Abstract
In this paper, some new sufficient conditions for the oscillation of all solutions of nonlinear neutral delay differential equations are established aiming at extending and/or improving some well known results in the literature. Our main results are obtained by employing the Riccati transformation aiming to transfer the neutral equation to a nonneutral type and then using some inequality techniques. Some illustrative examples are also included.
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1 Introduction
When delays appear in additional terms involving the highest order derivative of the unknown function in a differential equation, we are dealing with a neutral type differential equation. The study of the asymptotic and oscillatory behavior of solutions of neutral differential equations is of importance in applications. This is due to the fact that such equations appear in various phenomena including networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar, as the Euler equations for the minimization of functionals involving a time delay in some variational problems and in the theory of automatic control (see Hale [1], Driver [2] and Boe and Chang [3] and references cited therein). The construction of these models using delays is complemented by the mathematical investigation of nonlinear equations.
In this paper, we mainly consider the nonlinear neutral delay differential equations of the form
where
and f satisfies
and there exists a positive constant \(k_{0}\) such that
Many authors have considered linear neutral delay differential equations and established sufficient conditions for oscillation of all solutions. We refer to Ahmed et al. [4], Karpuz and Ocalan [5], Gopalsamy et al. [6] and Zhou [7].
In 2002, Saker and Kubiaczyk [8] considered the nonlinear neutral equation
where the function f satisfies condition (1.3) with \(\lim_{u\rightarrow0} \frac{f(u)}{u}=k_{1}\) exists and there exists \(t^{*}\geq t_{0}\) such that
In 2003, Kubiaczyk et al. [9] studied the nonlinear equations of the form
where \(\alpha_{i}>0\) and \(\sum_{i=1}^{n}\alpha_{i}=1\), and they have given some sharp sufficient conditions for the oscillation of all solutions of (1.7).
In 2004, Graef et al. [10] considered (1.1) when \(a(t)\equiv1\), \(r(t)\equiv1\) and developed some sufficient conditions for the oscillation of all solutions. For further results on the oscillation of various classes of neutral differential equations one can see [11–17].
A primary purpose of this paper is to establish new integral conditions that guarantee the oscillation of all solutions of (1.1) when \(p(t)\) is constant and equal \(p_{0}\). In some sense, the obtained results here extend and generalize several of well known results in the literature.
Let \(m=\max\{\tau, \sigma\}\). By a solution of (1.1) we mean a function \(x\in C[[t_{1}-m, \infty), \mathbb{R}]\) for some \(t_{1}\geq t_{0}\) such that \(a(t)x(t)+p(t)x(t-\tau)\) is continuously differentiable for \(t\geq t_{1}\) and such that (1.1) is satisfied for \(t\geq t_{1}\).
Let \(t_{1}\geq t_{0}\) be a given initial point and let \(\Phi\in C[[t_{1}-m, t_{1}], \mathbb{R}]\) be a given initial function. Then one can show by using the method of steps that (1.1) has a unique solution on \([t_{1}, \infty)\) satisfying the initial function
As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise the solution is said to be nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
In the sequel, unless otherwise specified, when we write a functional inequality we shall assume that it holds for all sufficiently large values of t.
2 Auxiliary lemmas
We give here some useful lemmas which will play an important role in the study of the oscillation of (1.1).
Lemma 2.1
[18]
Assume that \(P_{i}(t)\), \(\tau_{i}(t) \in C[[t_{0}, \infty), [0, \infty)]\), \(i=1, 2, \ldots \) .
Then the differential inequality
has an eventually positive solution if and only if the equation
has an eventually positive solution.
Lemma 2.2
[19]
Assume that
If \(x(t)\) is an eventually positive solution of the delay differential equation
then, for the same i,
Lemma 2.3
Assume that
Let \(x(t)\) be an eventually positive solution of the equation
Set
then
Proof
In view of (2.7) and our hypothesis, we see that \(z(t)>0\) eventually. From (2.8) we have
and
Since \(z'(t)<0\), we have \(z(t)>z(t+\tau)\), which implies with (2.10) that
Substituting in (2.11), we obtain
or
Hence
By Lemma 2.1, we find that
has an eventually positive solution as well.
As a result, by Lemma 2.2 and (2.6), we have
which is the desired result. The proof is complete. □
Lemma 2.4
Assume that conditions (2.5) hold. If (2.7) has an eventually positive solution, then
for all sufficiently large t.
Proof
Proceeding as in the proof of Lemma 2.3, we again obtain the inequality (2.13). Integrating (2.13) from t to \(t+\sigma -\tau\), it follows that
Using Bonnet’s theorem, it follows that
Then
Since \(z(t)>0\), (2.16) implies
Hence, for all sufficiently large t, we have
which is the desired result. The proof is complete. □
Remark 2.5
Lemma 2.3 and Lemma 2.4 extend results of Graef et al. [10].
3 Oscillation of solutions
Theorem 3.1
Assume that conditions (2.5) and (2.6) hold. If
Then every solution of (2.7) oscillates.
Proof
For the sake of obtaining a contradiction, assume that there is an eventually positive solution \(x(t)\) of (2.7). Set \(z(t)\) as in (2.8). Then \(z(t)\) is eventually positive and decreasing and satisfies the inequality (2.13). That is,
Let
Then \(\psi(t)\) is continuous and nonnegative. So, there exists \(t_{1}\geq t_{0}\) with \(z(t_{1})>0\) such that
Moreover, \(\psi(t)\) satisfies
By using the inequality (cf. Erbe et al. [20], p.32)
we have from (3.2)
where
Therefore,
Hence, for \(\xi>T+\sigma-\tau\),
By interchanging the order of integration, we have
Combining (3.3) and (3.4), leads to
Using (2.15) of Lemma 2.4 in (3.5), we obtain
i.e.,
This result along with condition (3.1) leads to
which contradicts (2.9) and completes the proof. □
Example 3.2
Consider the equation
Here we have
Then we have
Let \(k_{0}=1\),
Hence, according to Theorem 3.1, all solutions of (3.6) oscillate.
Remark 3.3
Theorem 3.1 extends results of Graef et al. ([10], Theorem 2.1 and Theorem 2.2), where \(a(t)\equiv1\).
Theorem 3.4
Assume that conditions (2.5) hold. If
and
Then every solution of (1.1) oscillates.
Proof
For the sake of obtaining a contradiction, assume that there is an eventually positive solution \(x(t)\) of (1.1). Set \(z(t)\) as in (2.8). Then \(z(t)\) is eventually positive and decreasing. From (2.12) and (1.1), we have
Dividing (3.9) by \(r(t)>0\), we obtain
Let
which implies that \(y(t)>0\). Substituting in (3.10) yields for all \(t\geq t_{0}\),
Set
Then \(\lambda(t)\) is continuous and positive. So, there exists \(t_{1}\geq t_{0}\) with \(y(t_{1})>0\) such that
Moreover, \(\lambda(t)\) satisfies
Applying the inequality
to (3.13) yields that
where
Therefore,
Hence, for \(\eta>T_{1}+\sigma-\tau\)
By interchanging the order of integration, we have
From (3.14) and (3.15), we obtain
Employing (2.15) in (3.16), it follows that
or
From (3.17) and (3.9), we have
On the other hand, from condition (3.8), there exists a sequence \(\{t_{n}\}\), \(t_{n}\rightarrow\infty\) as \(n\rightarrow\infty\), and there exists \(\mu_{n}\in(t_{n}-\sigma, t)\) for every n such that
Integrating both sides of (3.12) over the interval \([t_{n}, \mu _{n}]\) and \([\mu_{n}, t_{n}+\sigma-\tau]\), we have
and
From (3.19), (3.20), and (3.21), we have
and
This implies eventually
which is a contradiction with (3.18). The proof is complete. □
Example 3.5
Consider the equation
Here we have,
Then we have
Let \(k_{0}=1\); we have
and
Hence, according to Theorem 3.1, all solutions of (3.22) oscillate.
Remark 3.6
Theorem 3.4 extends results of Graef et al. [10], Ahmed et al. [4], Saker and Elabbasy [21] and Saker and Kubiaczyk [8].
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Acknowledgements
This research has been completed with the support of these grants: FRGS/2/2013/SG04/UKM/02/3 and DLP-2014-012.
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This research has been done by FA under the general supervision of authors Prof. RA, Dr. UD, and Prof. MN, who conceived of the study, participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
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Ahmed, F.N., Ahmad, R.R., Din, U.K.S. et al. Oscillation criteria for nonlinear functional differential equations of neutral type. J Inequal Appl 2015, 97 (2015). https://doi.org/10.1186/s13660-015-0608-5
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DOI: https://doi.org/10.1186/s13660-015-0608-5