Abstract
In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.
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1 Introduction
The importance of studying delay differential equations DDEs is not limited to the theoretical side only, but the applications of this type of equations extend to many branches of applied science. In fact, the neutral DDEs arise in the examination of vibrating masses attached to an elastic bar, in the solution of variational problems with time delays, and in problems concerning electric networks containing lossless transmission lines (as in high speed computers), see [1, 2].
The great development in the study of asymptotic behavior of DDEs is easily noted in many works in recent times. Some of these works that are concerned with improving the oscillation criteria of DDEs are [3–6]. In addition, many improved methods and interesting results can be found in studies [7–23], which study the oscillatory behavior of the NDDEs of different order.
In this work, we discuss the oscillation properties of the second-order NDDE with distributed deviating arguments
where \(t\geq t_{0}\) and
Throughout this work, we assume that
and the following hypotheses hold:
- (\(\mathrm{H}_{1}\)):
-
α is a ratio of odd natural numbers, \(r\in C ( [ t_{0},\infty ) , ( 0,\infty ) ) \), and
$$ \int _{t_{0}}^{\infty }r^{-1/\alpha } ( s ) \,\mathrm{d}s= \infty . $$(1.2) - (\(\mathrm{H}_{2}\)):
-
\(p\in C ( [ t_{0},\infty ) ) \), \(q\in C ( [ t_{0},\infty ) \times [ a,b ] ) \), \(0\leq p ( t ) <1\), and \(q ( t,s ) >0\) is not zero on any half line \([ t_{\ast },\infty ) \times [ a,b ] \) for all \(t_{\ast }\geq t_{0}\).
- (\(\mathrm{H}_{3}\)):
-
\(\tau \in C ( [ t_{0},\infty ) ,\mathbb{R} ) \), \(g\in C ( [ t_{0},\infty ) \times [ a,b ] ,\mathbb{R} ) \), \(\tau ( t ) \leq t\), \(\lim_{t\rightarrow \infty }\tau ( t ) =\infty \), \(g ( t,s ) \leq t\), \(\lim_{t\rightarrow \infty }g ( t,s ) =\infty \) for \(s\in [ a,b ] \), and g is strictly increasing with respect to t and s for all \(s\in ( a,b ) \).
- (\(\mathrm{H}_{4}\)):
-
\(f\in C ( ( -\infty ,\infty ) ) \) and \(f ( x ) /x^{\alpha }\geq \kappa \) for \(x\neq 0\), where κ is a positive constant.
For a solution of (1.1), we mean a function \(x\in C ( [t_{x},\infty ) ) \), \(t_{x}\geq t_{0}\), which has the property \(\aleph ( t ) \) and \(r ( t ) ( \aleph ^{\prime } ( t ) ) ^{\alpha }\) are continuously differentiable for \(t\in {}[ t_{x},\infty )\) and satisfies (1.1) on \([t_{x},\infty )\). We focus only on the solutions of (1.1) which satisfy \(\sup \{ \vert x ( t ) \vert :t_{x}\leq t\}>0\) for \(t\geq t_{x}\). A solution x of (1.1) is called nonoscillatory if it is either eventually positive or eventually negative; otherwise it is called oscillatory.
In the next part of the introduction, we provide some related work that contributed to the development of the study of oscillatory behavior of NDDEs.
In 1985, Grammatikopoulos et al. [7] studied the asymptotic behavior of NDDE
They proved that all solutions of (1.3) are oscillatory if \(p(t)\in [ 0,1 ] \) and
However, Erbe et al. [8] established the oscillation condition when \(q(t)\geqslant q_{0}>0\), \(p ( t ) \in [ p_{1},p_{2} ] \) and \(p(t)\) is not eventually negative. Posteriorly, Grace and Lalli [9] studied the oscillation of the NDDE
under the condition
where \(\rho \in C^{1} ( [ t_{0},\infty ) , ( 0, \infty ) ) \).
In the previous decade, under the hypothesis \(\tau \circ g=g\circ \tau \), Han et al. [10] presented the oscillation criteria for the NDDE
In 2012, by using the Riccati transformation technique, Liu et al. [11] and Wu et al. [12] obtained the oscillation conditions for the NDDE
where \(w ( t ) :=x(t)+p(t)x(\tau (t))\), \(\alpha \geq \beta \), \(r^{\prime } ( t ) >0\), and \(g^{\prime } ( t ) >0\). Based on establishing new comparison theorems that compare the second-order equation with a first-order DDE, Baculikova and Dzurina [13] studied the NDDE
under the conditions
Of interesting works recently, Moaaz et al. in [14, 15] studied the oscillatory properties of (1.4) and improved the results in [13].
For NDDE with distributed deviating arguments (1.1), Candan [16] studied the sufficient conditions for the oscillation of solutions.
In this work, we are creating an improved relationship between the corresponding function ℵ and its first derivative. This new relationship helps us to get sharp criteria for testing the oscillation. Based on the Riccati transformation and comparison principles, we obtain new and different criteria for the oscillation of solutions of (1.1). The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.
To prove our main results, we need the following auxiliary lemmas. The proof of the first lemma is similar to that of [13, Lemma 3] and hence we omit it.
Lemma 1.1
If x is a positive solution of (1.1) on \([ t_{0},\infty ) \), then there exists \(t_{1}\geq t_{0}\) such that
for \(t\geq t_{1}\).
Lemma 1.2
([14, Lemma 1.2])
Suppose that \(F ( s ) =As-Bs^{(\alpha +1)/\alpha }\), where \(A,B>0\) are constants. Then F attains its maximum value on \(\mathbb{R} \) at \(s^{\ast }= ( \alpha A\diagup ( ( \alpha +1 ) B ) ) ^{\alpha }\) and
Remark 1.1
All functional inequalities are assumed to hold eventually, that is, they are satisfied for all \(t>t_{1}\), where \(t_{1}\) is large enough.
2 Main results
For convenience, we denote the class of all eventually positive solutions by \(S^{+}\). Moreover, we assume the following notations: \(\beta \overset{\mathrm{def}}{=} ( \alpha +1 ) ^{\alpha +1}\), \(g_{a} ( t ) \overset{\mathrm{def}}{=}g ( t,a ) \),
and \(F_{+}(t)\overset{\mathrm{def}}{=}\max \{ 0,F(t) \} \), where \(t_{1}\geq t_{0}\).
The following theorem gives a criterion for the oscillation of (1.1), depending on the comparison with a first-order DDE.
Theorem 2.1
Every solution of (1.1) is oscillatory if the first-order DDE
is oscillatory.
Proof
Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume \(x\in S^{+}\), and so \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). It follows from Lemma 1.1 that (1.5) holds. Using the fact that \(r ( t ) ( \aleph ^{\prime } ( t ) ) ^{\alpha }\) is a nonincreasing function, we get
and hence
It follows from (1.1) and (H1) that
From the definition of ℵ, we have
which with (2.3) gives
which is a direct result of the facts that \(\aleph ^{\prime } ( t ) >0\) and \(\partial _{s}g ( t,s ) >0\). Combining
and
we get
Thus, from (2.4), we find
Integrating (2.5) from \(t_{1}\rightarrow t\), we have
Now, we set \(\phi ( t ) :=r ( t ) ( \aleph ^{\prime } ( t ) ) ^{\alpha }\). Then, from (2.2) and (2.6), we obtain
Combining (2.4) and (2.7), we have that ϕ is a positive solution of the first-order DD inequality
From [24, Theorem 1], DDE (2.1) also has a positive solution, which is a contradiction. This contradiction completes the proof. □
Applying a well-known condition [25, Theorem 2.1.1] for oscillation of first-order DDE (2.1), we get immediately the following criteria for oscillation of (1.1).
Corollary 2.1
Every solution of (1.1) is oscillatory if one of the following conditions is satisfied:
or
The next theorem gives another criterion for the oscillation of (1.1), depending on the Riccati transformation technique.
Theorem 2.2
Every solution of (1.1) is oscillatory if there is a positive function \(\rho \in C^{1} ( [ t_{0},\infty ) ) \) satisfying
where \(t_{1}\) is sufficiently large.
Proof
Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume \(x\in S^{+}\), and so \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). It follows from Lemma 1.1 that (1.5) holds. Now, we set
Thus, we note that \(w ( t ) >0\) for \(t\geq t_{1}\). By differentiating w, we get
Next, as in the proof of Theorem 2.1, we obtain (2.4) and (2.7). Then, from (2.7), we obtain \(\aleph (t)\geq \widetilde{\mu }(t)r^{1/\alpha } ( t ) \aleph ^{\prime } ( t ) \) for \(t\geq t_{1}\). Then, applying the Grönwall inequality, we find
which with (2.4) gives
Combining (2.12) and (2.13), we arrive at
Using Lemma 1.2 with \(A=\rho _{+}^{\prime }/\rho \) and \(B=\alpha ( \rho r ) ^{-1/\alpha }\), we get
Integrating (2.15) from \(t_{1}\rightarrow t\), we find
which contradicts (2.10). This contradiction completes the proof. □
It is easy to see that Corollary 2.1 cannot be applied in the case where
However, if \(x\in S^{+}\) and (2.16) holds, then we can get a sharp estimate of \(z ( g ( t ) ) /z ( t ) \). Thus, we can obtain a sharp criteria for the oscillation of (1.1).
Lemma 2.1
Assume that \(x\in S^{+}\) and
for some \(\delta >0\). Then
for every \(n\geq 0\), where
Proof
The proof of the first lemma is similar to that of [26, Lemma 1], and hence we omit it. □
Theorem 2.3
Assume that (2.17) holds for some \(\delta <0\). Every solution of (1.1) is oscillatory if there is a positive function \(\rho \in C^{1} ( [ t_{0},\infty ) ) \) satisfying
for some \(m\geq 0\), where \(\vartheta _{m}(\delta )\) is defined as (2.19).
Proof
Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume \(x\in S^{+}\), and so \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). It follows from Lemma 1.1 that (1.5) holds. As in the proof of Theorem 2.1, we obtain (2.4). Now, we set
Thus, we note that \(\omega (t)>0\) for \(t\geq t_{1}\). By differentiating ω and using (2.4), we obtain
Thus, it follows from Lemma 2.1 that
Using Lemma 1.2 with \(A=\rho _{+}^{\prime }/\rho \) and \(B=\alpha \vartheta _{m}^{1/\alpha } ( \delta ) / ( \rho r ( g ) ) ^{1/\alpha }\), we get
Integrating (2.22) from \(t_{1}\rightarrow t\), we find
which contradicts (2.20). This contradiction completes the proof. □
3 Further results
It is easy to notice that (2.7) is a sharper estimate than (2.2) for the relationship between ℵ and \(\aleph ^{\prime }\). By repeating the same steps that improved (2.2), we obtain iterative criteria that can be applied even when the other criteria fail.
Lemma 3.1
Assume that \(x\in S^{+}\). Then
for \(k=0,1,\ldots \) , where \(U_{0} ( t ) :=\widetilde{\mu }(t)\) and
Proof
Assume that \(x\in S^{+}\). Then \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). It follows from Lemma 1.1 that (1.5) holds. By induction, we will prove (3.1).
Now, as in the proof of Theorem 2.1, we obtain (2.4) and (2.7). From (2.7), we obtain
Next, for \(k=n\), we suppose that \(\aleph \geq U_{n}r^{1/\alpha }\aleph ^{\prime }\). Hence, we get
which with (2.4) gives
Letting \(H:=r ( \aleph ^{\prime } ) ^{\alpha }\), (3.3) reduces to
Applying the Grönwall inequality in (3.4), we find
for \(t\geq s\geq t_{1}\), and so
Integrating (3.5) from \(t_{1}\rightarrow t\), we see that
This completes the proof. □
Theorem 3.1
Assume that \(U_{k}\) are defined as in Lemma 3.1. Every solution of (1.1) is oscillatory if
for some \(k=0,1,\ldots \) .
Proof
Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume \(x\in S^{+}\), and so \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). It follows from Lemma 1.1 and 3.1 that (1.5) and (3.1) hold. As in the proof of Theorem 2.1, we obtain (2.4).
Now, we define w as in (2.11) with \(\rho \equiv 1\). Proceeding as in the proof of Theorem 2.2 and substituting (2.7) with (3.1), we arrive at
or
Integrating (3.8) from \(t_{1}\rightarrow t\), we get
Therefore, \(w ( t ) \rightarrow -\infty \) as \(t\rightarrow \infty \), which is a contradiction. This contradiction completes the proof. □
Theorem 3.2
Assume that \(U_{k}\) are defined as in Lemma 3.1. Every solution of (1.1) is oscillatory if
where
Proof
Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume \(x\in S^{+}\), and so \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). By the same procedure as in the proof of Theorem 3.1, we arrive at (3.7). Then, integrating (3.7) from \(t\rightarrow v\), we find
Letting \(v\rightarrow \infty \), we get
or equivalently,
If we set \(\varrho =\inf_{t\geq t_{1}} ( w(t)/G ( t ) ) \), then we note that \(\varrho \geq 1\). However, from (3.9) and (3.11), we get
Therefore, relationship (3.12) can be modeled on the form
which contradicts the possible values of ϱ and α. This contradiction completes the proof. □
For the following theorem, we need to define the sequence \(\{ \phi _{n} ( t ) \} _{n=0}^{\infty }\) as
and
for all \(t\geq t_{1}\geq t_{0}\), where G is defined as in Theorem 3.2.
Lemma 3.2
Assume that \(x\in S^{+}\) and w is defined as in (2.11) with \(\rho \equiv 1\). Then \(w ( t ) \geq \phi _{n} ( t ) \). In addition, there is a function \(\phi \in C ( [ t_{1},\infty ) , ( 0,\infty ) ) \) such that \(\lim_{n\rightarrow \infty }\phi _{n} ( t ) =\phi ( t ) \) and
Proof
Assume that \(x\in S^{+}\). Then \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). By the same procedure as in the proof of Theorem 3.2, we arrive at (3.10), and hence
Thus, from the definition of \(\phi _{n}(t)\), we note that \(w(t)\geq \phi _{n}(t)\) for all \(n>1\) and \(t\geq t_{1}\). Since \(\{\phi _{n}(t)\}_{n=0}^{\infty }\) is an increasing sequence and bounded from above, \(\phi _{n}(t)\) converges to \(\phi (t)\). By using Lebesgue’s monotone convergence theorem, if we take the limit of (3.13) as \(n\rightarrow \infty \), then we obtain that (3.14) hold. The proof is complete. □
Theorem 3.3
Assume that \(U_{k}\) are defined as in Lemma 3.1. Every solution of (1.1) is oscillatory if
for some positive integers n.
Proof
Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume \(x\in S^{+}\), and so \(x ( t )\), \(x ( \tau ( t ) ) \), and \(x ( g ( t,s ) ) \) are positive for \(t\geq t_{1}\geq t_{0}\) and \(s\in [ a,b ] \). Now, we define w as in (2.11) with \(\rho \equiv 1\). From the fact that \(( r ( s ) ( \aleph ^{\prime } ( s ) ) ^{\alpha } ) ^{\prime }\leq 0\), we have
which with (2.11) gives
Thus,
for \(t\geq t_{1}\). Taking into account Lemma 3.2, we get a contradiction with (3.15). This contradiction completes the proof. □
4 Examples
In this section, we apply our main results to some special cases of (1.1) and also compare our results with the previous related results.
Example 4.1
Consider the second-order NDDE
where \(p_{0}\in [ 0,1 ) \), \(q_{0}>0\), \(\delta \in [ 0,1 ) \), \(\tau ( t ) =\eta t\), \(\eta \in ( 0,1 ) \), and \(g ( t,s ) =st\) for \(s\in [ a,1 ] \). Obviously, we see that
Therefore, it is easy to verify that
where
Next, to apply Corollary 2.1, we must first check either condition (2.9) or (2.8). By substitution and a simple computation, we obtain
Thus, by using Corollary 2.1, (4.1) is oscillatory if
On the other hand, condition (3.9) with \(k=1\) reduces to
From Theorem 3.2, equation (4.1) is oscillatory if
Example 4.2
Consider the second-order NDDE
where \(\delta \in ( 0,1 ] \), \(q_{0}>0\), and \(\eta ,\lambda \in ( 0,1 ) \). Obviously, we see that
Therefore, it is easy to verify that
and
From Theorem 3.2, equation (4.1) is oscillatory if
Remark 4.1
Consider the special case of (4.2) where \(\delta =1\), \(\alpha =1/3\), and \(\lambda =0.9\). The oscillation criteria in [13, Corollary 2] and [17, Corollary 2.1] reduce to \(q_{0}>3.61643\) and \(q_{0}>1.92916\), respectively. However, (4.3) reduces to \(q_{0}>0.16131\). So, our results improve and extend some of the previous results.
5 Conclusion
The oscillation theory of DDEs has many applications in applied sciences. Thus, studying the oscillation of the solutions of these equations has practical importance besides the theoretical importance. In this study, we obtained different oscillation criteria with different techniques. These new criteria enable us to test the oscillation of a class of NDDEs with continuous delay. Our results extended to recently published works [14, 15], and also improved [13, 17].
Modeling by fractional-order differential equations has more advantages than by classical integer-order ones as it considers the effects of existence of time memory or long-range space interactions. So, it would be interesting to extend the results of this paper to the fractional delay differential equations. Moreover, it is interesting to study the periodicity behavior of solutions of the studied equation as an extension of the works [19, 23].
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The author is grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out some inaccuracies.
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This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937).
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Moaaz, O., Park, C., Elabbasy, E.M. et al. New oscillation criteria for second-order neutral differential equations with distributed deviating arguments. Bound Value Probl 2021, 35 (2021). https://doi.org/10.1186/s13661-021-01512-x
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DOI: https://doi.org/10.1186/s13661-021-01512-x