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Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type
Advances in Difference Equations volume 2018, Article number: 26 (2018)
Abstract
We investigate oscillatory behavior of solutions to a class of second-order nonlinear neutral delay dynamic equations with nonpositive neutral coefficients. In particular, we study the corresponding noncanonical neutral differential equations. New oscillation criteria are established that complement and improve related contributions to the subject. An example is given to illustrate the main results.
1 Introduction
Differential, difference equations, and dynamic equations on time scales have an enormous potential for applications in biology, engineering, economics, physics, neural networks, social sciences, etc. Recently, significant attention has been devoted to the oscillation theory of various classes of equations; see, e.g., [1–21]. In this paper, we are concerned with the oscillatory behavior of solutions to a second-order neutral dynamic equation
where \(\alpha\geq1\) is a ratio of odd integers and \(z(t)=x(t)-p(t)x(\tau(t))\). Throughout, the following assumptions are tacitly satisfied:
- \((I_{1})\) :
-
\(r\in\mathrm{C}_{\mathrm{rd}}([t_{0},\infty )_{\mathbb{T}},(0,\infty))\), \(R(t)=\int_{t_{1}}^{t}r^{-\frac{1}{\alpha}}(s)\Delta s\), where \(t_{1}\in[t_{0},\infty)_{\mathbb{T}}\) is sufficiently large;
- \((I_{2})\) :
-
\(p, q \in\mathrm{C}_{\mathrm{rd}}([t_{0},\infty )_{\mathbb{T}},\mathbb{R})\), \(0\leq p(t)\leq p_{0}<1\), \(q(t)\geq0\), and \(q(t)\) is not identically zero for large t;
- \((I_{3})\) :
-
\(\tau, \delta\in\mathrm{C}_{\mathrm {rd}}([t_{0},\infty)_{\mathbb{T}},\mathbb{T})\), \(\tau(t)\leq t\), \(\delta(t)\leq t\), and \(\lim_{t\rightarrow\infty}\tau (t)=\lim_{t\rightarrow\infty}\delta(t)=\infty\);
- \((I_{4})\) :
-
\(f \in\mathrm{C}(\mathbb{R}, \mathbb{R})\), \(xf (x) > 0\) for all \(x\neq0\), and there exists a positive constant k such that \({f (x)}/{x ^{\alpha}}\geq k \) for all \(x\neq0\).
We consider the following case:
By a solution of (1.1), we mean a function \(x \in \mathrm{C}_{\mathrm{rd}}[T_{x}, \infty)_{\mathbb{T}}\), \(T_{x} \in[t_{0}, \infty)_{\mathbb{T}}\), which has the property \(r(z^{\Delta})^{\alpha}\in\mathrm{C}^{1}_{\mathrm {rd}}[T_{x}, \infty)_{\mathbb{T}}\) and satisfies (1.1) on \([T_{x}, \infty)_{\mathbb{T}}\). We consider only those solutions x of (1.1) which satisfy \(\sup\{ \vert x(t) \vert : t \in[T, \infty)_{\mathbb{T}}\} > 0\) for all \(T\in[T_{x}, \infty)_{\mathbb{T}}\). We assume that (1.1) possesses such solutions. A solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is termed nonoscillatory.
In recent years, many studies have been devoted to the oscillatory behavior of solutions to different classes of equations with nonnegative neutral coefficients; see, e.g., [2, 4, 5, 12, 13, 15, 20] and the references cited therein. However, for equations with nonpositive neutral coefficients, there are relatively fewer results in the literature; see [3, 4, 6, 11, 14, 16–18]. For instance, in the particular case of (1.1) when \(\mathbb{T}=\mathbb{R}\), Li et al. [14] studied the differential equation
under the assumption that \(\int_{t_{0}}^{\infty}r^{-\frac{1}{\alpha} }(s)\,\mathrm{d}s=\infty\). Their results were improved by Arul and Shobha [3] who established new oscillation results for the solutions of (1.3). Seghar et al. [16] discussed the difference equation
where \(0\leq p_{n}\leq p<1\), \(q_{n}>0\), and \(k, l\) are positive integers, and they obtained several oscillation criteria for (1.4) assuming that \(\sum_{n=n_{0}}^{\infty}\frac{1}{a_{n}}<\infty\). Karpuz [11] established some sufficient conditions which guarantee that every solution of the second-order dynamic equation
oscillates or tends to zero, where \(0\leq p(t)\leq1\) and \(\int _{t_{0}}^{\infty }q(t)\Delta t=\infty\). Bohner and Li [6] gave new oscillation criteria for a class of second-order p-Laplace dynamic equations
where \(z(t)=x(t)-p(t)x(\tau(t))\), \(p>1\) is a constant, \(0\leq p(t)\leq p_{0}<1\), \(q(t)>0\), and \(\int_{t_{0}}^{\infty}r^{-\frac{1}{p-1}}(s)\Delta s=\infty\).
The aim of this paper is not only to improve some results in the cited papers but also to present new oscillation criteria for (1.3) in the noncanonical case
In what follows, all functional inequalities are assumed to hold eventually. Without loss of generality, we can deal only with eventually positive solutions of (1.1) and (1.3).
2 Auxiliary results
The following auxiliary results may play a major role throughout the proofs of our main results.
Lemma 2.1
(Bohner and Peterson [7])
Assume that \(v : \mathbb{T}\rightarrow\mathbb{R}\) is strictly increasing and \(\tilde{\mathbb {T}}:= v(\mathbb{T})\) is a time scale. Let \(y : \tilde{\mathbb{T}}\rightarrow \mathbb{R}\). If \(y^{\tilde{\Delta}}(v(t))\) and \(v^{\Delta}(t)\) exist for \(t \in \mathbb{T}^{\kappa}\), then
The following results can be obtained by similar techniques to those used in [3, 14].
Lemma 2.2
Let \(x(t)\) be an eventually positive solution of (1.1) and assume that (1.2) holds. Then \(z(t)\) satisfies one of the following two possibilities:
-
(I)
\(z(t)>0\), \(z^{\Delta}(t)>0\), \((r(t)(z^{\Delta}(t))^{\alpha})^{\Delta}\leq0\);
-
(II)
\(z(t)<0\), \(z^{\Delta}(t)>0\), \((r(t)(z^{\Delta}(t))^{\alpha})^{\Delta}\leq0\),
for \(t\in[t_{1},\infty)_{\mathbb{T}}\), where \(t_{1}\in[t_{0},\infty )_{\mathbb{T}}\) is sufficiently large.
Lemma 2.3
Let \(x(t)\) be an eventually positive solution of (1.1) and assume that the corresponding \(z(t)\) has property (II) of Lemma 2.2. Then
Lemma 2.4
If \(x(t)\) is an eventually positive solution of (1.1) such that case (I) of Lemma 2.2 holds, then \(x(t)\geq z(t)\) and \({z(t)}/{R(t)}\) is strictly decreasing for large t.
3 Main results
Theorem 3.1
Assume that (1.2) holds, \(\delta([t_{0},\infty )_{\mathbb{T}})=[\delta(t_{0}),\infty)_{\mathbb{T}}\), and \(\delta^{\Delta}(t)>0\). If
where \(Q(t)=\int_{t}^{\infty}k q(u)\Delta u\), then every solution \(x(t)\) of (1.1) is either oscillatory or satisfies \(\lim_{t\rightarrow\infty} x(t) = 0\).
Proof
Suppose that (1.1) has a nonoscillatory solution \(x(t)\) such that \(x(t)>0\), \(x(\tau(t))>0\), and \(x(\delta(t))>0\) for \(t\in[t_{1},\infty )_{\mathbb{T}}\). Then, by virtue of Lemma 2.2, \(z(t)\) satisfies one of the two cases (I) and (II) for \(t\in[t_{1},\infty )_{\mathbb{T}}\).
Case 1. Assume first that \(z(t)\) satisfies case (I). From the definition of \(z(t)\), we have
and therefore (1.1) takes the form
Define the Riccati substitution
It is clear that \(\nu(t)>0\) and
Note that
Using the fact that \(r(t)(z^{\Delta}(t))^{\alpha}\) is nonincreasing and \(\delta(t)\leq t \leq\sigma(t)\), (3.4) yields
Substituting (3.5) into (3.3), we get
Integrating (3.6) on \([t, s]\), we have
which implies that
Letting \(s\rightarrow\infty\), we obtain
An application of (3.7) yields
By (3.2), we conclude that
i.e.,
It follows now from (3.8) and (3.9) that
which contradicts (3.1).
Case 2. Assume now that \(z(t)\) satisfies case (II). By virtue of Lemma 2.3, \(\lim_{t\rightarrow\infty}x(t)=0\). The proof is complete. □
Theorem 3.2
Assume that (1.2) holds. If there exists a positive function \(\beta\in \mathrm{C}_{\mathrm{rd}}^{1}([t_{0},\infty)_{\mathbb{T}}, \mathbb{R})\) such that, for all sufficiently large \(t_{1}\in[ t_{0}, \infty)_{\mathbb{T}}\) and for some \(t_{2}\in[t_{1},\infty)_{\mathbb{T}}\),
then every solution \(x(t)\) of (1.1) is either oscillatory or satisfies \(\lim_{t\rightarrow\infty} x(t) = 0\).
Proof
Let \(x(t)\) be a nonoscillatory solution of (1.1) on \([t_{0},\infty )_{\mathbb{T}}\) such that \(x(t)>0\), \(x(\tau(t))>0\), and \(x(\delta(t))>0\) for \(t\in[t_{1},\infty )_{\mathbb{T}}\). Then, by Lemma 2.2, \(z(t)\) satisfies one of the two cases (I) and (II) for \(t\in[ t_{1},\infty)_{\mathbb{T}}\).
Case 1. Assume that \(z(t)\) satisfies case (I). Now, define the Riccati substitution
It is clear that \(\omega(t)>0\) and
By Lemma 2.4, we get
Applying the inequality
with
and using (3.11), we conclude that
Integrating (3.13) from \(t_{2}\) (\(t_{2}\in[t_{1}, \infty)_{\mathbb{T}}\)) to t, we have
which contradicts (3.10).
Case 2. If \(z(t)\) satisfies case (II), then, by Lemma 2.3, \(\lim_{t\rightarrow\infty}x(t)=0\). This completes the proof. □
Now, to discuss the oscillatory behavior of equation (1.3) under the assumption (1.5) (which is called a noncanonical neutral differential equation), we need the following lemma.
Lemma 3.1
Let \(x(t)\) be an eventually positive solution of (1.3). Then one of the following four cases holds for all sufficiently large t:
-
(i)
\(z(t)>0\), \(z^{\prime}(t)>0\), \((r(t)(z^{\prime}(t))^{\alpha})^{\prime}\leq0\);
-
(ii)
\(z(t)<0\), \(z^{\prime}(t)>0\), \((r(t)(z^{\prime}(t))^{\alpha})^{\prime}\leq0\);
-
(iii)
\(z(t)<0\), \(z^{\prime}(t)<0\), \((r(t)(z^{\prime}(t))^{\alpha})^{\prime}\leq0\);
-
(iv)
\(z(t)>0\), \(z^{\prime}(t)<0\), \((r(t)(z^{\prime}(t))^{\alpha})^{\prime}\leq0\).
Proof
The proof is similar to that of [14, Lemma 2.1], and hence is omitted. □
Theorem 3.3
Let conditions \((I_{1})\)-\((I_{4})\) be satisfied for \(\mathbb{T}=\mathbb{R}\) and assume that (1.5) and \(\delta^{\prime}(t)>0\) hold. Suppose further that there exists a positive function \(\beta\in \mathrm{C}^{1}([t_{0},\infty)_{\mathbb{T}},\mathbb{R})\) such that (3.10) holds for \(\mathbb{T}=\mathbb{R}\), for all sufficiently large \(t_{1}\geq t_{0}\) and for some \(t_{2}\geq t_{1}\). If
where \(\vartheta(t)=\int_{t}^{\infty}r^{-\frac{1}{\alpha }}(s)\,\mathrm{d} s\), then every solution \(x(t)\) of (1.3) is either oscillatory or satisfies \(\lim_{t\rightarrow\infty}x(t)=0\).
Proof
Let \(x(t)\) be a nonoscillatory solution of (1.3) on \([t_{0},\infty )\) such that \(x(t)>0\), \(x(\tau(t))>0\), and \(x(\delta(t))>0\) for \(t\geq t_{1}\). From Lemma 3.1, we have the following four possible cases.
Case 1. \(z(t)\) satisfies case (i). Using \(\mathbb{T}=\mathbb{R}\) in the proof of Theorem 3.2, we get a contradiction with (3.10).
Case 2. \(z(t)\) satisfies case (ii). By Lemma 2.3, we see that \(\lim_{t\rightarrow\infty}x(t)=0\).
Case 3. \(z(t)\) satisfies case (iii). Similar analysis to that in [4, Theorem 3, case (jjj)] leads to the conclusion that \(\lim_{t\rightarrow\infty}x(t)=0\).
Case 4. \(z(t)\) satisfies case (iv). Define
It is clear that \(\nu(t)<0\). Since \((r(t)(z^{\prime}(t))^{\alpha})^{\prime}\leq0\), we have, for \(s\geq t\) \((s, t\in[T_{1}, \infty))\),
i.e.,
Integrating the latter inequality from t to l, we obtain
Letting \(l\rightarrow\infty\), we get
It follows from (3.15) and (3.16) that
On the other hand, we have (3.6) with \(\sigma(t)=t\), and so
Multiplying (3.18) by \(\vartheta^{\alpha}(t)\) and integrating the resulting inequality from \(T_{1}\) to t, we deduce that
Applying inequality (3.12) with \(\omega=-\nu(t)\), \(A={\alpha \delta^{\prime}(t) \vartheta^{\alpha}(t)}/{r^{\frac{1}{\alpha }}(\delta(t))}\), and \(B= {\alpha\vartheta^{\alpha-1}(t)}/{r^{\frac {1}{\alpha}}(t)}\), we arrive at
Combining (3.19) and (3.20), we obtain
Then, by virtue of (3.17) and (3.21),
which contradicts (3.14). This completes the proof. □
Example 3.1
Assume that \(\mathbb{T}=\mathbb{R}\). Consider the second-order neutral delay differential equation
Here, \(\alpha=3\), \(z(t)=x(t)-x(t/3)/2\), \(\gamma>0\) is a constant, \(k=\gamma\), \(r(t)=t^{2}\), \(q(t)=t^{-2}\), and \(\delta(t)=t/2\). Now we have
Therefore, by Theorem 3.1, every solution \(x(t)\) of equation (3.22) is oscillatory or satisfies \(\lim_{t\rightarrow\infty}x(t)=0\) when \(\gamma>0.041\). However, [14, Theorem 3.1] yields the same conclusion if \(\gamma>{2}/{(27k_{0}^{3})}\) for some \(k_{0}\in(0,1)\) which means that \(\gamma>{2}/{27}\approx0.0741\). Hence, Theorem 3.1 improves [14, Theorem 3.1].
Remark 3.1
Oscillation criteria established in this paper for equation (1.3) complement, on one hand, the results reported by Arul and Shobha [3] and Li et al. [14] because we use assumption (1.5) rather than \(\int _{t_{0}}^{\infty}r^{-\frac{1}{\alpha}}(s)\,\mathrm{d}s=\infty\) and, on the other hand, those by Džurina and Jadlovská [8] since our criteria can be applied to the case where \(0\leq p(t)\leq p_{0}<1\).
Remark 3.2
As fairly noticed by the referees, technique used in this paper does not allow a straightforward extension of Theorem 3.3 to equation (1.1); this remains an open problem for further research.
References
Agarwal, RP, Bohner, M, Li, T, Zhang, C: Oscillation criteria for second-order dynamic equations on time scales. Appl. Math. Lett. 31, 34-40 (2014)
Agarwal, RP, O’Regan, D, Saker, SH: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. Math. Anal. Appl. 300, 203-217 (2004)
Arul, R, Shobha, VS: Improvement results for oscillatory behavior of second order neutral differential equations with nonpositive neutral term. Br. J. Math. Comput. Sci. 12, 1-7 (2016)
BaculÃková, B, Džurina, J: Oscillation of third-order neutral differential equations. Math. Comput. Model. 52, 215-226 (2010)
Bohner, M, Grace, SR, Jadlovská, I: Oscillation criteria for second-order neutral delay differential equations. Electron. J. Qual. Theory Differ. Equ. 2017, 60 (2017)
Bohner, M, Li, T: Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient. Appl. Math. Lett. 37, 72-76 (2014)
Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Džurina, J, Jadlovská, I: A note on oscillation of second-order delay differential equations. Appl. Math. Lett. 69, 126-132 (2017)
Erbe, L, Peterson, A, Saker, SH: Oscillation criteria for second-order nonlinear delay dynamic equations. J. Math. Anal. Appl. 333, 505-522 (2007)
Hilger, S: Analysis on measure chain - a unified approach to continuous and discrete calculus. Results Math. 18, 18-56 (1990)
Karpuz, B: Sufficient conditions for the oscillation and asymptotic behaviour of higher-order dynamic equations of neutral type. Appl. Math. Comput. 221, 453-462 (2013)
Li, T, Agarwal, RP, Bohner, M: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat. 41, 715-721 (2012)
Li, T, Saker, SH: A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales. Commun. Nonlinear Sci. Numer. Simul. 19, 4185-4188 (2014)
Li, Q, Wang, R, Chen, F, Li, T: Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients. Adv. Differ. Equ. 2015, 35 (2015)
Li, T, Zhang, C, Thandapani, E: Asymptotic behavior of fourth-order neutral dynamic equations with noncanonical operators. Taiwan. J. Math. 18, 1003-1019 (2014)
Seghar, D, Thandapani, E, Pinelas, S: Oscillation theorems for second order difference equations with negative neutral term. Tamkang J. Math. 46, 441-451 (2015)
Thandapani, E, Balasubramanian, V, Graef, JR: Oscillation criteria for second order neutral difference equations with negative neutral term. Int. J. Pure Appl. Math. 87, 283-292 (2013)
Thandapani, E, Mahalingam, K: Necessary and sufficient conditions for oscillation of second order neutral difference equations. Tamkang J. Math. 34, 137-145 (2003)
Wang, J, El-Sheikh, MMA, Sallam, RA, Elimy, DI, Li, T: Oscillation results for nonlinear second-order damped dynamic equations. J. Nonlinear Sci. Appl. 8, 877-883 (2015)
Zhang, C, Agarwal, RP, Bohner, M, Li, T: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. 38, 761-778 (2015)
Zhang, C, Li, T: Some oscillation results for second-order nonlinear delay dynamic equations. Appl. Math. Lett. 26, 1114-1119 (2013)
Acknowledgements
The authors thank the editors and two anonymous referees for the careful reading of the paper and for pointing out several inaccuracies in the text. This research is supported by Project of Shandong Province Independent Innovation and Achievement Transformation (No. 2014ZZCX02702), Shandong Province Key Research and Development Project (No. 2016GGX109001), Shandong Provincial Natural Science Foundation (Nos. ZR2017MF050, ZR2014FL008, and ZR2015FL014), and Project of Shandong Province Higher Educational Science and Technology Program (No. J17KA049).
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Zhang, M., Chen, W., El-Sheikh, M. et al. Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type. Adv Differ Equ 2018, 26 (2018). https://doi.org/10.1186/s13662-018-1474-5
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DOI: https://doi.org/10.1186/s13662-018-1474-5
MSC
- 34K11
- 34N05
Keywords
- oscillation
- second-order
- delay dynamic equation
- neutral dynamic equation