Abstract
The aim of this work is to improve the oscillation results for second-order neutral differential equations with damping term. We consider the noncanonical case which always leads to two independent conditions for oscillation. We are working to improve related results by simplifying the conditions, based on taking a different approach that leads to one condition. Moreover, we obtain different forms of conditions to expand the application area. An example is also given to demonstrate the applicability and strength of the obtained conditions over known ones.
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1 Introduction
This work is concerned with studying the oscillation properties of the second-order neutral delay differential equations with damping term of the form
where
Throughout this work, we assume \(\alpha \in Q_{\mathrm{odd}}^{+}:= \{ a/b: a,b\in \mathbf{Z} ^{+}\text{ are odd} \} \), r, c, τ, σ, p, \(q\in C ( [t_{0},\infty ) ) \), r is positive, c, p and q are nonnegative, \(\tau ( t ) \leq t\), \(\sigma ( t ) \leq t \), \(\sigma ^{\prime } ( t ) >0\), \(\lim_{t\rightarrow \infty }\tau ( t ) =\lim_{t \rightarrow \infty }\sigma ( t ) =\infty \), \(f\in C ( \mathbf{R} ,\mathbf{R} ) \), and it satisfies the following property:
where \(k>0\) is a constant and \(\beta \in Q_{\mathrm{odd}}^{+}\). Furthermore, this study requires that
and
where
A real-valued function \(x\in C ( [t_{x},\infty ),\mathbf{R} ) \), \(t_{x}\geq t_{0}\) is a solution of (1) if x satisfies (1) on \([t_{x},\infty )\), and it has the property that \(z ( t ) \) and \(r ( t ) ( z^{\prime } ( t ) ) ^{ \alpha }\) are continuously differentiable for \(t\in {}[ t_{x},\infty )\). A nontrivial solution x is said to be oscillatory, if it has arbitrary large zeros. Otherwise, it is said to be non-oscillatory. The set of all eventually positive solutions of (1) is denoted by ℜ, that is, if \(x\in \Re \), then there exists a \(t_{1}\geq t_{0}\) large enough such that \(x ( t ) >0\) for all \(t\geq t_{1}\). We only focus on solutions of (1), which exist on \([ t_{0},\infty ) \) and satisfy \(\sup \{ \vert x ( t ) \vert :t_{x}\leq t\}>0\) for every \(t\geq t_{x}\).
Half-linear differential equations arise in real problems; for instance, in the study of non-Newtonian fluid theory and the turbulent flow of a polytropic gas in a porous medium; see [1, 2].
In the past few years, there has been research studying the asymptotic properties and the oscillatory behavior of solutions of differential equations with different order. This research focused on developing and improving the oscillation criteria for differential equations. References [3–9] improved the oscillation criteria for noncanonical second-order equations with delay and advanced argument. For canonical second-order delay equations, Refs. [10, 11] developed the oscillation criteria. The results in [12–21] dealt with the issue of oscillation of equations of higher order. For differential equations with damping, we present the following results that are closely related to this paper.
Tunc and Kaymaz [22] studied the oscillatory behavior of equations with damping term
under the condition
They are extended to more general second-order linear and/or nonlinear neutral differential equations with damping in [23–25]. Saker et al. [26] established Kamenev-type and Philos-type theorems for oscillation of equation with damping term
Results in [26] are extended and improved results in [27–29].
In this paper, we try to improve the oscillation criteria for solutions of (1) by creating new and more effective criteria. In a noncanonical case, we always have two cases of derivatives signs for corresponding function z, which often leads to two independent conditions to ensure oscillatory. As an extension of the results of [4], we create a new criterion for oscillation of (1), which in turn is a simplification of the previous results in [26].
Although the theoretical advantage lies in reducing the number of conditions that are sufficient to verify the oscillation of solutions of differential equations, but sometimes a single condition is less effective in practical applications. So, we also follow the usual approach to creating two independent criteria of oscillation.
Remark 1
The functional inequalities in this paper are supposed to hold eventually, that is, they are satisfied for all t large enough.
2 Preliminaries
In the following, we provide some notations which help us to easily display the results. Moreover, we present the auxiliary lemmas which help in validating the main results.
Notation 1
For the sake of brevity, we define the operators \(\gamma := ( \alpha / ( \alpha +1 ) ) ^{ \alpha +1}\),
and
where \(l_{1}\), \(l_{2}\) are positive constants.
Notation 2
For the sake of brevity, we define the functions
and
Lemma 1
Let \(\Theta ( \theta ) =A_{1}\theta -A_{2} ( \theta -A_{3} ) ^{ ( \alpha +1 ) /\alpha }\), where \(A_{2}>0, A_{1}\) and \(A_{3}\) are constants. Then the maximum value of Θ on R at \(\theta ^{\ast }=A_{3}+ ( \alpha A_{1}/ ( ( \alpha +1 ) A_{2} ) ) ^{\alpha }\) is
Lemma 2
Assume that \(x\in \Re \). Then:
-
(i)
The function z satisfies \(z ( t ) >0\), \(( \mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha } ) ^{\prime }\leq 0\) and one of the next cases:
$$\begin{aligned}& ( \mathbf{D}_{1} )\quad z^{\prime } ( t ) > 0; \\& ( \mathbf{D}_{2} )\quad z^{\prime } ( t ) < 0. \end{aligned}$$ -
(ii)
If z satisfies (\(\mathbf{D}_{2} \)), then
$$ z^{\beta -\alpha } ( t ) \geq \varrho ( t ) $$(6)and
$$ \bigl( \mu ( t ) r ( t ) \bigl( z^{\prime } ( t ) \bigr) ^{\alpha } \bigr) ^{\prime }\leq -Q ( t ) z^{\beta } \bigl( \sigma ( t ) \bigr) . $$(7)
Proof
Since \(x\in \Re \), it is obvious that there exists a \(t_{1}\geq t_{0}\) such that \(x ( t )\), \(x ( \tau ( t ) ) \) and \(x ( \sigma ( t ) ) \) are positive functions. As a direct conclusion from the definition (2), we find that \(z ( t ) >0\). Next, from (1) and (3), we obtain
which means that \(\mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha }\) is a nonincreasing function and has a fixed sign. Since \(\mu ( t ) r ( t ) >0\), we get either \(z^{\prime } ( t ) >0\) or \(z^{\prime } ( t ) <0\).
Now, let \(z^{\prime } ( t ) <0\) for all \(t\geq t_{2}\geq t_{1}\), where \(t_{2}\) is large enough. Then there exists a constant \(K_{1}>0\) such that
On the other hand, using the fact that \(( \mu ( t ) r ( t ) ( z^{\prime } ( t ) ) ^{\alpha } ) ^{\prime }\leq 0\), we get
and therefore
Integrating (10) from t to ∞ we obtain \(-z ( t ) \leq -K_{2}^{1/\alpha }\eta ( t ) \). Thus,
i.e., (6) holds. Moreover, we have
and so
Hence,
which, in view of (8), gives (7). The proof of the lemma is complete. □
Lemma 3
Let \(x\in \Re \), the function z satisfy (\(\mathbf{D}_{2} \)) and
Then
Proof
Suppose that \(x\in \Re \) and the function z satisfies (\(\mathbf{D}_{2} \)). Since z is a positive decreasing function, we have \(\lim_{t\rightarrow \infty }z ( t ) =\epsilon \geq 0\).
Let \(\epsilon >0\). From Lemma 2, there exists a \(t_{1}\geq t_{0}\) such that
Integrating this inequality from \(t_{1}\) to t, we have
Integrating (14) from \(t_{1}\) to t, we get
Taking \(\lim_{t\rightarrow \infty }\) of this inequality and using (12), we are led to a contradiction with positivity of z. Therefore, \(\epsilon =0\) and (13) hold. This completes the proof. □
Lemma 4
Let \(x\in \Re \), the function z satisfy (\(\mathbf{D}_{2} \)) and (12) hold. If, moreover there exists a constant \(\delta \in [ 0,1 ) \) such that
then
Proof
Suppose that \(x\in \Re \) and the function z satisfies (\(\mathbf{D}_{2} \)). From Lemma 2, we see that (7) holds. Integrating (7) from \(t_{1}\) to t, we obtain
From Lemma 3, clearly (13) holds. Thus,
for \(t\geq t_{2}\), where \(t_{2}\) is large enough. Combining (17) and (18) and using the fact that \(z^{\beta -\alpha } ( t ) \geq \varrho ( t ) \), we find
which, in view of (15), gives
Thus,
This completes the proof. □
3 The one criterion theorems
In the following theorems, we obtain a criterion which ensure oscillation of (1) without verifying the extra condition.
Theorem 1
If
for any \(t_{1}\geq t_{0}\), then all solutions of (1) are oscillatory.
Proof
Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).
Let (\(\mathbf{D}_{2} \)) hold. As in the proof of Lemma 2, we arrive at (19). Integrating (19) from t to ∞, we find
for all \(t\geq t_{2}\geq t_{1}\), where \(t_{2}\) is large enough. From (7) in Lemma 2, we have
Integrating this inequality from \(t_{2}\) to t, we get
Integrating again from \(t_{2}\) to t, we obtain
Taking \(\lim_{t\rightarrow \infty }\) of this inequality and using (20), we are led to a contradiction with positivity of z.
Next, we let (\(\mathbf{D}_{1} \)) hold. Since \(z ( t ) \geq x ( t ) \), \(\tau ( t ) \leq t\) and \(z^{\prime } ( t ) >0\), we get \(x ( t ) \geq ( 1-c ( t ) ) z ( t ) \). Hence, (1) becomes
Since \(\eta ( \tau ( \sigma ( t ) ) ) \geq \eta ( \sigma ( t ) ) \), we obtain
We note that the function \(\int ^{t}\eta ^{\beta } ( \sigma ( v ) ) Q ( v ) \,\mathrm{d}v\) is unbounded due to (20) and (4). Taking into account \(\eta ^{\prime } ( t ) \leq 0 \) and (23), it is easy to see that
Combining (22) and (23), we have
which, in view of (24), contradicts the positivity \(z^{\prime } ( t ) \). The proof of the theorem is complete. □
Theorem 2
If
for any \(t_{1}\geq t_{0}\), then all solutions of (1) are oscillatory.
Proof
Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).
Let (\(\mathbf{D}_{2} \)) hold. From the proofs of Lemmas 2 and 4, we arrive at (11) and (17), respectively. Combining (11) and (17), we get
and so
which contradicts (25).
Next, we let \(z^{\prime } ( t ) >0\) for \(t\geq t_{1}\). We note that (25) along with (4) imply (24). The rest of the proof is similar to the proof of Theorem 1. The proof of the theorem is complete. □
4 Theorems of two independent criteria
Theorem 3
Assume that (12) holds. If
then all solutions of (1) are oscillatory.
Proof
For the proof of this lemma, it suffices to use (19) [from the proof of Lemma 4] instead of (17) in the proof of Theorem 2. □
Using Lemma 4, we obtain the new results which improve the previous theorems when
Theorem 4
Assume that (12) and there exists a \(\delta \in [ 0,1 ) \) such that (15) holds. If
then all solutions of (1) are oscillatory.
Proof
Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).
Let (\(\mathbf{D}_{2} \)) hold. Using Lemma 4, we arrive at (16) and (19). From (16), we conclude that
which with (19) gives
As in the proof of Lemma 2, we arrive at (11). Combining (11) and (29), we obtain
then
This contradicts (27).
Next, we let (\(\mathbf{D}_{1} \)) hold. We note that (12) along with (4) imply (24). The rest of the proof is similar to the proof of Theorem 1. The proof of the theorem is complete. □
Theorem 5
Assume that (12) and there exists a \(\delta \in [ 0,1 ) \) such that (15) holds. If there exist a function \(\rho \in C^{1} ( [ t_{0},\infty ) , ( 0, \infty ) ) \) and a \(t_{1}\in [ t_{0},\infty ) \) such that
then all solutions of (1) are oscillatory.
Proof
Assuming that the required result is not fulfilled. We assume that (1) has a solution \(x\in \Re \). From Lemma 2, we see that z satisfies either (\(\mathbf{D}_{1} \)) or (\(\mathbf{D}_{2} \)) for all \(t\geq t_{1}\).
Let (\(\mathbf{D}_{2} \)) hold. From Lemma 2, we see that (6), (7) and (11) hold. We define the function
Using (11), we see that \(w ( t ) \geq 0\) for all \(t\geq t_{2}\geq t_{1}\). Differentiating (31), we get
Combining (6), (7) and (31), we obtain
Using inequality (5) with \(A_{1}:=\rho ^{\prime }/\rho \), \(A_{2}:=\alpha / ( \rho \mu r ) ^{-1/\alpha }\), \(A_{3}=\rho /\eta ^{\alpha }\) and \(\theta :=w\), we get
which, in view of (32), gives
As in the proof of Theorem 4, we arrive at (28). Thus, (33) becomes
Integrating this inequality from \(t_{2}\) to t, we have
From (11), we have
which, in view of (34), implies
Taking the lim sup on both sides of this inequality, we are led to a contradiction with (30).
The rest of proof is the same and hence we omit it. The proof of the theorem is complete. □
Example 1
Consider the differential equation
where \(t\geq 1\), \(q_{0}\in ( 0,\infty ) \), \(c_{0}\in ( 0,1/16 ) \) and \(\lambda \in ( 0,1 ] \). From Theorem 2, we deduce that Eq. (35) is oscillatory if
To apply Theorem 4, we first note that (12) and (15) are satisfied with
Therefore, (35) is oscillatory if
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Moaaz, O., Chatzarakis, G.E., Abdeljawad, T. et al. Amended oscillation criteria for second-order neutral differential equations with damping term. Adv Differ Equ 2020, 553 (2020). https://doi.org/10.1186/s13662-020-03013-0
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DOI: https://doi.org/10.1186/s13662-020-03013-0