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Simplified and improved criteria for oscillation of delay differential equations of fourth order
Advances in Difference Equations volume 2021, Article number: 295 (2021)
Abstract
An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.
1 Introduction and preliminaries
Delay differential equations (DDEs) are of great importance in modeling many phenomena and problems in various applied sciences, see [13]. The mounting interest in studying the qualitative properties of solutions of DDEs is easy to notice, see for example [1–12] and [14–25]. However, the equations with noncanonical operator did not receive the same attention as the equations in the canonical case. One can trace the evolution in the study of the oscillatory properties of higher-order DDEs with noncanonical operator through works of Baculikova et al. [7], Zhang et al. [23–25], and, recently, Moaaz et al. [16, 18].
This study is concerned with finding sufficient oscillation conditions for the solutions of the DDE
in the noncanonical case, that is,
In this study, we suppose that \(\kappa >0\) is a ratio of odd integers, \(a\in C^{1} ( I_{0},\mathbb{R} ^{+} ) \), \(a^{\prime } ( l ) \geq 0\), \(g\in C ( I_{0},\mathbb{R} ^{+} ) \), \(g ( l ) \leq l\), \(g^{\prime } ( l ) >0\), \(\lim_{l\rightarrow \infty }g ( l ) =\infty \), \(I_{\vartheta }:= [ l_{\vartheta },\infty ) \), \(f\in C ( I_{0}\times \mathbb{R} ,\mathbb{R} ) \), and there exists a function \(h\in C ( I_{0}, [ 0,\infty ) ) \) such that \(f ( l,v ) \geq h ( l ) v^{\kappa }\).
By a solution of (1.1), we mean a nontrivial real-valued function \(v\in C ( [ l_{\varkappa },\infty ) ,\mathbb{R} ) \) for some \(l_{\varkappa }\geq l_{0}\), which has the property \(a ( v^{\prime \prime \prime } ) ^{\kappa }\in C^{1} ( [ l_{0},\infty ) ,\mathbb{R} ) \) and satisfies (1.1) on \([ l_{0},\infty ) \). We will consider only those solutions of (1.1) which exist on some half-line \([ l_{\varkappa },\infty ) \) and satisfy the condition
If v is either positive or negative, eventually, then v is called nonoscillatory; otherwise it is called oscillatory. Equation (1.1) itself is termed oscillatory if all its solutions are oscillatory.
Zhang et al. [25] considered the higher-order DDE
where κ, γ are a ration of odd integers and \(0<\gamma \leq \kappa \). Moreover, Zhang et al. [23] studied the oscillation of solutions for (1.3) and improved the results [25]. For the convenience of the reader, we present some of their results below at \(\kappa =\gamma \) and \(n=4\).
Theorem 1.1
([25, Corollary 2.1])
If
and
for some \(\varepsilon _{1}\in ( 0,1 ) \), then every nonoscillatory solution of (1.1) tends to zero.
Theorem 1.2
([23, Corollary 2.1])
for some \(\varepsilon _{1}\in ( 0,1 ) \), where
and
then (1.1) is oscillatory.
Dzurina and Jadlovska [9] considered the second-order DDE
Moreover, Dzurina et al. [10] investigated the oscillation of solutions for (1.7) and improved the results [9].
Theorem 1.3
([9, Theorem 3])
Assume that
Then (1.7) is oscillatory.
Theorem 1.4
([10, Theorem 2.3])
Let
hold. If
or
where
then (1.7) is oscillatory.
The objective of this paper is to improve and simplify the oscillation criteria of the fourth-order DDE (1.1) in the noncanonical case. In the noncanonical case, it is usual to have oscillation criteria in the form of at least three independent conditions; however, in Sect. 2, we obtain only two independent conditions that guarantee the oscillation of all solutions. In Sect. 3, we take an approach that creates improved criteria for oscillation. Further, the examples provided illustrate the significance of the results.
Lemma 1.1
([5])
Assume that \(F\in C^{m} ( I_{0},\mathbb{R} ) \) and \(F^{ ( m ) } ( l ) \) is eventually of constant sign. Then there are \(l_{u}\geq l_{0}\) and \(\ell \in \mathbb{Z} \), \(0\leq \ell \leq m\), with \(m+\ell \) even for \(F^{ ( m ) } ( l ) \geq 0\) or \(m+\ell \) odd for \(F^{ ( m ) } ( l ) \leq 0\), such that
and
for all \(l\in I_{u}\).
2 Simplified criteria for oscillation
Lemma 2.1
Assume that \(v\in C ( [ l_{0},\infty ) , ( 0,\infty ) ) \) is a solution of (1.1). Then \(( a ( l ) ( v^{\prime \prime \prime } ( l ) ) ^{\kappa } ) ^{\prime }\leq 0\), and one of the following cases holds, eventually:
-
(a)
\(v^{\prime } ( l )\) and \(v^{\prime \prime \prime } ( l )\) are positive, and \(v^{(4)} ( l )\) is nonpositive;
-
(b)
\(v^{\prime } ( l )\) and \(v^{\prime \prime } ( l )\) are positive, and \(v^{\prime \prime \prime } ( l )\) is negative;
-
(c)
\(v^{\prime \prime } ( l )\) is positive, and \(v^{\prime } ( l )\) and \(v^{\prime \prime \prime } ( l )\) are negative.
Proof
Assume that \(v\in C ( [ l_{0},\infty ) , ( 0,\infty ) ) \) is a solution of (1.1). From (1.1), we have
From (1.1) and Lemma 1.1, there exist three possible cases (a), (b), and (c) for \(l\geq l_{1}\), \(l_{1}\) large enough. The proof is complete. □
Let us define
Theorem 2.1
Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If
then v satisfies case (b) in Lemma 2.1.
Proof
Assume on the contrary that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution (1.1) and satisfies either case (a) or case (c).
First, we suppose that (c) holds on \(I_{1}\). Since \(( a ( l ) ( v^{\prime \prime \prime } ( l ) ) ^{\kappa } ) ^{\prime }\leq 0\), we have
which is
If we divide (2.3) by \(a^{1/\kappa }\) and then integrate from lto ϱ, we find
Letting \(\varrho \rightarrow \infty \), we get
Integrating (2.4) from l to ∞, we obtain
Integrating (2.5) from l to ∞ implies that
Integrating (2.7) from \(l_{1}\) to l, we obtain
Integrating (2.8) from \(l_{1}\) to l, we get
At \(l\rightarrow \infty \), we arrive at a contradiction with (2.1).
Finally, let case (a) hold on \(I_{1}\). On the other hand, it follows from (2.1) and (1.2) that \(\int _{l_{1}}^{l}h ( s ) \psi _{2}^{\kappa }(s)\,\mathrm{d}s\) must be unbounded. Further, since \(\psi _{2}^{\prime }(s)<0\), it is easy to see that
Integrating (1.1) from \(l_{2}\) to l, we get
From (2.9) and (2.10), we get a contradiction with the positivity of \(a ( l ) (v^{\prime \prime \prime } ( l ) )^{ \kappa }\). This completes the proof. □
Theorem 2.2
Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If
then v satisfies case (b) in Lemma 2.1.
Proof
Assume on the contrary that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution (1.1) and satisfies case (a) or case (c).
First, we suppose that (c) holds on \(I_{1}\). Then
Integrating (2.12) twice from l to ∞, we arrive at
and
Integrating (1.1) from \(l_{1}\) to l, we get
since \(g^{\prime } ( l ) >0\) and \(s\leq l\), we obtain
Since \(g ( l ) \leq l\), we have
From (2.14) and (2.16), we find
Dividing both sides of inequality (2.17) by \(a ( l ) (v^{\prime \prime \prime } ( l ) )^{ \kappa }\) and taking the limsup, we arrive at
we arrive at a contradiction with (2.11).
Next, we suppose that case (a) holds on \(I_{1}\). From (2.11) and the fact that \(\psi _{2} ( l ) <\infty \), we get that (2.9) holds. Then, this part of the proof is similar to that of Theorem 2.1. This completes the proof. □
Theorem 2.3
Assume that (2.1) or (2.11) holds. If there is \(\rho \in C^{1} ( I_{0},\mathbb{R} ^{+} ) \) such that
holds for some \(\lambda _{1}\in ( 0,1 ) \), then all solutions of (1.1) are oscillatory.
Proof
Suppose that (1.1) has a nonoscillatory solution v in \(I_{0}\). Then we assume that v is eventually positive. From Lemma 2.1, we have three cases for v and its derivatives. Using Theorems 2.1 and 2.2, we have that condition (2.1) or (2.11) ensures that solution v satisfies case (b). On the other hand, using Theorem 2.2 in [18], we find that condition (2.18) contrasts with case (b). This completes the proof. □
Example 2.1
Consider the DDE
where \(h_{0}>0\) and \(\epsilon \in ( 0,1 ] \). Note that \(a ( l ) :=l^{3\kappa +1}\), \(g ( l ) :=\epsilon l\), \(f ( v ) :=v^{\kappa }\), and \(h ( l ) :=h_{0}\). Thus, we have that
and
Now, condition (2.11) reduces to
Furthermore, if \(\rho ( l ) :=1/l^{2\kappa +1}\), then condition (2.18) becomes
Using Theorem 2.3, we have that (2.19) is oscillatory if (2.20) and (2.21) hold.
Remark 2.4
Note that, we used two conditions only for testing the oscillation of the fourth-order DDEs. Moreover, our results can also be applied to ordinary DEs when \(g ( l ) =l\).
3 Improved criteria for oscillation
Theorem 3.1
Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If the DE
is oscillatory, then the solution v does not satisfy case (c).
Proof
Suppose the contrary that v satisfies case (c). As in the proof of Theorem 2.2, we get that (2.12) and (2.15) hold. From (2.12), we have
Thus, we get that
that is, \(-v^{\prime } ( l ) \psi ( l ) \geq v^{\prime \prime } ( l ) \psi _{1} ( l ) \). Therefore,
Using (3.2), we obtain that
that is, \(-\psi _{1} ( l ) v ( l ) \leq v^{\prime } ( l ) \psi _{2} ( l ) \). Hence,
Now, integrating (2.15) from l to ∞ and using (3.3), we get
Integrating (3.4) from l to ∞, we find
Thus, it is easy to see that v is a positive solution of the first-order delay differential inequality
Using [22], we have that (3.1) has also a positive solution, a contradiction. This completes the proof. □
Corollary 3.1
Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1). If
then the solution v does not satisfy case (c).
Proof
Using [22], we note that condition (3.5) ensures the oscillation of (3.1). This completes the proof. □
Lemma 3.1
Assume that \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1) and case (c) holds. If
then \(\lim_{l\rightarrow \infty }v ( l ) =0\).
Proof
Suppose that v satisfies case (c). Then we obtain that \(\lim_{l\rightarrow \infty }v ( l ) =c\geq 0\). We claim that \(\lim_{l\rightarrow \infty }v ( l ) =0\). Suppose the contrary that \(c>0\). Thus, there exists \(l_{1}\geq l_{0}\) such that \(v ( g ( l ) ) \geq c\) for \(l\geq l_{1}\), and hence
for \(l\geq l_{1}\). Integrating (3.7) twice from \(l_{1}\) to l, we obtain
and
Letting \(l\rightarrow \infty \) and using (3.6), we obtain that \(\lim_{l\rightarrow \infty }v^{\prime \prime }(l)=-\infty \), which contradicts \(v^{\prime \prime }(l)>0\). Thus, the proof is complete. □
Lemma 3.2
Assume that (3.6) holds, \(v\in C ( I_{0}, ( 0,\infty ) ) \) is a solution of (1.1), and case (c) holds. If there exists a constant \(\mu \geq 0\) such that
then
Proof
Suppose that v satisfies case (c). As in the proof of Theorem 2.2, we get that (2.13) holds. Integrating (1.1) from \(l_{1}\) to l and using \(v^{\prime } ( l ) <0\), we find
Using Lemma 3.1, we get that \(\lim_{l\rightarrow \infty }v ( l ) =0\). Thus, there is \(l_{2}\geq l_{1}\) such that
which, with (3.10), gives
Next, we have that
Combining (2.13) and (3.11), we get
This implies
It follows from (3.8) that \(\psi _{2}^{\mu } ( l ) v^{\prime } ( l ) +\mu \psi _{2}^{\mu -1} ( l ) \psi _{1} ( l ) v ( l ) \leq 0\), which, with (3.12), implies that the function \(v ( l ) /\psi _{2}^{\mu } ( l ) \) is nonincreasing. This completes the proof. □
Theorem 3.2
Assume that (3.6) holds. If there exists a constant \(\mu \geq 0\) such that (3.8) holds, and the equation
is oscillatory, then the solution v does not satisfy case (c).
Proof
Assume on the contrary that (1.1) has a positive solution v which satisfies case (c). Using Theorem 2.2 and Lemma 3.2, we get that (2.13) and (3.9) hold, respectively. Integrating (3.9) from \(g ( l ) \) to l, we obtain
which with (1.1) gives
Integrating (2.13) from l to ∞ provides
Next, we define
From (3.14) and (3.16), we conclude that
which, in view of (2.13), gives
In view of [6], differential equation (3.13) is nonoscillatory if and only if there exists a function \(w\in C ( [ l_{1},\infty ) ,\mathbb{R} ) \) satisfying inequality (3.17) for \(l\geq l_{1}\), \(l_{1}\) large enough, which is a contradiction. This completes the proof. □
Using Theorems 3.2, 1.3, and 1.4, we establish the following oscillation criteria for (1.1) under the assumption \(\psi _{2} ( l_{0} ) <\infty \).
Corollary 3.2
Assume that (3.6) holds and there exists a constant \(\mu \geq 0\) such that (3.8) holds. If \(\psi _{2} ( l_{0} ) <\infty \) and
or
hold, then the solution v does not satisfy case (c).
Theorem 3.3
Assume that (1.4), (1.5), and (3.5) hold, then all solutions of equation (1.1) are oscillatory.
Proof
Suppose to the contrary that there exists a nonoscillatory solution v of (1.1). Without loss of generality, we suppose that there exists \(l_{1}\in [ l_{0},\infty ) \) such that \(v ( l ) >0\) and \(v ( g ( l ) ) >0\) for \(l\geq l_{1}\). Using Lemma 2.1, there exist three possible cases (a)–(c). Obviously, one can show that Theorem 1.1 together with (a) and (b) leads to a contradiction with (1.4) and (1.5). Therefore, v satisfies (c). From Corollary 3.1, we get a contradiction with condition (3.5). This completes the proof. □
Theorem 3.4
Assume that (3.6), (1.4), and (1.5hold and there exists a constant \(\mu \geq 0\) such that (3.8) holds. If \(\psi _{2} ( l_{0} ) <\infty \) and (3.19) hold, then all solutions of equation (1.1) are oscillatory.
Proof
Suppose to the contrary that there exists a nonoscillatory solution v of (1.1). Without loss of generality, we suppose that there exists \(l_{1}\in [ l_{0},\infty ) \) such that \(v ( l ) >0\) and \(v ( g ( l ) ) >0\) for \(l\geq l_{1}\). Using Lemma 2.1, there exist three possible cases (a)–(c). Obviously, one can show that Theorem 1.1 together with (a) and (b) leads to a contradiction with (1.4) and (1.5). Therefore, v satisfies (c). From Corollary 3.2, we get a contradiction with condition (3.19). This completes the proof. □
Example 3.1
Consider the delay differential equation
where \(h_{0}>0\). We note that \(a ( l ) :=\mathrm{e}^{3l}\), \(h ( l ) :=h_{0}\mathrm{e}^{3l}\), \(f ( v ) :=v^{3}\), and \(g ( l ) :=l-1\). Thus, we have that
It is easy to verify that \(\psi _{2} ( l_{0} ) <\infty \), (3.6), (1.4), and (1.5) are satisfied. Now, (3.5) holds if \(h_{0}>0.149 36\). Moreover, if we choose \(\mu := ( h_{0}/3 ) ^{1/3} \), then we see that (3.8) is satisfied and (3.19) holds if \(h_{0}>0.115 05\).
Hence, by Theorem 3.3, every solution of (3.20) is oscillatory if \(h_{0}>0.149 36\). Further, by Theorem 3.4, every solution of (3.20) is oscillatory if \(h_{0}>0.115 05\).
Remark 3.5
By using [23, Corollary 2.1], equation (3.20) is oscillatory when \(h_{0}>0.316 41\). Thus, we note that Theorem 3.4 provides a better criterion for the oscillation of (3.20). Moreover, our oscillation criteria take into account the influence of \(g ( l ) \), which has not been taken care of in the related results [18, 25].
4 Conclusion
In this work, we simplified and improved the oscillation criteria for a class of even-order delay differential equations. In the noncanonical case, it always sets three conditions to check the oscillation of even-order DDEs. First, we obtained a criterion with only two conditions to check the oscillation. Furthermore, we improved the three-condition oscillation criteria by creating a better estimate of the ratio \(v ( g ( l ) ) /v ( l ) \). Through the example, we compared our results with the previous results and explained the importance of our new oscillation criteria. It will be interesting to extend our results of this study to the neutral and mixed case.
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The authors present their sincere thanks to the two anonymous referees. (E.E. Mahmoud) Taif University Research Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.
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Moaaz, O., Muhib, A., Baleanu, D. et al. Simplified and improved criteria for oscillation of delay differential equations of fourth order. Adv Differ Equ 2021, 295 (2021). https://doi.org/10.1186/s13662-021-03449-y
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DOI: https://doi.org/10.1186/s13662-021-03449-y
MSC
- 34C10
- 34K11
Keywords
- Delay argument
- Noncanonical operator
- Fourth-order
- Oscillation
- Differential equations