Finally, (22), together with (24), implies that

$$\begin{array}{cc}\hfill y\left(\tau \right(t\left)\right)& \ge (1+\ell ){\int }_{\tau \left(t\right)}^{t}p\left(s\right)y\left(\tau \left(s\right)\right)(s-\tau \left(t\right))\mathrm{d}s\hfill \\ \hfill & \ge (1+\ell ))\alpha \left(t\right)y\left(\tau \left(t\right)\right){\int }_{\tau \left(t\right)}^t{\displaystyle \frac{p\left(s\right)(s-\tau (t\left)\right)}{\alpha \left(s\right)}}\mathrm{d}s.\hfill \end{array}$$

(26)

where (19) has been also employed. Since (26) contradicts (21), the proof is finished. □

This is the standard way to illustrate the progress achieved by means of Euler differential equations.

Example 1.

Consider the differential equation

$$y^{″}\left(t\right)={\displaystyle \frac{p_0}{t^2}}y\left(\lambda t\right),p_0>0,\lambda \in (0,1).$$

(E₁)

By Theorem 1, Equation (E₁) for $\lambda =0.5$ has no noscillatory solutions from class ${\mathcal{N}}_0$, provided that $p_0>5.177.$ On the other hand, it is easy to verify that

$$\tilde{P}\left(t\right)={\displaystyle \frac{p_0}{t}}(1-\lambda )and\widetilde{{\alpha }_0}\left(t\right)=p_0(1-\lambda )lnt$$

Hencez,

$$\tilde{\alpha }\left(t\right)=t^{{\beta }_0},where{\beta }_0=p_0(1-\lambda ).$$

We set $\xi \left(t\right)=\sqrt{\lambda }t$. Consequently,

$$P_1\left(t\right)=p_0{\lambda }^{-{\displaystyle \frac{{\beta }_0}{2}}}\left({\displaystyle \frac{1-{\lambda }^{\frac{{\beta }_0}{2}}}{{\beta }_0}}+{\displaystyle \frac{{\lambda }^{\frac{1+{\beta }_0}{2}}-1}{1+{\beta }_0}}\right),$$

$$P_2\left(t\right)=p_0{\lambda }^{-{\displaystyle \frac{{\beta }_0}{2}}}\left({\displaystyle \frac{1-{\lambda }^{\frac{{\beta }_0}{2}}}{{\beta }_0}}+{\displaystyle \frac{{\lambda }^{\frac{2+{\beta }_0}{2}}-{\lambda }^{\frac{1}{2}}}{1+{\beta }_0}}\right),$$

$$P_3\left(t\right)=p_0{\lambda }^{-{\displaystyle \frac{{\beta }_0}{2}}}\left({\displaystyle \frac{1-{\lambda }^{\frac{{\beta }_0}{2}}}{{\beta }_0}}+{\displaystyle \frac{{\lambda }^{\frac{3+{\beta }_0}{2}}-\lambda }{1+{\beta }_0}}\right).$$

Then,

$$\ell ={\displaystyle \frac{\lambda }{K-\lambda }}$$

and

$$\alpha \left(t\right)=t^{\gamma },where\gamma ={\displaystyle \frac{p_0(1+\ell )\left({\lambda }^{-{\beta }_0}-\lambda \right)}{1+{\beta }_0}}.$$

Condition (21) reduces for (E₁) to

$$p_0\left[{\displaystyle \frac{{\lambda }^{-\gamma }-1}{\gamma }}+{\displaystyle \frac{\lambda -{\lambda }^{-\gamma }}{1+\gamma }}\right]>1+\ell .$$

(27)

Consequently, for $\lambda =0.5$, condition (27) is satisfied for $p_0=2.496$, which by Theorem 1 yields ${\mathcal{N}}_0=\varnothing$ for (E₁). On the other hand, Baculikova’s way [3] requires $p_0=2.56$. The progress is obvious.

4. Discussion

In this paper we have introduced a new method for investigating the properties of linear second-order delay differential equations. There may be two ways for generalizing or improving the presented result. The first way would be to try one’s hand at establishing the iteration process for improving the monotonicity (19) (see the technique presented in [12]). The second way would involve trying to extend the results to higher-order differential equations.