Dynamics of a stochastic hybrid delay one-predator-two-prey model with harvesting and jumps in a polluted environment

1.   Introduction

In this article, we study the oscillatory behavior of the fourth-order neutral nonlinear differential equation of the form

where $w\left(t\right) : = u\left(t\right) +a\left(t\right) u\left(\tau \left(t\right) \right)$ and the first term means the $p$-Laplace type operator ($1 < p < \infty$). The main results are obtained under the following conditions:

$L1:$ $\Phi _{p_{i}}[s] = |s|^{p_{i}-2}s, \ i = 1, 2,$

$L2:$ $r\in C[t_{0}, \infty)$ and under the condition

$L3:$ $a, q\in C[t_{0}, \infty),$ $q\left(t\right) > 0,$ $0\leq a\left(t\right) < a_{0} < \infty,$ $\tau, \vartheta \in C[t_{0}, \infty),$ $\tau \left(t\right) \leq t,$ $\lim_{t\rightarrow \infty }\tau \left(t\right) = \lim_{t\rightarrow \infty }\vartheta \left(t\right) = \infty$

By a solution of (1.1) we mean a function $u$ $\in C^{3}[t_{u}, \infty), \ t_{u}\geq t_{0}, \ $which has the property $r\left(t\right) \left(w^{\prime \prime \prime }\left(t\right) \right) ^{p_{1}-1}\in C^{1}[t_{u}, \infty), \ $and satisfies (1.1) on $[t_{u}, \infty).\ $We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on $[t_{u}, \infty),$ and otherwise it is called to be nonoscillatory. (1.1) is said to be oscillatory if all its solutions are oscillatory.

We point out that delay differential equations have applications in dynamical systems, optimization, and in the mathematical modeling of engineering problems, such as electrical power systems, control systems, networks, materials, see ^[1]. The $p$-Laplace equations have some significant applications in elasticity theory and continuum mechanics.

During the past few years, there has been constant interest to study the asymptotic properties for oscillation of differential equations with $p$-Laplacian like operator in the canonical case and the noncanonical case, see ^[2,3,4,11] and the numerical solution of the neutral delay differential equations, see ^[5,6,7]. The oscillatory properties of differential equations are fairly well studied by authors in ^{[16,17,18,19,20,21,22,23,24,25,26,27]}. We collect some relevant facts and auxiliary results from the existing literature.

Liu et al. ^[4] studied the oscillation of even-order half-linear functional differential equations with damping of the form

where $n\ $is even. This time, the authors used comparison method with second order equations.

The authors in ^[9,10] have established sufficient conditions for the oscillation of the solutions of

where$\ n\ $is even$\ $and $p > 1\ $is a real number$,$ in the case where $\vartheta _{i}\left(t\right) \geq \upsilon$ (with $r\in C^{1}\left((0, \infty), \mathbb{R}\right)$, $q_{i}\in C\left([0, \infty), \mathbb{R} \right), \ i = 1, 2, .., j$).

We point out that Li et al. ^[3] using the Riccati transformation together with integral averaging technique, focuses on the oscillation of equation

Park et al. ^[8] have obtained sufficient conditions for oscillation of solutions of

As we already mentioned in the Introduction, our aim here is complement results in ^[8,9,10]. For this purpose we discussed briefly these results.

In this paper, we obtain some new oscillation criteria for (1.1). The paper is organized as follows. In the next sections, we will mention some auxiliary lemmas, also, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of (1.1), and we will give some examples to illustrate the main results.

2.   Main results

For convenience, we denote

for some $\mu \in \left(0, 1\right)$ and every $M_{1}, M_{2}\ $are positive constants.

Definition 1. A sequence of functions $\left\{ \delta _{n}\left(t\right) \right\} _{n = 0}^{\infty }$ and $\left\{ \sigma _{n}\left(t\right) \right\} _{n = 0}^{\infty }\ $as

We see by induction that $\delta _{n}\left(t\right) \leq \delta _{n+1}\left(t\right)$ and $\sigma _{n}\left(t\right) \leq \sigma _{n+1}\left(t\right)$ for$\ t\geq t_{0}, \ n > 1.$

In order to discuss our main results, we need the following lemmas:

Lemma 2.1. ^[12] If the function $w$ satisfies $w^{(i)}\left(\nu \right) > 0,$ $i = 0, 1, ..., n,$ and $w^{\left(n+1\right) }\left(\nu \right) < 0\ $ eventually. Then, for every $\varepsilon _{1}\in \left(0, 1\right), \ w\left(\nu \right) /w^{\prime }\left(\nu \right) \geq \varepsilon _{1}\nu /n\ $eventually.

Lemma 2.2. ^[13] Let $u\left(t\right)$ be a positive and $n$-times differentiable function on an interval $\left[ T, \infty \right)$ with its $n$th derivative $u^{\left(n\right) }\left(t\right) \ $non-positive on $\left[ T, \infty \right)$ and not identically zero on any interval of the form $\left[ T^{\prime }, \infty \right), \ T^{\prime }\geq T\ $and $u^{\left(n-1\right) }\left(t\right) u^{\left(n\right) }\left(t\right) \leq 0, \ t\geq t_{u}\ $then there exist constants $\theta, \ 0 < \theta < 1\ $ and $\varepsilon > 0\ $such that

for all sufficient large $t$.

Lemma 2.3 ^[14] Let $u\in C^{n}\left(\left[ t_{0}, \infty \right), \left(0, \infty \right) \right).$ Assume that $u^{\left(n\right) }\left(t\right)$ is of fixed sign and not identically zero on $\left[ t_{0}, \infty \right)$ and that there exists a $t_{1}\geq t_{0}$ such that $u^{\left(n-1\right) }\left(t\right) u^{\left(n\right) }\left(t\right) \leq 0$ for all $t\geq t_{1}$. If $\lim_{t\rightarrow \infty }u\left(t\right) \neq 0,$ then for every $\mu \in \left(0, 1\right)$ there exists $t_{\mu }\geq t_{1}$ such that

Lemma 2.4. ^[15] Assume that (1.2) holds and $u\ $is an eventually positive solution of (1.1). Then, $\left(r\left(t\right) \left(w^{\prime \prime \prime }\left(t\right) \right) ^{p_{1}-1}\right) ^{\prime } < 0$ and there are the following two possible cases eventually:

Theorem 2.1. Assume that

Then (1.1) is oscillatory.

proof. Assume that $u$ be an eventually positive solution of (1.1). Then, there exists a $t_{1}\geq t_{0}$ such that $u\left(t\right) > 0,$ $u\left(\tau \left(t\right) \right) > 0$ and $u\left(\vartheta \left(t\right) \right) > 0$ for $t\geq t_{1}$. Since $r^{\prime }\left(t\right) > 0$, we have

for $t\geq t_{1}$. From definition of $w$, we get

which with (1.1) gives

Define

for some a constant $\zeta \in \left(0, 1\right).$ By differentiating and using (2.4), we obtain

From Lemma 2.2, there exist constant $\varepsilon > 0$, we have

Which is

by using (2.5) we have

Since $w^{\prime }\left(t\right) > 0$, there exist a $t_{2}\geq t_{1}$ and a constant $M > 0$ such that

Then, (2.6), turns to

that is

Integrating the above inequality from $t$ to $l$, we get

Letting $l\rightarrow \infty$ and using $\varpi > 0$ and $\varpi ^{\prime } < 0$, we have

This implies

Let $\lambda = \inf_{t\geq T}\varpi \left(t\right) /\phi _{1}\left(t\right)$ then obviously $\lambda \geq 1$. Thus, from (2.2) and (2.7) we see that

or

which contradicts the admissible value of $\lambda \geq 1\ $and $\left(p_{1}-1\right) > 0$.

Therefore, the proof is complete.

Theorem 2.2. Assume that

and

Then (1.1) is oscillatory.

proof. Assume to the contrary that (1.1) has a nonoscillatory solution in $\left[ t_{0}, \infty \right)$. Without loss of generality, we let $u$ be an eventually positive solution of (1.1). Then, there exists a $t_{1}\geq t_{0}$ such that $u\left(t\right) > 0,$ $u\left(\tau \left(t\right) \right) > 0$ and $u\left(\vartheta \left(t\right) \right) > 0$ for $t\geq t_{1}$. From Lemma 2.4 there is two cases $\left(\bf{G}_{1}\right) \ $and $\left(\bf{G}_{2}\right)$.

For case $\left(\bf{G}_{1}\right)$. Define

By differentiating $\omega$ and using (2.4), we obtain

From Lemma 2.1, we get

Integrating again from $t$ to $\vartheta \left(t\right)$, we find

It follows from Lemma 2.3 that

for all $\mu _{1}\in \left(0, 1\right)$ and every sufficiently large $t$. Since $w^{\prime }\left(t\right) > 0$, there exist a $t_{2}\geq t_{1}$ and a constant $M > 0$ such that

for $t\geq t_{2}$. Thus, by (2.10), (2.11), (2.12) and (2.13), we get

that is

Integrating (2.14) from $t$ to $l$, we get

Letting $l\rightarrow \infty$ and using $\omega > 0$ and $\omega ^{\prime } < 0$, we have

This implies

Let $\lambda = \inf_{t\geq T}\omega \left(t\right) /\xi _{\ast }\left(t\right)$ then obviously $\lambda \geq 1$. Thus, from (2.8) and (2.16) we see that

or

which contradicts the admissible value of $\lambda \geq 1\ $and $\left(p_{1}-1\right) > 0$.

For case $\left(\bf{G}_{2}\right).\ $Integrating (2.4) from $t$ to $m$, we obtain

From Lemma 2.1, we get that

For (2.17), letting $m\rightarrow \infty \,$and using (2.18), we see that

Integrating this inequality again from $t$ to $\infty$, we get

for all $\varepsilon _{1}\in \left(0, 1\right)$. Define

By differentiating $y$ and using (2.13) and (2.19), we find

hence

The proof of the case where $\left(\bf{G}_{2}\right)$ holds is the same as that of case $\left(\bf{G}_{1}\right)$. Therefore, the proof is complete.

Theorem 2.3. Let $\delta _{n}\left(t\right) \ $and $\sigma _{n}\left(t\right) \ $be defined as in (2.1). If

and

for some $n$, then (1.1)is oscillatory.

proof. Assume to the contrary that (1.1) has a nonoscillatory solution in $\left[ t_{0}, \infty \right)$. Without loss of generality, we let $u$ be an eventually positive solution of (1.1). Then, there exists a $t_{1}\geq t_{0}$ such that $u\left(t\right) > 0,$ $u\left(\tau \left(t\right) \right) > 0$ and $u\left(\vartheta \left(t\right) \right) > 0$ for $t\geq t_{1}$. From Lemma 2.4 there is two cases.

In the case $\left(\bf{G}_{1}\right)$, proceeding as in the proof of Theorem 2.2, we get that (2.12) holds. It follows from Lemma 2.3 that

From definition of $\omega \left(t\right)$ and (2.24), we have

Thus,

Therefore,

which contradicts (2.22).

The proof of the case where $\left(\bf{G}_{2}\right)$ holds is the same as that of case $\left(\bf{G}_{1}\right)$. Therefore, the proof is complete.

Corollary 2.1. Let $\delta _{n}\left(t\right) \ $and $\sigma _{n}\left(t\right) \ $be defined as in (2.1). If

and

for some $n$, then (1.1) is oscillatory.

proof. Assume to the contrary that (1.1) has a nonoscillatory solution in $\left[ t_{0}, \infty \right)$. Without loss of generality, we let $u$ be an eventually positive solution of (1.1). Then, there exists a $t_{1}\geq t_{0}$ such that $u\left(t\right) > 0,$ $u\left(\tau \left(t\right) \right) > 0$ and $u\left(\vartheta \left(t\right) \right) > 0$ for $t\geq t_{1}$. From Lemma 2.4 there is two cases $\left(\bf{G}_{1}\right) \ $and $\left(\bf{G}_{2}\right)$.

In the case $\left(\bf{G}_{1}\right)$, proceeding as in the proof of Theorem 2, we get that (2.15) holds. It follows from (2.15) that $\omega \left(t\right) \geq \delta _{0}\left(t\right)$.$\ $ Moreover, by induction we can also see that $\omega \left(t\right) \geq \delta _{n}\left(t\right) \ $for $t\geq t_{0}, \ n > 1$. Since the sequence $\left\{ \delta _{n}\left(t\right) \right\} _{n = 0}^{\infty }$ monotone increasing and bounded above, it converges to $\delta \left(t\right)$. Thus, by using Lebesgue's monotone convergence theorem, we see that

and

Since $\delta _{n}\left(t\right) \leq \delta \left(t\right)$, it follows from (2.27) that

Hence, we get

This implies

which contradicts (2.25). The proof of the case where $\left(\bf{G} _{2}\right)$ holds is the same as that of case $\left(\bf{G} _{1}\right)$. Therefore, the proof is complete.

Example 2.1. Consider the differential equation

where $q_{0} > 0$ is a constant. Let $p_{1} = p_{2} = 2,$ $r\left(t\right) = 1,$ $a\left(t\right) = 1/2, \ \tau \left(t\right) = t/2,$ $\vartheta \left(t\right) = t/3$ and $q\left(t\right) = q_{0}/t^{4}$. Hence, it is easy to see that

and

also, for some $\varepsilon > 0$, we find

Hence, by Theorem 2.1, every solution of Eq (2.28) is oscillatory if $q_{0} > 121.5\varepsilon.$

Example 2.2. Consider a differential equation

where $q_{0} > 0$ is a constant. Note that $p = 2, \ t_{0} = 1, \ r\left(t\right) = 1, \ a\left(t\right) = a_{0}, \ \tau \left(t\right) = \tau _{0}t, \ \vartheta \left(t\right) = \vartheta _{0}t\ $ and $q\left(t\right) = q_{0}/t^{n}.$

Easily, we see that condition (2.8) holds and condition (2.9) satisfied$.$

Hence, by Theorem 2.2, every solution of Eq (2.29) is oscillatory$.$

Remark 2.1. Finally, we point out that continuing this line of work, we can have oscillatory results for a fourth order equation of the type:

3.   Conclusion

The paper is devoted to the study of oscillation of fourth-order differential equations with $p$-Laplacian like operators. New oscillation criteria are established by using a Riccati transformations, and they essentially improves the related contributions to the subject.

Further, in the future work we get some Hille and Nehari type and Philos type oscillation criteria of (1.1) under the condition $\int_{\upsilon _{0}}^{\infty }\frac{1}{r^{1/\left(p_{1}-1\right) }\left(s\right) }\mathrm{ d}s < \infty.$

Acknowledgments

The authors express their debt of gratitude to the editors and the anonymous referee for accurate reading of the manuscript and beneficial comments.

Conflict of interest

The author declares that there is no competing interest.