An oscillation criterion for delay differential equations with several non-monotone arguments

TL;DR

This paper establishes a new oscillation criterion for delay differential equations with multiple non-monotone delay arguments, broadening understanding of solution behaviors in complex dynamic systems.

Contribution

It introduces a novel sufficient oscillation condition involving lim sup for equations with several non-monotone delays, which was not previously available.

Findings

New oscillation criterion involving lim sup

Applicable to equations with multiple non-monotone delays

Illustrated with a relevant example

Abstract

The oscillatory behavior of the solutions to a differential equation with several non-monotone delay arguments and non-negative coefficients is studied. A new sufficient oscillation condition, involving lim sup, is obtained. An example illustrating the significance of the result is also given.

Equations

$$x^{\prime }(t)+\sum _{i=1}^m{p}_i(t)x\left({\tau }_i(t)\right)=0,\forall t\ge 0\text{,}$$

$${\tau }_i(t)<t\text{, }t\ge 0\text{ and }\underset{t\to \infty }{\lim }{\tau }_i(t)=\infty \text{, }1\le i\le m\text{.}$$

$$\underset{t\to \infty }{\mathrm{liminf}}\underset{{\tau }_{\max }(t)}{\overset{t}{\int }}\sum _{i=1}^m{p}_i(s)ds>\frac{1}{e},$$

$$\underset{t\to \infty }{\mathrm{liminf}}\sum _{i=1}^m{p}_i(t)\left(t-{\tau }_i(t)\right)>\frac{1}{e},$$

$$x^{\prime }(t)+p(t)x\left(\tau (t)\right)=0,\forall t\ge 0,$$

$$0<\alpha :=\underset{t\to \infty }{\mathrm{liminf}}{\int }_{\tau (t)}^{t}p(s)ds\le 1/e,$$

$$\tau (t)\text{ is decreasing and }\underset{t\to \infty }{\mathrm{limsup}}\underset{\tau (t)}{\overset{t}{\int }}p(s)ds>\frac{1+\ln {\lambda }_0}{{\lambda }_0},$$

$$h(t)=\underset{1\le i\le m}{\max }{h}_i(t),\text{ where }{h}_i(t)=\underset{0\le s\le t}{\sup }{\tau }_i(s),t\ge 0$$

$$a_1(t,s):=\exp \left\{{\int }_s^t\sum _{i=1}^m{p}_i(\zeta )d\zeta \right\}\text{, }{a}_{r+1}(t,s):=\exp \left\{{\int }_s^t\sum _{i=1}^m{p}_i(\zeta )a_r(\zeta ,{\tau }_i(\zeta ))d\zeta \right\}.$$

$$\underset{t\to \infty }{\mathrm{limsup}}{\int }_{h(t)}^t\sum _{i=1}^m{p}_i(\zeta )a_r(h(t),{\tau }_i(\zeta ))d\zeta >1,$$

$$0<\alpha :=\underset{t\to \infty }{\mathrm{liminf}}{\int }_{h(t)}^t\sum _{i=1}^m{p}_i(s)ds\le 1/e,$$

$$\underset{t\to \infty }{\mathrm{limsup}}{\int }_{h(t)}^t\sum _{i=1}^m{p}_i(\zeta )a_r(h(t),{\tau }_i(\zeta ))d\zeta >1-\frac{1-\alpha -\sqrt{1-2\alpha -{\alpha }^2}}{2},$$

$$x(t)a_r(t,s)\le x(s),t\ge s\ge 0.$$

$$0<\alpha :=\underset{t\to \infty }{\mathrm{liminf}}{\int }_{\tau (t)}^t\sum _{i=1}^m{p}_i(s)ds\le \frac{1}{e}\text{,}$$

$$\underset{t\to \infty }{\mathrm{liminf}}\frac{x(h(t))}{x(t)}\ge {\lambda }_0\text{,}$$

$$x^{\prime }(t)=-\sum _{i=1}^m{p}_i(t)x\left({\tau }_i(t)\right)\le 0,\text{ for all }t\ge t_1\text{,}$$

$$\underset{t\to \infty }{\mathrm{liminf}}{\int }_{h(t)}^t\sum _{i=1}^m{p}_i(s)ds=\underset{t\to \infty }{\mathrm{liminf}}{\int }_{\tau (t)}^t\sum _{i=1}^m{p}_i(s)ds=\alpha .$$

$${\int }_{h(t)}^t\sum _{i=1}^m{p}_i(s)ds\ge \alpha -\epsilon \text{ for }t\ge t_{\epsilon }\ge t_1\text{.}$$

$$\underset{t\to \infty }{\mathrm{liminf}}\frac{x(h(t))}{x(t)}\ge {\lambda }_0\left(\epsilon \right)\text{,}$$

$$e^{\left(\alpha -\epsilon \right)\lambda }=\lambda \text{.}$$

$$\frac{e^{\left(\alpha -\epsilon \right)\gamma }}{\gamma }\ge 1+{\epsilon }_0\text{,}$$

$$\gamma =\underset{t\to \infty }{\mathrm{liminf}}\frac{x(h(t))}{x(t)}<{\lambda }_0\left(\epsilon \right).$$

$$\frac{x(h(t))}{x(t)}\ge \gamma -\delta \text{ for }t\ge t_{\delta }\text{.}$$

$$-\frac{x^{\prime }(t)}{x(t)}=\sum _{i=1}^m{p}_i(t)\frac{x\left({\tau }_i(t)\right)}{x(t)}\ge \sum _{i=1}^m{p}_i(t)\frac{x\left(h(t)\right)}{x(t)}\ge \left(\gamma -\delta \right)\sum _{i=1}^m{p}_i(t)\text{.}$$

$$-{\int }_{h(t)}^t\frac{x^{\prime }(s)}{x(s)}ds\ge \left(\gamma -\delta \right){\int }_{h(t)}^t\sum _{i=1}^m{p}_i(s)ds\ge \left(\gamma -\delta \right)\left(\alpha -\epsilon \right)\text{,}$$

$$\frac{x\left(h(t)\right)}{x(t)}\ge e^{\left(\alpha -\epsilon \right)\left(\gamma -\delta \right)}\text{ for large }t\text{.}$$

$$\gamma =\underset{t\to \infty }{\mathrm{liminf}}\frac{x(h(t))}{x(t)}\ge e^{\left(\alpha -\epsilon \right)\left(\gamma -\delta \right)}$$

$$\gamma \ge e^{\left(\alpha -\epsilon \right)\gamma }\text{.}$$

$$\underset{t\to \infty }{\mathrm{limsup}}{\int }_{h(t)}^t\sum _{i=1}^m{p}_i(\zeta )a_r(h(\zeta ),{\tau }_i(\zeta ))d\zeta >\frac{1+\ln {\lambda }_0}{{\lambda }_0},$$

$$x^{\prime }(t)=-\sum _{i=1}^m{p}_i(t)x\left({\tau }_i(t)\right)\le 0,\text{ for all }t\ge t_1\text{,}$$

Full text

Keywords: differential equation, non-monotone delay argument, oscillatory solutions, nonoscillatory solutions.

2010 Mathematics Subject Classification: 34K11, 34K06.

*▼Corresponding author : George E. Chatzarakis; email address: geaxatz@otenet.gr; gea.xatz@aspete.gr; tel. +30-210-2896774; Greece *

1. INTRODUCTION

The paper deals with the differential equation with several non-monotone delay arguments of the form

[TABLE]

where $p_{i}$, $1\leq i\leq m$, are functions of nonnegative real numbers, and $\tau_{i}$, $1\leq i\leq m$, are non-monotone functions of positive real numbers such that

[TABLE]

Let $T_{0}\in[0,+\infty)$, $\tau(t)=\min\{\tau_{i}(t):i=1,\dots,m\}$ and $\tau_{(-1)}(t)=\sup\{s:\tau(s)\leq t\}$. By a solution of the equation (1.1) we understand a function $x\in C([T_{0},+\infty);\mathbb{R})$, continuously differentiable on $[\tau_{(-1)}(T_{0}),+\infty)$ and that satisfies (1.1) for $t\geq\tau_{(-1)}(T_{0})$.

A solution $x(t)$ of (1.1) is oscillatory, if it is neither eventually positive nor eventually negative. If there exists an eventually positive or an eventually negative solution, the equation is nonoscillatory. An equation is oscillatory if all its solutions oscillate.

The problem of establishing sufficient conditions for the oscillation of all solutions of equation (1.1) has been the subject of many investigations. See, for example, [2, 3, 5$-$13, 15,17,18] and the references cited therein. Most of these papers concern the special case where the arguments are nondecreasing, while a small number of these papers are dealing with the general case where the arguments are non-monotone. See, for example, [2,3, 16] and the references cited therein. For the general oscillation theory of differential equations the reader is referred to the monographs [1, 4, 14].

In 1978 Ladde [13] and in 1982 Ladas and Stavroulakis [12] proved that if

[TABLE]

where $\tau_{\max}(t)=\max_{1\leq i\leq m}\{\tau_{i}(t)\},$ then all solutions of (1.1) oscillate.

In 1984, Hunt and Yorke [7] proved that if $t-\tau_{i}(t)\leq\tau_{0}$, $1\leq i\leq m,$ and

[TABLE]

then all solutions of (1.1) oscillate.

When $m=1,$ that is in the special case of the equation

[TABLE]

in 1991, Kwong [11], proved that if

[TABLE]

[TABLE]

where $\lambda_{0}$ is the smaller root of the equation $\lambda=e^{\alpha\lambda}$, then all solutions of $(1.1)^{\prime}$ oscillate.

Recently, Braverman, Chatzarakis and Stavroulakis [2], established the following theorem in the general case that the arguments $\tau_{i}(t)$, $1\leq i\leq m$ are non-monotone.

Theorem 1**.**

Assume that $p_{i}(t)\geq 0$, $1\leq i\leq m$,

[TABLE]

and $a_{r}(t,s)$, $r\in{\mathbb{N}}$ are defined as

[TABLE]

If for some $r\in\mathbb{N}$

[TABLE]

or

[TABLE]

and

[TABLE]

then all solutions of $(1.1)$ oscillate.

An oscillation criterion involving $\lim\sup$, which essentially improves the above results is established. An example illustrating the result is also given.

2. MAIN RESULT

The proof of our main result is essentially based on the following lemmas.

Lemma 1**.**

$[$2, Lemma 1$]$ Assume that $x(t)$ is a positive solution of $(1.1)$ and $a_{r}(t,s)$ are defined by $(1.7)$. Then

[TABLE]

Lemma 2**.**

$[$cf. 8$]$ Assume that $x(t)$ is a positive solution of $(1.1)$, and

[TABLE]

where $\tau(t)=\max_{1\leq i\leq m}\tau_{i}(t)$. Then we have

[TABLE]

where $h(t)$ is defined by $(1.6)$ and $\lambda_{0}$ is the smaller root of the equation $\lambda=e^{\alpha\lambda}$.

Proof.

Assume that $x(t)$ is an eventually positive solution of $(1.1)$. Then there exists $t_{1}>0$ such that x(t),\ x\left(\tau_{i}(t)\right)>0,\for all $t\geq t_{1}.$ Thus, from $(1.1)$ we have

[TABLE]

which means that $x(t)$ is an eventually nonincreasing function of positive numbers.

Also, by a similar procedure as in the proof of Lemma 2.1.1 [4], we have

[TABLE]

In view of this, for any $\varepsilon\in\left(0,\alpha\right)$, there exists $t_{\varepsilon}\in\mathbb{R}_{+}$ such that

[TABLE]

We will show that

[TABLE]

where $\lambda_{0}\left(\varepsilon\right)$ is the smaller root of the equation

[TABLE]

Assume, for the sake of contradiction, that (2.6) is not satisfied. Then there exists $\varepsilon_{0}>0$ such that

[TABLE]

where

[TABLE]

On the other hand, for any $\delta>0$ there exists $t_{\delta}$ such that

[TABLE]

Dividing $(1.1)$ by $x(t)$ we obtain

[TABLE]

Integrating last inequality from $h(t)$ to $t$ for sufficiently large $t$, and taking into account (2.5), we have

[TABLE]

or

[TABLE]

Therefore

[TABLE]

which implies

[TABLE]

This contradicts (2.7) and therefore (2.6) is true. Thus, as $\varepsilon\rightarrow 0$, (2.6) implies (2.3). The proof of the lemma is complete.

Remark 1**.**

If $\alpha>1/e$ then equation $\lambda=e^{\alpha\lambda}$ has no real roots. In this case, lemma is inappropriate since $(1.1)$ does not have nonoscillatory solutions at all.

Theorem 2**.**

Assume that $(2.2)$ holds and for some $r\in\mathbb{N}$

[TABLE]

where $h(t)$ is defined by $(1.6)$, $a_{r}(t,s)$ is defined by $(1.7)$, and $\lambda_{0}$ is the smaller root of the equation $\lambda=e^{\alpha\lambda}$. Then all solutions of $(1.1)$ oscillate.

Proof.

Assume, for the sake of contradiction, that there exists a nonoscillatory solution $x(t)$ of (1.1). Since $-x(t)$ is also a solution of (1.1), we can confine our discussion only to the case where the solution $x(t)$ is eventually positive. Then there exists $t_{1}>0$ such that x(t),\ x\left(\tau_{i}(t)\right)>0,\for all $t\geq t_{1}.$ Thus, from (1.1) we have

[TABLE]

which means that $x(t)$ is an eventually nonincreasing function of positive numbers.

By Lemma 2, inequality (2.3) is fulfilled. Therefore

[TABLE]

where $\varepsilon$ is an arbitrary real number with $0<\varepsilon<\lambda_{0}$. Thus, there exists a $t^{\ast}\in(h(t),t)$ such that

[TABLE]

Integrating (1.1) from $t^{\ast}$ to $t$ and using Lemma 1, we have

[TABLE]

Hence

[TABLE]

or

[TABLE]

which, in view of (2.10), gives

[TABLE]

Dividing (1.1) by $x\left(t\right),$ integrating from $h(t)$ to $t^{\ast}$ and using Lemma 1, we have

[TABLE]

Taking into account the fact that $\zeta\geq h(t)$ the inequality (2.9) guarantees that

[TABLE]

In view of this, (2.12) gives

[TABLE]

or

[TABLE]

i.e.,

[TABLE]

Combining the inequalities (2.11) and (2.14), we have

[TABLE]

The last inequality holds true for all real numbers $\varepsilon$ with $0<\varepsilon<\lambda_{0}$. Hence, for $\varepsilon\rightarrow 0,$ we have

[TABLE]

which contradicts (2.8). The proof of the theorem is complete.

Example 1**.**

Consider the delay differential equation

[TABLE]

with

[TABLE]

where $\mathbb{N}_{0}$ is the set of non-negative integers.

By $(1.6)$, we see that

[TABLE]

and consequently

[TABLE]

Observe that the function $f:\mathbb{R}_{0}\rightarrow\mathbb{R}_{+}$ defined as $f_{r}(t)=\int_{h(t)}^{t}\sum_{i=1}^{m}p_{i}(\zeta)a_{r}(h(\zeta),\tau_{i}(\zeta))~{}d\zeta$ attains its maximum at $t=3k+2.6,$ $k\in\mathbb{N}_{0}$, for every $r\in\mathbb{N}$. Specifically,

[TABLE]

where

[TABLE]

Thus

[TABLE]

Now, we see that

[TABLE]

[TABLE]

and

[TABLE]

that is, none of the conditions $(1.3)$, $(1.4)$ and $(1.8)$ is satisfied.

Observe, however, that the smaller root of $e^{\alpha\lambda}=\lambda$ is $\lambda_{0}=1.49883$. Thus

[TABLE]

That is, condition $(2.8)$ of Theorem 2 is satisfied for $r=1$, and therefore all solutions of $(2.15)$ oscillate.

Acknowledgement 1**.**

The authors would like to thank both referees for the constructive remarks which improved the presentation of the paper.