Oscillatory Behavior of Second-Order Neutral Differential Equations

Abstract

In this paper, we study oscillatory properties of neutral differential equations. Moreover, we discuss some examples that show the effectiveness and the feasibility of the main results.

1 Introduction

Delay differential equations are widely used in mathematical modeling to describe physical and biological systems, by inducing oscillatory behavior.

In the literature, several mathematical models with different levels of complexity have been proposed for delay differential equations in order to represent for example the cardiovascular system (CVS).

The pioneering and remarkable paper of Ottesen (1997) shows how to use delay differential equations to solve a cardiovascular model that has a discontinuous derivative. Ottesen (1997) also illustrated that complex dynamic interactions between nonlinear behaviors and delays associated with the autonomic-cardiac regulation may cause instability (Ataeea et al. 2015).

Moreover, a model-based approach to stability analysis of autonomic-cardiac regulation was studied in Ataeea et al. (2015); specifically, it is important to underline that the autonomic-cardiac regulation operates by the interaction between autonomic nervous system (ANS) and cardiovascular system (CVS) (Ataeea et al. 2015).

It is clear that mathematical analysis related to physics-based models can be a versatile tool in examining delay differential equations from the point of view of medical and biological systems.

In this paper we consider the following equation of neutral type

$$\begin{aligned} \left( a(y)\left( w'(y)\right) ^\gamma \right) ' +\sum _{j=1}^{m_2}q_j(y)x^{\beta _j}\left( \vartheta _j(y)\right) =0, \quad y\ge y_0, \end{aligned}$$

(1.1)

belonging to those families used to model problems that arise in the biological sciences. Our aim is to study the oscillatory behavior of (1.1) where $w(y)=x(y)+\sum _{i=1}^{m_1}p_i(y)x^{\alpha _i}\left( \varsigma _i(y)\right)$, $\alpha _i$, $\gamma$ and $\beta _j$, for all $i=1,\ldots ,m_1$ and $j=1,\ldots ,m_2$, are quotients of odd positive integers.

Moreover, many researchers study qualitative properties of delay mathematical models examining oscillation and nonoscillation properties of different delay logistic models and their modifications (Agarwal et al. 2014c). These studies are concerned also with the investigation of local and global stability. Mainly the oscillation properties are investigated for models with delayed feedback, hyperlogistic models and models with varying capacity. For further details regarding the techniques and other applications to Biology we refer the reader to Agarwal et al. (2014a, 2014b, 2014c, 2015, 2016), Baculíková et al. (2011), Džurina et al. (2020), Fisnarova and Marik (2017), Grace et al. (2018), Li and Rogovchenko (2014, 2015, 2017), Li et al. (2015), Pinelas and Santra (2018), Qian and Xu (2011), Santra (2016, 2017, 2019a, 2019b, 2020a, 2020b); Santra and Dix (2020) Tripathy and Santra (2020), Zhang et al. (2015); Bazighifan (2020a, 2020b); Chatzarakis et al. (2019b), Moaaz et al. (2017), Bazifghifan and Ramos (2020) and Bazighifan et al. (2020a).

For a recent review on the asymptotic properties for functional differential equations (FDEs), we suggest to the reader the interesting book Berezansky et al. (2020).

2 Mathematical Background and Hypotheses

Throughout this work, we assume that the following assumptions are fulfilled for Eq. (1.1):

• (A1) $\vartheta _j, \varsigma _i \in C([y_0,\infty ),\mathbb {R_+})$, $\varsigma _i \in C^2([y_0,\infty ),\mathbb {R_+})$, $\vartheta _j(y)<y$, $\varsigma _i(y)<y$, $\lim _{y\rightarrow \infty }\vartheta _j(y)=\infty$, $\lim _{y\rightarrow \infty }\varsigma _i(y)=\infty$ for all $i=1,2,\ldots ,m_1$ and $j=1,2,\ldots ,m_2$;

• (A2) $a\in C^1([y_0,\infty ),\mathbb {R_+})$, $q_j \in C([y_0,\infty ),\mathbb {R_+})$; $0\le q_j(y)$, for all $y\ge 0$ and $j=1,2,\ldots ,m_2$; $\sum _{j=1}^{m_2} q_j(y)$ is not identically zero in any interval $[b,\infty )$;

• (A3) $\lim _{y \rightarrow \infty }A(y)=\infty$, where $A(y)=\int _{y_0}^y a^{-1/\gamma }(\eta )\,d\eta$;

• (A4) $p_i : [y_0,\infty ) \rightarrow \mathbb {R^+}$ are continuous functions for $i=1,2,\ldots ,m$;

• (A5) there exists a differentiable function $\vartheta _0(y)$ satisfying the properties $\displaystyle 0<\vartheta _0(y)=\min \nolimits _{j=1,\ldots ,m_2}\{\vartheta _j(y): y \ge y^*>y_0\}$ and $\vartheta _0'(y) \ge \vartheta _0$ for $y\ge y^*>y_0$, $\vartheta _0>0$.

Now we recall some basic definitions.

Definition 2.1

A function $x(y):[y_{x},\infty )\rightarrow \mathbb {R} ,y_{x}\ge y_{0}$ is said to be a solution of (1.1) if x(y) and $a(y)\left( w'(y)\right) ^\gamma$ are continuously differentiable for all $y\in [y_{x},\infty )$ and it satisfies the equation (1.1) for all $y\in [y_{x},\infty )$.

We assume that (1.1) admits a solution in the sense of Definition 2.1.

Definition 2.2

A solution x(y) of (1.1) is said to be non-oscillatory if it is eventually positive or eventually negative; otherwise, it is said to be oscillatory.

Definition 2.3

Equation (1.1) is said to be oscillatory if all of its solutions are oscillatory.

In this paper, we restrict our attention to study oscillation and non-oscillation of (1.1). First of all, it is interesting to make a review in the context of functional differential equation.

Brands (1978) proved that for each bounded delay $\vartheta (y)$, the equation

$$\begin{aligned} x''(y)+q(y)x(y-\vartheta (y)) =0 \end{aligned}$$

is oscillatory if and only if the equation

$$\begin{aligned} x''(y)+q(y)x(y)=0 \end{aligned}$$

is oscillatory. Chatzarakis et al. (2019a) and Chatzarakis and Jadlovská (2019) considered the following more general equation

$$\begin{aligned} \left( a(x^{\prime })^\beta \right) ^{\prime }(y)+q(y)x^{\beta }(\vartheta (y))=0 \end{aligned}$$

(2.1)

and established new oscillation criteria for (2.1) when $\lim _{y \rightarrow \infty }A(y)=\infty$ and $\lim _{y \rightarrow \infty }A(y)<\infty$.

Wong (2000) has obtained oscillation conditions of

$$\begin{aligned} \bigl (x(y)+px(y-\varsigma )\bigr )''+ q(y)f\left( x(y-\vartheta )\right) =0, \quad -1<p<0 \end{aligned}$$

in which the neutral coefficient and delays are constants. In Baculíková and Džurina (2011a) and Džurina (2011), the authors studied the equation

$$\begin{aligned} \left( a(y)\left( w'(y)\right) ^\gamma \right) ' + q(y)x^{\beta }(\vartheta (y))=0, \quad w(y)=x(y)+p(y)x(\varsigma (y)), \quad y\ge y_0, \end{aligned}$$

(2.2)

and established the oscillation of solutions of (2.2) using comparison techniques when $\gamma =\beta =1$, $0\le {}p(y)<\infty$ and $\lim _{y \rightarrow \infty }A(y)=\infty$. Using the same technique, Baculíková and Džurina (2011b) considered (2.2) and obtained oscillation conditions of (2.2) considering the assumptions $0\le {}p(y)<\infty$ and $\lim _{y \rightarrow \infty }A(y)=\infty$. Tripathy et al. (2016), studied (2.2) and established several conditions of the solutions of (2.2) considering the assumptions $\lim _{y \rightarrow \infty }A(y)=\infty$ and $\lim _{y \rightarrow \infty }A(y)<\infty$ for different values of the neutral coefficient p. Bohner et al. (2017) obtained sufficient conditions for the oscillation of the solutions of (2.2) when $\gamma =\beta$, $\lim _{y \rightarrow \infty }A(y)<\infty$ and $0 \le p(y)<1$. Grace et al. (2018) studied the oscillation of (2.2) when $\gamma =\beta _j$, assuming that $\lim _{y \rightarrow \infty }A(y)<\infty$, $\lim _{y \rightarrow \infty }A(y)=\infty$ and $0 \le p(y)<1$. Li et al. (2019) established sufficient conditions for the oscillation of the solutions of (2.2), under the assumptions $\lim _{y \rightarrow \infty }A(y)<\infty$ and $p(y) \ge 0$. Karpuz and Santra (2019) studied the equation

$$\begin{aligned} \bigl (a(y)(x(y)+p(y)x(\varsigma (y)))^\prime \bigr )^\prime +q(y)f\bigl (x(\vartheta (y))\bigr )=0, \end{aligned}$$

considering the assumptions $\lim _{y \rightarrow \infty }A(y)<\infty$ and $\lim _{y \rightarrow \infty }A(y)=\infty$, for different values of p.

For any positive, continuous and decreasing to zero function $\rho : [y_0, \infty ) \rightarrow \mathbb {R^{+}}$, we set

$$\begin{aligned}&P(y)= \left( 1-\sum _{i=1}^{m}\alpha _i p_i(y)-\frac{1}{\rho (y)}\sum _{i=1}^{m}(1-\alpha _i)p_i(y)\right) ; \\&Q_1(y)=\sum _{j=1}^{m_2}q_j(y)P^{\beta _j}\left( \vartheta _j(y)\right) ; \\&Q_2(y)=\sum _{j=1}^{m_2}q_j(y) P^{\beta _j}\left( \vartheta _j(y)\right) \rho ^{\beta _j-1}\left( \vartheta _j(y)\right) ; \\&Q_3(y)=\sum _{j=1}^{m_2}q_j(y) P^{\beta _j}\left( \vartheta _j(y)\right) A^{\beta _j-1}\left( \vartheta _j(y)\right) ; \\&Q_4(y)=\sum _{j=1}^{m_2}q_j(y) P^{\beta _j}\left( \vartheta _j(y)\right) A^{\beta _j}(\vartheta _j(y)); \\&U(y)=\int _y^\infty \sum _{j=1}^{m_2}q_j(\zeta )x^{\beta _j}(\vartheta _j(\zeta ))\,d\zeta \,. \end{aligned}$$

Let us assume that P(y) and U(y) are non-negative in $[y_0,\infty )$.

We now recall the technical lemmas and the main results contained in Bazighifan et al. (2020b).

Lemma 2.1

Let (A1)–(A4) hold for $y\ge y_0$. If a solution x of (1.1) is eventually positive, then w satisfies

$$\begin{aligned} w(y)>0, \quad w'(y)>0, \quad \text {and} \quad \left( a(w')^\gamma \right) '(y) \le 0 \quad \text {for}\quad y\ge y_1. \end{aligned}$$

(2.3)

Lemma 2.2

Let (A1)–(A4) hold for $y\ge y_0$. If a solution x of (1.1) is eventually positive, then w satisfies

$$\begin{aligned} w(y) \ge \left( a(y)\right) ^{1/\gamma }w'(y) A(y)\quad \text {for } y\ge y_1. \end{aligned}$$

and

$$\begin{aligned} \frac{w(y)}{A(y)} \text { is decreasing for }y\ge y_1. \end{aligned}$$

Lemma 2.3

Let (A1)–(A4) hold for $y\ge y_0$. If a solution x of (1.1) is eventually positive, then w satisfies

$$\begin{aligned} x(y) \ge P(y)w(y) \quad \text {for}\quad y\ge y_1. \end{aligned}$$

(2.4)

Lemma 2.4

Let (A1)–(A4) hold for $y \ge y_0$. If a solution x of (1.1) is eventually positive, then there exist $y_1 > y_0$ and $\delta >0$ such that

$$\begin{aligned}&0<w(y)\le \delta A(y) \text { and } \end{aligned}$$

(2.5)

$$\begin{aligned}&\quad A(y) U^{1/\gamma }(y) \le w(y) \end{aligned}$$

(2.6)

hold for all $y\ge y_1$.

Theorem 2.4

Assume that there exists a constant $\delta _1$, quotient of odd positive integers, such that $0<\beta _j<\delta _1<\gamma$, and (A1)–(A4) hold for $y \ge y_0$. If

• (A6) $\int _0^\infty Q_4(\eta ) \,d\eta =\infty \,.$

holds, then every solution of (1.1) is oscillatory.

Theorem 2.5

Assume that there exists a constant $\delta _2$, quotient of odd positive integers, such that $\gamma<\delta _2<\beta _j$. Furthermore, assume that (A1)–(A5) hold for $y \ge y_0$ and a(y) is non-decreasing. If

• (A7) $\int _{0}^\infty \left[ \frac{1}{a(\eta )}\int _\eta ^\infty Q_1(\zeta )\,d\zeta \right] ^{1/\gamma }\,d\eta =\infty$

holds, then every solution of (1.1) is oscillatory.

3 Oscillation Criteria for (1.1)

In this section we discuss our main results. The oscillation criteria in this paper complete the study started in Bazighifan et al. (2020b) but it is important to underline that the criteria discussed in Bazighifan et al. (2020b) differ from those examined in this work in terms of assumptions. Precisely, both the main results of Bazighifan et al. (2020b) (Theorem 1 and 2), require the existence of two constants $\delta _{1}$ and $\delta _{2}$ that are quotients of odd positive integers and the bounds for $b_{j}$ involve such constants. The results presented in this paper do not involve the existence of auxiliary constants and under fewer hypotheses guarantee the oscillatory behavior of the equations under consideration.

Theorem 3.1

Let (A1)–(A4) hold for $y \ge y_0$. If

• (A6) $\int _{0}^{\infty }Q_1(\eta )d\eta =\infty$

holds, then every solution of (1.1) is oscillatory.

Proof

Let the solution x be eventually positive. Then there exists $y_0>0$ such that $x(y)>0$, $x(\varsigma _i(y))>0$ and $x\big (\vartheta _j(y)\big )>0$ for all $y\ge y_0$ and for all $i=1,2,\ldots ,m_1$ and $i=1,2,\ldots ,m_2$. Applying Lemmas 2.1 and 2.3 for $y \ge y_1>y_0$ we conclude that w satisfies (2.3), w is increasing and $x(y) \ge P(y)w(y)$ for all $y\ge y_1$. From (1.1), we have

$$\begin{aligned} \Big (a(y)\big (w'(y)\big )^\gamma \Big )'+\sum _{j=1}^{m_2}q_j(y)P^{\beta _j}\big (\vartheta _j(y)\big )w^{\beta _j}\big (\vartheta _j(y)\big ) \le 0 \end{aligned}$$

(3.1)

for $y \ge y_1$. Applying (2.3) we conclude that $\lim _{y \rightarrow \infty } \Big (a(y)\big (w'(y)\big )^\gamma \Big )$ exists, and there exist $y_2>y_1$ and a number $c>0$ such that $w(y) \ge c$ for $y \ge y_2$. Integrating (3.1) from $y_2$ to y, for a suitable constant ${\tilde{c}}$, we have

$$\begin{aligned} {\tilde{c}}\int _{y_2}^{y}\sum _{j=1}^{m_2}q_j(\eta )P^{\beta _j}\big (\vartheta _j(\eta )\big ) d\eta \le -\Big [a(\eta )\big (w'(\eta )\big )^\gamma \Big ]_{y_2}^{y} < \infty \quad \text {as} \quad y \rightarrow \infty , \end{aligned}$$

which is a contradiction to (A6).

The case where x is an eventually negative solution is similar and we omit it here. Thus, the proof is complete. $\square$

Remark

Theorem 3.1 holds for any $\beta _j$ and $\gamma$.

Theorem 3.2

Let (A1)–(A4) hold for $y \ge y_0$ and $\beta _j>1$. If

• (A7) $\int _{0}^{\infty }Q_2(\eta )d\eta =\infty$

holds, then every solution of (1.1) is oscillatory.

Proof

Proceeding as in the proof of Theorem 3.1 we obtain (3.1). Since w(y) is positive and increasing, $\rho (y)$ is positive and decreasing to zero, there exists $y_0 \ge y_1$ such that

$$\begin{aligned} w(y) \ge \rho (y) \quad \text {for}\quad y \ge y_1. \end{aligned}$$

(3.2)

Applying (3.2) in (3.1) we have

$$\begin{aligned} \Big (a(y)\big (w'(y)\big )^\gamma \Big )'+\sum _{j=1}^{m_2}q_j(y)P^{\beta _j}\big (\vartheta _j(y)\big )\rho ^{\beta _j-1}\big (\vartheta _j(y)\big )w\big (\vartheta _j(y)\big ) \le 0. \end{aligned}$$

(3.3)

The rest of the proof is similar to that of Theorem 3.1 and hence it is omitted. $\square$

Theorem 3.3

Let (A1)–(A4) hold for $y \ge y_0$ and $0<\beta _j<1$. If

• (A8) $\int _{0}^{\infty }Q_3(\eta )d\eta =\infty$

holds, then every solution of (1.1) is oscillatory.

Proof

Proceeding as in the proof of Theorem 3.1 we obtain (3.1). Now (3.1) can be written as

$$\begin{aligned} \Big (a(y)\big (w'(y)\big )^\gamma \Big )'\!+\!\sum _{j\!=\!1}^{m_2}q_j(y)P^{\beta _j}\big (\vartheta _j(y)\big )A^{\beta _j\!-\!1}\big (\vartheta _j(y)\big )\frac{w^{\beta _j\!-\!1}\big (\vartheta _j(y)\big )}{A^{\beta _j\!-\!1}\big (\vartheta _j(y)\big )}w\big (\vartheta _j(y)\big ) \le 0 \end{aligned}$$

(3.4)

for $y \ge y_2 >y_1$. Since $\frac{w(y)}{A(y)}$ is decreasing, there exists a constant k such that

$$\begin{aligned} \frac{w(y)}{A(y)} \le k \quad \text {for} \quad y\ge y_2. \end{aligned}$$

(3.5)

Using (3.5) and $\beta _j<1$ in (3.4), we have

$$\begin{aligned} \Big (a(y)\big (w'(y)\big )^\gamma \Big )'+\sum _{j=1}^{m_2}q_j(y)\frac{P^{\beta _j}\big (\vartheta _j(y)\big )A^{\beta _j-1}\big (\vartheta _j(y)\big )}{k^{1-\beta _j}}w\big (\vartheta _j(y)\big ) \le 0. \end{aligned}$$

The rest of the proof is similar to that of Theorem 3.2 and hence it is omitted. $\square$

4 Examples

We conclude the paper presenting some examples that show the effectiveness and the feasibility of the main results.

Example 4.1

Let us consider the differential equation

$$\begin{aligned} \Big (y\Big (\Big (x(y)+\frac{1}{y}x^\frac{1}{3}\left( \frac{y}{2}\right) +\frac{1}{y^2}x^\frac{1}{5}\left( \frac{y}{3}\right) \Big )'\Big )^3\Big )' +y^6 x^{3}\left( \frac{y}{3}\right) +y^7 x^{3}\left( \frac{y}{4}\right) =0 \quad \text {for}\quad y\ge 4, \end{aligned}$$

(4.1)

where $a(y) :\equiv y$, $p_i(y) :\equiv \frac{1}{y^i}$, $\alpha _i :\equiv \frac{1}{2i+1}$, $\varsigma _i(y) :\equiv \frac{y}{i+1}$, $\beta _j=\gamma =3$, $q_j(y) :\equiv y^{j+5}$ and $\vartheta _j(y) :\equiv \frac{y}{j+2}$ for $i=1,2$, $j=1,2$ and $y \ge 4$. All the assumptions of Theorem 3.1 are fulfilled with $i=1,2$, $j=1,2$. Hence, due to Theorem 3.1, equation (4.1) is oscillatory in the sense of Definition of 2.3.

Example 4.2

Let us consider the differential equation

$$\begin{aligned} \Big (y \Big (\Big (x(y)+\frac{1}{y}x^\frac{1}{3}\left( \frac{y}{3}\right) +\frac{1}{y^2}x^\frac{1}{5}\left( \frac{y}{4}\right) \Big )'\Big )^5\Big )' +y^{\frac{6}{5}} x\left( \frac{y}{2}\right) +y^{\frac{7}{6}} x\left( \frac{y}{3}\right) =0 \quad \text {for}\quad y\ge 2, \end{aligned}$$

(4.2)

where $a(y) :\equiv y$, $p_i(y) :\equiv \frac{1}{y^i}$, $\alpha _i :\equiv \frac{1}{2i+1}$, $\varsigma _i(y) :\equiv \frac{y}{i+2}$, $\beta _j=1<\gamma =5$, $q_j(y) :\equiv y^{\frac{j+5}{j+4}}$ and $\vartheta _j(y) :\equiv \frac{y}{j+1}$ for $i=1,2$, $j=1,2$ and $y \ge 2$. All the assumptions of Theorem 3.1 are fulfilled with $i=1,2$, $j=1,2$. Hence, due to Theorem 3.1, equation (4.2) is oscillatory in the sense of Definition of 2.3.

Example 4.3

Let us consider the differential equation

$$\begin{aligned} \Big (y^2\Big (\Big (x(y)+\frac{1}{y^2}x^\frac{1}{5}\left( \frac{y}{3}\right) +\frac{1}{y^4}x^\frac{1}{9}\left( \frac{y}{5}\right) \Big )'\Big )^3\Big )' +y^7 x^{3}\left( \frac{y}{4}\right) +y^9x^{3}\left( \frac{y}{6}\right) =0 \quad \text {for}\quad y\ge 6, \end{aligned}$$

(4.3)

where $a(y) :\equiv y^2$, $p_i(y) :\equiv \frac{1}{y^{2i}}$, $\alpha _i :\equiv \frac{1}{4i+1}$, $\varsigma _i(y) :\equiv \frac{y}{2i+1}$, $\beta _j=3>1$, $\gamma =3$, $q_j(y) :\equiv y^{2j+5}$ and $\vartheta _j(y) :\equiv \frac{y}{2j+2}$ for $i=1,2$, $j=1,2$ and $y \ge 6$. All the assumptions of Theorem 3.2 are fulfilled with $i=1,2$, $j=1,2$ and $\rho (y)=\frac{1}{y}$. Hence, due to Theorem 3.2, equation (4.3) is oscillatory in the sense of Definition of 2.3.

Example 4.4

Let us consider the differential equation

$$\begin{aligned} \Big (y\Big (\Big (x(y)+\frac{1}{y^{1/2}}x^\frac{1}{5}\left( \frac{y}{3}\right) +\frac{1}{y}x^\frac{1}{9}\left( \frac{y}{5}\right) \Big )'\Big )^3\Big )' +y^5 x^{1/5}\left( \frac{y}{4}\right) +y^6 x^{1/5}\left( \frac{y}{5}\right) =0 \quad \text {for}\quad y\ge 5, \end{aligned}$$

(4.4)

where $a(y) :\equiv y$, $p_i(y) :\equiv \frac{1}{y^{i/2}}$, $\alpha _i :\equiv \frac{1}{4i+1}$, $\varsigma _i(y) :\equiv \frac{y}{2i+1}$, $\beta _j=\frac{1}{5}<1$, $\gamma =3$, $q_j(y) :\equiv y^{j+4}$ and $\vartheta _j(y) :\equiv \frac{y}{j+3}$ for $i=1,2$, $j=1,2$ and $y \ge 5$. All the assumptions of Theorem 3.3 are fulfilled with $i=1,2$, $j=1,2$ and $A(y)=\frac{5}{2}(y^{2/5}-y_{0}^{2/5})$. Hence, due to Theorem 3.3, equation (4.4) is oscillatory in the sense of Definition of 2.3.

Example 4.1 and 4.2 show that Theorem 3.1 can be applied for any $\gamma$ and $\beta _j$. Example 4.3 is valid for $\gamma >1$ and $\rho (y)=\frac{1}{y}$, and Example 4.4 is valid for $\gamma <1$.

5 Conclusions

In this work we established several oscillation criteria for second-order nonlinear neutral differential equations. Our results complete the research started in Bazighifan et al. (2020b). For the sake of completeness, we presented some examples related to the main results of the paper.