Abstract
In the present article, we considered a class of nth order impulsive neutral differential equations. The study on the oscillatory and asymptotic behavior of solutions for the higher-order neutral differential equation is theoretical and practical. Various techniques appeared for these studies. We reduced this class into a class of non-impulsive neutral differential equations by using suitable substitutions. Through a comparison strategy involving first-order differential equations, we studied the oscillatory and asymptotic behavior of solutions. Sufficient conditions are obtained for asymptotic as well as oscillatory bounded solutions. Several examples have illustrated the effectiveness of the requirements.
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Introduction
The differential equations having the higher-order derivatives with and without delay are called neutral differential equations. Higher-order neutral differential equations are used to model many mathematical phenomena in natural science and technology. Initially, the existence and uniqueness of solutions for different types of neutral equations have been studied. In recent years, extensive considerations have been given to their oscillatory nature by many researchers [1, 3, 5, 11, 12, 17, 18, 22, 23, 28, 33, 34, 36] and in the last few decades the characteristics of such neutral equations with even/odd order have been studied [18, 22, 37]. The asymptotic and oscillation properties of higher-order neutral equations with some relaxed conditions on coefficients are investigated in [1, 3]. Yildiz et al. [33, 34] have considered neutral type nonlinear higher-order functional differential equations with oscillating coefficients. Basic definitions and results on oscillation for neutral type differential equations are given in [2].
It is a well-known fact that the motions on the earth are not always uniform as various kinds of resistance appear during the motions. If high-intensity forces act for a short duration, then the motions caused by these forces are called impulsive motions. In mathematical models, these types of motions are described by impulsive differential equations. The differential equations with impulsive effect can be used to simulate those discontinuous processes in which impulses occur. So, it becomes an important tool to handle the natural function of mathematical models and phenomena such as in optimal control, electric circuit, biotechnology, population dynamics, fractals, neural network, viscoelasticity, and chemical technology. One of the main advantages of the impulses can be seen in the paper of Sugie and Ishihara [26]. They provided the model in which the mass point might oscillate with impulsive effect; however, the mass point didn’t oscillate without an impulsive effect. For more work on impulsive impact, refer to the article of Feng et al. [6] as well as Raheem and Maqbul [24].
In 1989, some researchers started to investigate the oscillatory nature of differential equations with impulses and were at the initial stage of its development. Later on, authors in papers [6, 8,9,10, 18] have extended the study of oscillation to parabolic and hyperbolic impulsive partial differential equations. The oscillatory and asymptotic nature of the solutions for a higher-order delay differential equation with impulses were examined by some researchers using comparison results with associated delay differential equations without impulses[7, 19, 37]. We have often seen that even non-impulsive neutral delay differential equations may have solutions of oscillatory nature due to some additional controls.
In literature, we noticed Riccati techniques are widely used to obtain Kamenev, and Philos-type oscillation criteria [6, 27, 31]. Oscillation theory extended to the first-order impulsive differential equations with variable delays in [11]. Several results for third-order delay differential equations were discussed by Tiryaki and Aktas [29]. For the oscillation results on second and fourth-order dynamical systems, refer to the paper [12, 38]. Oscillatory and non-oscillatory solutions play a significant role in many applied problems in natural sciences and engineering. The research on oscillation and asymptotic behavior of impulsive differential equation is emerging as an important area of study and is developing rapidly [4, 20, 25, 30, 35].
After considering the above formulations, we study the oscillation and asymptotic behavior of solutions for higher-order neutral differential equations with impulsive conditions. We converted the impulsive differential equations into non-impulsive differential equations by using suitable substitutions. Moreover, we reduced the nth order neutral differential equation into the first-order equation using generalized Riccati transformation. It allows using the comparison theorems to establish the oscillation results. The obtained conditions are sufficient for asymptotic as well as oscillatory bounded solutions. Philos-type oscillation criteria are proved for taking n as an even integer.
Necessary lemmas and fundamental assumptions are provided in “Preliminaries and Assumptions” section. Main results are obtained in “Main Results” section for the problem (1) by using generalized Riccati transformations and comparison theorems. And in “Frequency-Amplitude Formulation” section, the applicability of the main results is demonstrated by several examples.
Here, we established the oscillation results for the following model of impulsive neutral differential equation of order n:
where \( v (x)=u(x)+\alpha u(\eta _1(x)),\) \(\eta _1(x) \le x,\) \(\eta _2(x) \le x,\) \(x>x_0,\) \(\alpha >0,\) \(d_p>0,\) \( v ^{(r)}(x)\) denote the rth (\(r \ge 1\)) order derivatives.
Preliminaries and Assumptions
Throughout the paper, we consider the following assumptions:
-
(C)
\(\eta _r:(x_0,\infty ) \rightarrow \mathbb {R},\) \(r=1,2\) are continuous functions with the following conditions:
-
(i)
\(\eta _r(x) \le x,\) \(\eta _1(\eta _2(x))=\eta _2(\eta _1(x))\)
-
(ii)
\(\eta '_r(x)=1\) and \(\eta ''_r(x)=0\)
-
(iii)
\(\lim \limits _{x \rightarrow \infty } \eta _r(x)=\infty .\)
-
(i)
Lemma 1
\(U( x)=\prod \limits _{x_0<x_p \le x} (1+d_p)^{-1}u(x)\) satisfies
where
and
if and only if u(x) satisfies (1) on the interval \((x_0, \infty )\) .
Proof
Let \(U(x)=\prod \limits _{x_0<x_p \le x} (1+d_p)^{-1}u(x)\) satisfies (2). Then we will show that u(x) satisfies (1) on the interval \((x_0, \infty )\).
Obviously,
Thus
Using (2), we get
Therefore, for \(x \ne x_p,\) we have
Also, we obtain
which implies that
This shows that \( u (x)\) satisfied (1).
Conversely, we assume \( u (x)=\prod _{x_0<x_p \le x} (1+d_p) U (x)\) satisfies (1). Then we will show that \( U (x)\) satisfies (2) on \((x_0,\infty ).\)
As \( V (x)=\prod \limits _{x_0<x_p \le x}(1+d_p)^{-1} v (x),\) we have
Using (1), we obtain
Now, we can easily show that \( U ^{(r)}(x_p^{-})= U ^{(r)}(x_p).\) This shows that \( U (x)\) satisfied (2). \(\square \)
Lemma 2
[19] A non zero solution \( u (x)\) of (1) is oscillatory on \((x_0,\infty )\) if and only if the corresponding solution \( U (x)=\prod _{x_0<x_p \le x} (1+d_p)^{-1} u (x)\) of (2) is oscillatory on \((x_0,\infty ).\) Moreover, \(\lim {x \rightarrow \infty } u (x)=0\) if and only if \(\lim \limits _{x \rightarrow \infty } U (x)=0.\)
Lemma 3
[5] Let the nth order derivative of \( V (x)\) has a constant sign and not identically zero on a subinterval of \([x_0,\infty ).\) If \( V (x)\) and its derivatives up-to order \(n-1\) are of constant sign in \([x_0,\infty ),\) then there exists an integer \(q>0\) and \(\tau \ge x_0\) such that \(0 \le q \le n-1,\) and \((-1)^{n+q} V (x) V ^{(n)}(x)>0,\)
and
on \([\tau , \infty ).\)
Lemma 4
[18] Let \( V \) be a function defined in Lemma 3. If \(\lim _{x \rightarrow \infty } V (x) \ne 0,\) then for every \(\mu \in (0,1),\) there exists \(x_\mu \in [\tau ,\infty )\) such that
on \([\tau _\mu ,\infty ).\)
Lemma 5
[18] Let \( V \) be a function defined in Lemma 3. If \( V ^{(n-1)}(x) V ^n(x) \le 0,\) then for any constant \(\nu \in (0,1)\) and sufficiently large x, there exists a constant \(M>0\) satisfying
Lemma 6
[18] If \( U \) is a positive solution of (2), then corresponding function
satisfies \( V (x)>0,\) \( V ^{(n-1)}(x)>0,\) \( V ^{(n)}(x)<0\) eventually.
Main Results
Theorem 7
If first order neutral differential inequality:
where
has no eventually positive solution then every non zero solution of (1) is oscillatory.
Proof
Let on contrary \( U (x)\) be an eventually positive solution of (2). From (2), we have
Using (3), we get
From Lemma 4, we have
Using above inequality, we get
If we assume \(Y(x)= V ^{(n-1)}(x),\) then first order neutral differential inequality
has an eventually positive solution which is a contradiction to the condition of theorem. Applying Lemma 2, result follows. \(\square \)
Corollary 8
Let n be an even integer and there exists a constant \(K>0\) such that
If first order differential inequality:
has no eventually positive solution, then every bounded solution of (1) is oscillatory.
Proof
Let \( U (x)\) be a bounded and non-oscillatory solution of (2). We may assume that \( U (x)\) is eventually positive. Since \( U (x)>0\) is bounded, \( V (x)\) is also bounded and \( V (x)>0\) eventually. As n is even and \( V (x)\) is bounded, by using Lemma 3, we have \(q=1\) i.e.
In particular \( V '(x)>0.\) From (3), we have
Using above inequality in (2), we get
Using Lemma 4, we get
As \( V ^{(n-1)}(x)\) is decreasing, we have
Using above inequality in (6), we get
If we assume \(Y(x)= V ^{(n-1)}(x),\) then first order differential inequality
has an eventually positive solution which is a contradiction to the condition of theorem. Applying Lemma 2, result follows. \(\square \)
Theorem 9
Let n be an even integer and \(\frac{\eta _1(x)}{2} \le \eta _2(x)\). We assume that there exist real valued continuously differentiable functions \(\Psi (x,y), \phi (x,y)\) with domain \(D_1=\{(x,y)|x \ge y \ge x_0>0\}\), and continuously differentiable function \(\rho \) with the domain \([x_0, \infty )\) satisfying the following conditions:
-
(A1)
\(\Psi (x,x)=0\) for \(x \ge x_0\) and \(\Psi (x,y)>0\) for \(x>y\ge x_0;\)
-
(A2)
\(\frac{\partial }{\partial x}\Psi (x,y) \ge 0,\) \(\frac{\partial }{\partial y}\Psi (x,y) \le 0;\)
-
(A3)
\(\frac{\partial \Psi (x,y)}{\partial y}+\Psi (x,y) \frac{\rho '(y)}{\rho (y)}=\phi (x,y),\,\,\,(x,y)\in D_1.\)
Further, assume that
Then every bounded solution of (2) is oscillatory.
Proof
Let \( U (x)\) be a bounded and non-oscillatory solution of (2). We may assume that \( U (x)\) is an eventually positive. Since \( U (x)>0\) is bounded, \( V (x)\) is also bounded and \( V (x)>0\) eventually. As n is even and \( V (x)\) is bounded, by using Lemma 3, we have \(q=1\) i.e.
In particular, \( V '(x)>0.\)
From (2), we have
Using (3), we get
Define
Differentiating with respect to x, we get
Using Lemma 5, we obtain
Using above inequality in (10), we get
Using (9), we get
Define another function
Differentiating with respect to x, we get
Using (8), we get
Multiplying by \(\Psi (x,y)\) and integrating from T to x, we get
Using inequality \(P \psi -Q \psi ^2 \le \frac{P^2}{4Q},\) we get
\(\Rightarrow \)
which gives
which contradicts the condition (7). Applying Lemma 2, result follows. \(\square \)
Theorem 10
Let n be an odd integer and u(x) be an eventually positive bounded solution of (1). Further, if there exists a constant \(K_1>0\) such that
and
then \(\lim \limits _{x \rightarrow \infty } u (x)=0.\)
Proof
Since \( u (x)\) is eventually positive, consequently \( U (x)\) and \( V (x)\) are also eventually positive, therefore there exists \(\varrho \ge x_0\) such that \( V (x)>0,\) for \(x \ge \varrho .\) Let \(\lim \limits _{x \rightarrow \infty } V (x)=L.\) Then \(L \ge 0.\) Claim \(L=0,\) otherwise \(L>0.\) As n is an odd integer and \( V (x)\) is bounded, by using Lemma 3, we have \(q=0,\) (otherwise \( V (x)\) is unbounded) i.e.
In particular, \( V '(x)<0\) for \(x \ge \varrho .\) Since \( V (x)\) is decreasing for \(x \ge \varrho ,\) we have \(L+\epsilon> V (x)>L\) for all \(\epsilon >0\). Choose \(0<\epsilon <\frac{L(1-\alpha K_1)}{\alpha K_1}\). It is easy to see that
where \(P=\frac{L-\alpha K(L+\epsilon )}{L+\epsilon } >0.\)
Therefore, from (2), we have
Integrating from \(\varrho \) to x, we get
\( \Rightarrow \)
Taking limit as \(x \rightarrow \infty \), (15) contradicted. Hence \(\lim \limits _{x \rightarrow \infty } V (x)=0.\) Since \( U (x) \le V (x),\) we have
Therefore, by applying Lemma 2, we get
This completes the proof. \(\square \)
Corollary 11
Condition (15) of Theorem 10 can be replaced by the following condition
Proof
Multiplying (17) by \(y^{n-1}\) and integrating from \(\varrho \) to x, we get
\(\Rightarrow \)
where
Since \((-1)^r V ^{(r)}(x)>0,\) \(r=0,1,2, \ldots , (n-1)\) for \(x \ge \varrho ,\) \(G(x)>0\) for \(x \ge \varrho .\) From (19), we have
Taking limit as \(x \rightarrow \infty ,\) we get
which contradicts (18). Thus the proof is completed. \(\square \)
In the next section, following the idea used in papers [13,14,15,16], we elucidate the frequency-amplitude relationship.
Frequency-Amplitude Formulation
According to the Lemma 1, problem (1) is equivalent to the following problem:
where V, \(\pounds \) and U are defined in Lemma 1.
Equation (20) is of the following form:
where \(f(U)=k(x)\pounds (x) U (\eta _2(x)).\)
We assume that all the conditions of Theorem 7 hold. Therefore every non zero solution of (1) is oscillatory. To find frequency- amplitude relationship, we consider the following conditions:
where A is its initial amplitude.
According to He’s frequency-amplitude formulation, we use the following trial functions:
where \(w_1,\) \(w_2,\) ..., \(w_n\) are trial frequencies.
If n is even i.e. \(n=2m,\) \(m=1,2,3, \ldots ,\) then the residuals are given by:
If n is odd i.e.\(n=2m-1,\) \(m=1,2,3, \ldots ,\) then the residuals are given by:
According to classic procedure of He’s frequency-amplitude, the approximate frequency of non linear oscillator (21) can be calculated by using above residuals.
Application
Example 12
Consider the following system
Here \(n=6,\) \(\eta _1(x)=x-2\pi ,\) \(\eta _2(x)=x-\pi ,\) \(x>x_0,\) \(\alpha >0,\) \(k(x)=\frac{3}{x},\) \(J(x)=\frac{3}{x},\) \(d_p=\frac{1}{p},\) \(x_p=p\pi .\)
Let \(\Psi (x,y)=(x-y)^2,\) \(\rho (y)=y.\)
Clearly, \(\Psi (x,x)=(x-x)^2=0\) for \(x \ge x_0\) and \(\Psi (x,y)=(x-y)^2>0\) for \(x>y \ge x_0.\)
Therefore (A1) holds.
\(\frac{\partial }{\partial x}\Psi (x,y) =2(x-y)\ge 0,\) \(\frac{\partial }{\partial y}\Psi (x,y)=-2(x-y) \le 0,\) for \(x>y\).
Therefore (A2) holds.
We see that
We can easily show that
Thus all the conditions of Theorem 9 are fulfilled, therefore every bounded solution of (12) is oscillatory.
Example 13
Consider the following system
where n is any positive even integer.
Here \(\eta _1(x)=x-4\pi ,\) \(\eta _2(x)=x-2\pi ,\) \(x>x_0,\) \(\alpha >0,\) \(k(x)=e^{2x}\), \(d_p=\frac{1}{p},\) \(x_p=p\pi .\)
Let \(\Psi (x,y)=(x-y)^2,\) \(\rho (y)=e^{-2y}.\)
Clearly (A1), (A2) hold.
Now, we can easily show that
Thus all the conditions of Theorem 9 are fulfilled, therefore every bounded solution of (13) is oscillatory.
Example 14
Consider the following system
Here \(n=5,\) \(k(x)=x^2,\) \(d_p=\frac{1}{p},\) \(\eta _1(x)=x-2\pi ,\) \(\eta _2(x)=x-\pi .\) Clearly, there exits a constant \(K_1>0\) such that
and
Thus all the conditions of Theorem 10 are fulfilled, therefore by using Theorem 10, we get
Conclusion
Our main emphasis is analyzing of higher-order neutral impulsive differential equations using Riccati transformations and comparison theorems in the present work. Using these strategies, every solution of the studied equation [i.e., problem (1.1)] oscillates under certain assumptions. In addition to that, some sufficient conditions are obtained for bounded oscillatory solutions using the corresponding non-impulsive differential equation. Moreover, the asymptotic behavior of the oscillatory solutions is also discussed.
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Acknowledgements
The authors would like to thank the referees for their valuable suggestions. The second and third authors acknowledge UGC, India, for providing financial support through MANF F.82-27/2019 (SA-III)/ 4453 and F.82-27/2019 (SA-III)/191620066959, respectively.
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Raheem, A., Afreen, A. & Khatoon, A. On Oscillatory and Asymptotic Behavior of Higher Order Neutral Differential Equations with Impulsive Conditions. Int. J. Appl. Comput. Math 7, 155 (2021). https://doi.org/10.1007/s40819-021-01092-5
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DOI: https://doi.org/10.1007/s40819-021-01092-5
Keywords
- Neutral differential equations
- Higher order
- Impulsive conditions
- Oscillation criteria
- Asymptotic behavior