Abstract
The purpose of this work is to study the oscillation criteria for generalized Emden–Fowler neutral differential equation. We establish new oscillation criteria using both the technique of comparison with first order delay equations and the technique of Riccati transformation. Our new criteria are interesting as they improve, simplify, and complement some results that have been published recently in the literature. Moreover, we present an illustrating example.
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1 Introduction
In aeromechanical systems, where they have a significant role, in the theory of automatic control, in study of vibrating masses attached to an elastic bar (as the Euler equation), in the networks that have lossless transmission lines (as is the case in high-speed computers), and other applications, delay or neutral differential equations can be seen in the modeling of the mentioned phenomena, see [1, 2, 5, 15]. As a result of these applications, research groups including us still study the differential equations with delay. The theory of oscillation of delay differential equations comes at the forefront of topics that have received the attention of researchers in recent times, see [1–29]. In the last decade, there has been a research movement to improve and develop the oscillation criteria of solutions of second order differential equations with delay (see [9, 10]), neutral (see [3, 7, 13]) and advanced (see [4, 13]).
In this work, we present new oscillation criteria for second-order Emden–Fowler delay differential equations of neutral type
where \(\upsilon ( t ) =u ( t ) +p ( t ) u ( \tau ( t ) ) \) and α is a ratio of odd positive integers. We also assume that \(r\in C^{1} ( [t_{0},\infty), ( 0,\infty ) ) \), \(\tau,\sigma,p,q\in C^{1} ( [t_{0},\infty),\mathbb{R} ) \), \(\sigma ( t ) \leq t\), \(\tau ( t ) \leq t\), \(0\leq p ( t ) <\min \{ \pi ( t ) /\pi ( \tau ( t ) ) ,1 \} \), \(q ( t ) \geq0\), \(\lim_{t\rightarrow\infty}\tau ( t ) =\lim_{t \rightarrow\infty}\sigma ( t ) =\infty\), and
For the function f, we suppose that \(f\in C ( \mathbb{R} ,\mathbb{R} ) \) and satisfies the following condition:
where k is a positive constant and β is a quotient of odd positive integers.
A solution of (1.1) means \(u\in C ( [t_{0},\infty),[0,\infty) ) \) with \(t_{a}=\min\{\tau ( t_{b} ) ,\sigma ( t_{b} ) \}\), for some \(t_{b}>t_{0}\), which satisfies the property \(r ( \upsilon^{\prime} ) ^{\alpha}\in C^{1} ( [t_{a}, \infty),[0,\infty) ) \) and satisfies (1.1) on \([t_{b},\infty)\). We consider the nontrivial solutions of (1.1) existing on some half-line \([t_{b},\infty)\) and satisfying the condition
If u is neither positive nor negative eventually, then \(u ( t ) \) is called oscillatory, or it will be nonoscillatory.
For canonical form (if \(\eta ( t_{0} ) =\infty\)), there have been some studies that consider the oscillation and nonoscillation criteria of solutions of (1.1), see for example [19, 24].
For noncanonical form (if \(\eta ( t_{0} ) <\infty\)), Liu et al. [18] got necessary and appropriate conditions that ensure all solutions of (1.1) can be oscillatory, or they can tend to zero, following the conditions \(\lim_{t\rightarrow\infty}p ( t ) =C\),
Furthermore, Saker [23] developed the results of [18] in the sense that they established the conditions that assure all the solutions of Eq. (1.1) are oscillatory. The results of both [23] as well as [18] follow an approach that does lead to two conditions, and they are requested (1.2).
Wu et al. [28] established some criteria of oscillation for the neutral equation
under conditions (1.2),
This work aims at developing the oscillation theory of second order quasi-linear equations with delay argument. The use of the technique of comparison with first order delay equations and the technique of Riccati transformation helps us to get two various conditions, ensuring oscillation of (1.1) without requiring (1.2). In this paper, in the first two theorems, we simplify results in [18, 23, 28] and obtain new criteria for ensuring oscillation of (1.1) without checking the additional conditions. Our criteria complement and extend the results in [7, 8]. In [7, Theorem 2.2], Bohner et al. proved that equation (1.1) with \(\alpha=\beta\) is oscillatory if
In our paper, Theorems 2.5 and 2.6 substantially improve Theorem 2.2 in [7, Theorem 2.2], when \(W\leq1\).
The next lemma collects two useful inequalities that can be found in [29].
Lemma 1.1
Letαbe a ratio of two odd positive integers. Then
and
2 Main results
In this section, we shall establish new oscillation criteria for (1.1). Let us define
and
where \(t_{1}\in [ t_{0},\infty ) \) and \(a_{1}\), \(a_{2}\) are any positive constants.
Lemma 2.1
Assume thatuis an eventually nonincreasing positive solution of (1.1). Then\(\upsilon^{\beta-\alpha} ( t ) \geq\eta ( t ) \).
Proof
Let υ be an eventually positive solution of (1.1) and \(\upsilon^{\prime}(t)<0\). Then we have the following cases:
In the case where \(\alpha=\beta\), it is easy to see that \(\upsilon^{\beta-\alpha} ( t ) =1\).
Let \(\alpha>\beta\). Since \(\upsilon ( t ) \) is a nonincreasing positive function, there exists \(M_{1}>0\) such that \(\upsilon ( t ) \leq M_{1}\), which implies that
In the case \(\alpha<\beta\), by using the decreasing property of \(r ( \upsilon^{\prime} ) ^{\alpha}\), we obtain
hence
Integrating the last inequality from t to ∞, we get
Thus, we include that
Therefore, we have \(\upsilon^{\beta-\alpha} ( t ) \geq\eta ( t ) \). The proof of the lemma is complete. □
Lemma 2.2
Letube a positive solution of (1.1) on\([ t_{0},\infty ) \). If
for\(t_{1}\geq t_{0}\), then
- (H):
υis decreasing, \(r ( \upsilon ^{\prime} ) ^{\alpha}\)is nonincreasing, eventually.
Proof
Let u be a positive solution of (1.1) on \([ t_{0},\infty ) \). Then we suppose that there exists \(t_{1}\in [ t_{0},\infty ) \) such that \(u ( t ) >0\), \(u ( \tau ( t ) ) >0\), and \(u ( \sigma ( t ) ) >0\) for all \(t\in [ t_{1},\infty ) \). Obviously, we find \(\upsilon ( t ) \geq u ( t ) \) and
Therefore, \(\upsilon^{\prime}\) is either eventually negative or eventually positive.
Suppose now that \(\upsilon^{\prime}>0\) on \([ t_{1},\infty ) \). Then \(u ( t ) \geq ( 1-p ( t ) ) \upsilon ( t ) \), and (2.2) becomes
Since \(\pi ( \tau ( \sigma ( t ) ) ) \geq\pi ( \sigma ( t ) ) \), we get
Integrating (2.3) from \(t_{1}\) to t and using (2.4), we get
a contradiction with positivity of \(\upsilon^{\prime} ( t ) \). The proof of this lemma is complete. □
Lemma 2.3
Letube a positive solution of (1.1) on\([ t_{0},\infty ) \). If
for\(t_{1}\geq t_{0}\), then\((\mathbf{H})\)holds and
Proof
Let u be a positive solution of (1.1) on \([ t_{0},\infty ) \). From \(\pi ( t_{0} ) <\infty\) and (2.5), we have that (2.1) holds. Hence, from Lemma 2.2, we have that \(\upsilon^{\prime} ( t ) <0\), (2.3) and (2.4) hold.
Now, since \(\upsilon>0\) and \(\upsilon^{\prime}<0\), we get that \(\lim_{t\rightarrow\infty}\upsilon ( t ) =c\geq0\). Suppose that \(c>0\). Then there exists \(t_{2}\geq t_{1}\) such that \(\upsilon ( \sigma ( t ) ) \leq c\). From (2.3) and (2.4), we obtain
for \(t\geq t_{2}\). Integrating two times this inequality from \(t_{2}\) to t, we get, after the first integration,
After the second integration, we obtain
This implies that \(\lim_{t\rightarrow\infty}\upsilon(t)=-\infty\), which contradicts \(\upsilon>0\). The proof of this lemma is complete. □
Theorem 2.1
If
for\(t_{1}\geq t_{0}\), then (1.1) is oscillatory.
Proof
Let u be a positive solution of (1.1) on \([ t_{0},\infty ) \) (assume the converse). Then we suppose that there exists \(t_{1}\in [ t_{0},\infty ) \) such that \(u ( t ) >0\), \(u ( \tau ( t ) ) >0\) and \(u ( \sigma ( t ) ) >0\) for all \(t\in [ t_{1},\infty ) \). Since \(\pi ( t_{0} ) <\infty\) and (2.7), we have that \(\int_{t_{1}}^{t}Q ( \nu ) \pi^{\beta} ( \sigma ( \nu ) ) \,\mathrm{d}\nu\) must be unbounded. Thus, and from the fact \(\pi^{\prime} ( t ) <0\), it is easy to see that (2.1) holds. Hence, from Lemma 2.2, we have that \(\upsilon^{\prime} ( t ) <0\) and (2.2) holds. Since
it follows that
In view of the definition of υ, we deduce
Consequently, (2.2) becomes
From the monotonicity property of \(r ( t ) ( \upsilon^{\prime} ( t ) ) ^{\alpha}\), we have
which in view of (2.8) implies
Integrating (2.11) from \(t_{1}\) to t, we obtain
Integrating (2.12) from \(t_{1}\) to t and using (2.7), we get
which in view of (2.7) contradicts the positivity of \(\upsilon ( t ) \). The proof of the theorem is complete. □
Theorem 2.2
Assume that\(\sigma^{\prime} ( t ) >0\). If
for\(t_{1}\geq t_{0}\), then (1.1) is oscillatory.
Proof
To the contrary, we suppose that u is a positive solution of (1.1) on \([ t_{0},\infty ) \). Then there exists \(t_{1}\geq t_{0}\) such that \(u ( \tau ( t ) ) >0\) and \(u ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). From (2.13) and \(\pi ( t_{0} ) <\infty\), we get (2.1) holds. Using Lemma 2.2, we get that \(\upsilon^{\prime}<0\) on \([ t_{1},\infty ) \). As in the proof of Theorem 2.1, we get (2.8) and (2.9) hold. By integrating (2.9) from \(t_{1}\) to t, we get
Since \(\sigma ( t ) \leq t\) and \(\upsilon^{\prime} ( t ) <0\), we obtain
By Lemma 2.1 and (2.8) we arrive at
and so
a contradiction with (2.13). Then the proof is complete. □
Theorem 2.3
Assume that (2.1) holds. If the first order delay differential equation
is oscillatory, then (1.1) is oscillatory.
Proof
To the contrary, we suppose that u is a positive solution of (1.1) on \([ t_{0},\infty ) \). Then there exists \(t_{1}\geq t_{0}\) such that \(u ( \tau ( t ) ) >0\) and \(u ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). Using (2.1) and Lemma 2.2, we get that \(\upsilon^{\prime}<0\) on \([ t_{1},\infty ) \). As in the proof of Theorem 2.2, we get (2.14) holds. From (2.14), it is clear that υ is a positive solution of the first order differential inequality
In view of [25, Lemma 1], we see that the first-order delay differential equation (2.16) has a positive solution, a contradiction. Then the proof is complete. □
Corollary 2.1
Assume that\(\alpha=\beta\). If
then (1.1) is oscillatory.
Proof
In view of [16, Theorem 2], condition (2.17) implies oscillation of (2.16). On the other hand, if suffices to note that
is necessary for the validity of (2.1). Therefore, the proof is complete. □
Corollary 2.2
Assume that\(\alpha>\beta>0\). If
then (1.1) is oscillatory.
Proof
Since \(\beta/\alpha\in ( 0,1 ) \), it is shown that all the solutions of (2.16) oscillate if and only if (2.18) holds, see [12] and [17]. On the other hand, we see that (2.18) is necessary for the validity of (2.1). Therefore, the proof is complete. □
Corollary 2.3
Assume that\(\alpha<\beta\), \(\sigma ( t ) \)is continuously differentiable, \(\sigma^{\prime} ( t ) >0\)and (2.1) holds. If there exists a continuously differentiable function\(\xi ( t ) \)such that\(\xi^{\prime} ( t ) >0\), \(\lim_{t\rightarrow\infty}\xi ( t ) =\infty\),
and
then (1.1) is oscillatory.
Proof
In view of [25, Theorem 1], condition (2.19) implies oscillation of (2.16). □
The following results serve as an improvement of Theorems 2.2, when \(\alpha=\beta\) and
For the simplicity, we define the following notations:
and
Theorem 2.4
Assume that\(\alpha=\beta\)and (2.5) is satisfied. If
or
then (1.1) is oscillatory.
Proof
Suppose against the assumption of theorem that equation (1.1) has a nonoscillatory solution u on \([ t_{0},\infty ) \). Without loss of generality, we may assume that \(u ( t ) >0\) and \(u ( \sigma ( t ) ) >0\) for \(t\geq t_{1}\geq t_{0}\). Let
then
Combining (2.22) and (2.23), and using inequality (2.9), we get
Integrating (2.24) from t to ∞, we get
It follows that
and so
Now, let (2.20) hold. It follows from (2.20) that there exists \(\epsilon>0\) such that
By virtue of definition of m, we see that
this contradicts the positivity of υ.
Assume next that the case \(m\leq\alpha\) holds. Proceeding as in the proof of Theorem 2.2, we get (2.14). Thus, by (2.25), we get
that is,
Hence,
which implies
this contradicts (2.21). Then the proof is complete. □
Lemma 2.4
Assume that (1.1) has an eventually positive solutionuon\([ t_{0},\infty ) \). Then there exist\(T\geq t_{1}\)and\(\epsilon>0\)such that
where\(N=M-\epsilon\).
Proof
Assume that u is a positive solution of (1.1) on \([T,\infty)\). By Lemma 2.3, \(u ( t ) \) satisfies \(( \mathbf{H} ) \) and (2.6). Proceeding as in the proof of Theorem 2.2, we have (2.14) holds. Now, we see that
In view of (2.14), we note that
which implies
Therefore, (2.27) becomes
Then the proof is complete. □
Theorem 2.5
Assume that\(\alpha=\beta\)and (2.5) holds. If there is a constantκsuch that
for all\(t\geq t_{0}\)and
then (1.1) is oscillatory.
Proof
Proceeding as in the proof of Theorem 2.2, we get that (2.14) holds. By Lemma 2.4, we see that
From (2.14), we get
In view of (2.8), we obtain
It follows that
Taking the lim sup on both sides, we obtain a contradiction. Then the proof is complete. □
Theorem 2.6
Assume that\(\alpha=\beta\), (2.5) holds, and there is a constantκsuch that (2.28) holds. If (2.20) or
then (1.1) is oscillatory.
Proof
As in the proof of Theorem 2.4, if we replace (2.14) by (2.31), then we get
and so
Taking lim sup on both sides, we obtain
Therefore,
Then the proof is complete. □
In the next theorems, by using a generalized Riccati substitution, we establish new oscillation criteria of (1.1).
Theorem 2.7
Assume that\(\sigma^{\prime} ( t ) >0\)and\(\alpha\geq1\). If there exist functions\(\delta,\varphi\in C^{1} ( [ t_{0},\infty ) , ( 0,\infty ) ) \)such that
and
where
and\(H_{+} ( t ) =\max \{ H ( t ) ,0 \} \), then (1.1) is oscillatory.
Proof
To the contrary, we suppose that u is a positive solution of (1.1) on \([ t_{0},\infty ) \). Thus, there exists \(t_{1}\geq t_{0}\) such that \(u ( \tau ( t ) ) >0\) and \(u ( \sigma ( t ) ) >0\) for all \(t\geq t_{1}\). Then we get that \(\upsilon^{\prime} \) has one sign eventually.
Now, we let \(\upsilon^{\prime} ( t ) <0\) for \(t\geq t_{1}\). As in the proof of Theorem 2.1, we get (2.9) holds. Define the function \(\omega ( t ) \) by
From (2.8), we see that \(\omega ( t ) \geq0\). By differentiating (2.34), we get
Using inequality (1.6) with
we obtain
Using Lemma 2.1 with (2.9), we have
By inequality 1.5 with \(C:=\alpha ( \delta ( t ) r ( t ) ) ^{-1/\alpha}\), \(D:=\varPhi ( t ) \), and \(V:=\omega ( t ) \), we obtain
Integrating from \(t_{2}\) to t, we obtain
which contradicts (2.32).
On the other hand, let \(\upsilon^{\prime} ( t ) >0\) for all \(t\geq t_{2}\). It is easy to prove that \(u ( t ) \geq ( 1-p ( t ) ) \upsilon ( t ) \) and
Since \(( r ( \upsilon^{\prime} ) ^{\alpha} ) ^{ \prime} ( t ) <0\), we find
Define the function
Hence, \(R ( t ) \geq0\). By differentiating \(R ( t ) \) and using (2.39) and (2.40), we get
Proceeding as in the proof of the previous case, we obtain
which contradicts (2.33). This completes the proof. □
Theorem 2.8
Assume that\(\sigma^{\prime} ( t ) >0\). If there exist functions\(\delta,\varphi\in C^{1}( [ t_{0},\infty ) , ( 0,\infty ) )\)such that (2.33) holds and
for any\(t\in [ t_{0},\infty ) \), then (1.1) is oscillatory.
Proof
Proceeding as in the proof of Theorem 2.7, we obtain that \(\upsilon^{\prime}\) has one sign eventually. For the case where \(\upsilon^{\prime} ( t ) <0\) for all \(t\geq t_{1}\), let us define the function ω as in (2.34). From (2.35) we have
Using Lemma 1.1 with \(C=\delta^{\prime} ( t ) /\delta ( t ) \), \(D= \alpha ( \delta ( t ) r ( t ) ) ^{ \frac{-1}{\alpha}}\), and \(V=\delta ( t ) /\pi^{\alpha} ( t ) \), we obtain
Integrating (2.42) from \(t_{2}\) to t, we get
In view of the definition of \(\omega ( t ) \), we get
Therefore, from (2.8), it follows that
Substituting the above inequality into (2.43), we are led to
Now, taking the \(\lim\sup_{t\rightarrow\infty}\) on both sides of this inequality, we are led to contradiction.
On the other hand, let \(\upsilon^{\prime} ( t ) >0\) for all \(t\geq t_{1}\). The proof of this case is similar to that of Theorem 2.7, and so we omit it. Then the proof is complete. □
Example 2.1
Consider the equation
where \(\alpha>0\), \(\lambda,\delta\in(0,1)\), \(p_{0}\in{}[ 0,\lambda)\), \(q_{0}>0\), and \(\gamma=\max \{ \alpha,\beta \} \). We note that
and \(f ( u ) =u^{\beta}\). It is easy to calculate that
From Theorem 2.3, equation (2.44) is oscillatory if the first order delay differential equation
is oscillatory, where
For\(\alpha>\beta\), we see that \(\gamma=\alpha\) and hence
Then, by Corollary 2.2, equation (2.44) is oscillatory.
For\(\alpha<\beta\), we have (2.13) holds if
But this condition is not feasible as a result of constant \(a_{2}\). However, according to Theorem 2.2, if we take \(\gamma=\beta+1\), then (2.13) holds and hence equation (2.44) is oscillatory.
For\(\alpha=\beta\), we have the following criteria for oscillation:
By Theorem 2.2, we get the condition
By Corollary 2.1, we get the condition
By Theorem 2.6, we get the condition
By Theorem 2.8 with \(\delta ( t ) :=t^{-\alpha}\), we have that condition (2.41) holds if
As special cases, we consider the equations
and
From Table 1, we note that Condition (\(\mathrm {C}_{4}\)) supports the most efficient condition for (\(\mathrm {E}_{1}\)) and Condition (\(\mathrm {C}_{2}\)) supports the most efficient condition for (\(\mathrm {E}_{2}\)).
Moreover, for (\(\mathrm {E}_{1}\)), we see that Condition (\(\mathrm {C}_{3}\)) provides an improvement of Conditions (\(\mathrm {C}_{1}\)) and (\(\mathrm {C}_{2}\)), namely \(q_{0}>0.8532\).
Also, for Euler differential equation, if \(p_{0}=0\) and \(\alpha=1\), then condition (\(\mathrm {C}_{4}\)) reduces to \(q_{0}>1/4\), which is sharp for oscillation.
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Moaaz, O., Elabbasy, E.M. & Qaraad, B. An improved approach for studying oscillation of generalized Emden–Fowler neutral differential equation. J Inequal Appl 2020, 69 (2020). https://doi.org/10.1186/s13660-020-02332-w
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DOI: https://doi.org/10.1186/s13660-020-02332-w