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Oscillation of third order nonlinear damped dynamic equation with mixed arguments on time scales
Advances in Difference Equations volume 2018, Article number: 233 (2018)
Abstract
The objective of this paper is to offer sufficient conditions for the oscillation of all solutions of the third order nonlinear damped dynamic equation with mixed arguments of the form
on time scales, where \(a(t)\geq t\) and \(g(t)\leq t\). Using Riccati transformation, integral averaging technique, and comparison theorem, we give some new criteria for the oscillation of the studied equation. Our results essentially improve and complement the earlier ones.
1 Introduction
This paper deals with oscillatory behavior of all solutions of the third order nonlinear damped dynamic equation with mixed arguments of the form
where \(I=[t_{0},\infty)_{\mathbb{T}}\), \(\alpha\geq1\) is the ratio of positive odd integers. In the sequel, assume that the conditions are satisfied:
-
(H1)
\(r_{1},r_{2},p,q\in C_{rd}(I,\mathbb{R}^{+})\), \(a\in C_{rd}(I,\mathbb {R})\), \(g\in C_{rd}^{1}(I,\mathbb{R})\), where \(\mathbb{R}^{+}=(0,\infty)_{\mathbb{T}}\);
-
(H2)
\(a(t)\geq\sigma(t)\geq t\), \(g(t)\leq t\), \(g^{\Delta}(t)\geq0\) and \(g(t)\rightarrow\infty\) as \(t\rightarrow\infty\);
-
(H3)
\(\psi, f \in C(\mathbb{T}\times\mathbb{R},\mathbb{R})\) such that \(\psi(t,x(t))\geq k_{1}x^{\alpha}(t)\), \(\psi(t,-x(t))=-\psi(t,x(t))\), and \(f(t,x(t))\geq\max\{k_{2}x^{\beta}(t),k_{2}x^{\beta}(\sigma(t))\}\), \(f(t,-x(t))=-f(t,x(t))\), and \(x(t)\) is defined on \(\mathbb{T}\), \(k_{1}\), \(k_{2}\) are constants, β is the ratio of positive odd integers.
We define
and
for \(t_{0}\leq t_{1}\leq t\leq\infty\), and assume that
and
Let \(\mathbb{T}\) be a time scale with \(\sup\mathbb{T}=\infty\). We only consider these solutions of (1.1) which exist on some half-line \([t_{0},\infty)_{\mathbb{T}}\) and satisfy \(\sup\{|x(t)|:t_{1}\leq t<\infty\}>0\) for any \(t_{1} \geq t_{0}\). If \(y, r_{1}(y^{\Delta})^{\alpha}, r_{2}(r_{1}(y^{\Delta})^{\alpha})^{\Delta}\in C^{1}_{rd}([t_{y},\infty),\mathbb{R})\) and y satisfies (1.1) on \([t_{y},\infty)_{\mathbb{T}}\) for some \(t_{y}\geq t_{0}\), then the function y is called a solution of (1.1). A solution \(y(t)\) of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. The equation itself is called oscillatory if all of its solutions are oscillatory.
In recent years, there has been an increasing interest in studying the oscillation of solutions of the equations, we refer the readers to [1–13] and the references cited therein. The dynamic equations with deviating arguments are deemed to be adequate in modeling of the countless processes in all areas of science. As is well known, a distinguishing feature of delay dynamic equations under consideration is the dependence of the evolution rate of the processes described by such equations on the past history. This consequently results in predicting the future in a more reliable and efficient way, explaining at the same time many qualitative phenomena such as periodicity, oscillation or instability. Contrariwise, advanced dynamic equations can find use in many applied problems whose evolution rate depends not only on the present, but also on the future, it also plays a vital role. The dynamic equations with mixed arguments have both advanced arguments and delay arguments, and have both properties.
In 2017, BaculÃková [3] studied the oscillatory behavior of the second order advanced differential equation
where \(\sigma(t) \geq t\), and amended some oscillatory criteria for the second order advanced differential equation.
And there are many results on the oscillation of the delay dynamic equation, we refer the readers to [4–8, 10–13] and the references cited therein. The study of dynamic equation with mixed arguments is also of great significance, due to the comprehensive use in natural science and theoretical study.
In 2014, Adıvar et al. [8] studied the oscillation of the third order delay and advanced dynamic equations
and
on \([t_{0},\infty) \) such that \(t\in\mathbb{T}\) and \(t_{0}\geq0\), where α is the ratio of two positive odd integers.
However, to the best of our knowledge, there is very little known about the oscillatory behavior of dynamic equation with mixed arguments on time scales. And there are no known results regarding the oscillation of third order dynamic equation with mixed arguments of type (1.1). More exactly, the existing literature does not provide any criteria which ensure oscillation of all solutions of equation (1.1).
In view of the above motivation, our aim in this paper is to present sufficient conditions which ensure that all solutions of (1.1) are oscillatory. We give some new criteria for the oscillation of (1.1) by using the Riccati transformation and the integral averaging technique. Moreover, we present a new comparison theorem for deducing the oscillation of (1.1) from the oscillation of a suitable second order advanced dynamic equation. Thus, our method essentially simplifies the examination of the third order equation; and what is more, it supports backward the research on the second order advanced dynamic equation. For the study of oscillation of the advanced equation, we refer the readers to [3, 9, 10]. Indeed, there are no known results about the oscillation of the third order damped advanced dynamic equation in the form of (1.1) when \(q(t)\equiv0\). And there are the results of the third order delay dynamic equation in the form of (1.1) when \(p(t)\equiv0\). Our results essentially improve and complement the earlier ones. We also repair some of results of Bohner et al. [4].
2 Preliminaries
As usually, studying the properties of oscillatory solutions of (1.1), we can restrict our attention only to positive ones. In this section, we derive some new properties of oscillatory solutions of (1.1) that will be used for establishing new oscillatory criteria. Let
Definition 2.1
For function \(f:\mathbb{T} \rightarrow\mathbb{R}\), we define the derivative \(f^{\Delta}\) as follows: Let \(t\in\mathbb{T}\). If there exists a number \(\alpha\in\mathbb{R}\) such that for all \(\varepsilon>0\) there exists a neighborhood U of t with
for all \(s\in U\), then f is said to be differentiable at t, and we call α the delta derivative of f at t and denote it by \(f^{\Delta}(t)\).
Lemma 2.1
Assume that (1.1) is nonoscillatory and y is a nonoscillatory solution of (1.1) on \([t_{1},\infty)_{\mathbb{T}}\), \(t_{1}\geq t_{0}\). Then there exists \(t_{2}\in[t_{1},\infty)_{\mathbb{T}}\) such that one of the following cases holds for all sufficiently large \(t\geq t_{2}\):
Proof
If y is a nonoscillatory solution of (1.1) on \([t_{1},\infty)_{\mathbb{T}}\), say \(y(t)>0\), \(y(g(t))>0\) for \(t\geq t_{1}\geq t_{0}\). Since \(p,q\in C_{rd}(I,\mathbb{R}^{+})\), \(\psi, f \in C(\mathbb{T}\times\mathbb{R},\mathbb{R})\), then, it is easy to see that
then \(L_{2}y(t)\) is decreasing on \([t_{1},\infty)_{\mathbb{T}}\), which implies \(L_{2}y(t)\) does not change sign eventually, then there exists \(t_{2}\geq t_{1}\) such that either \(L_{2}y(t) >0\) or \(L_{2}y(t) <0\) for any \(t\geq t_{2}\).
Next, assume that \(L_{2}y(t)<0\), then \(L_{1}y(t)\) is decreasing and \(L_{1}y(t) >0\) or \(L_{1}y(t) <0\) for \(t\geq t_{3} \geq t_{2}\). If \(L_{1}y(t)>0\), we have
and (1.3) would imply \(L_{1}y(t)\rightarrow-\infty\) as \(t\rightarrow\infty\), which is a contradiction to the positivity of \(L_{1}y(t)\). Further, if \(L_{1}y(t)<0\), then by integration of
we obtain \(y(t)<0\) for all large t, which is a contradiction. Altogether, \(L_{2}y(t)>0\) on \([t_{3},\infty)_{\mathbb{T}}\). This completes the proof. □
Lemma 2.2
Suppose that (2.1) of Lemma 2.1 holds and y is a nonoscillatory solution of (1.1), \(t\geq t_{1}\geq t_{0}\). Then
and
Proof
If y is a nonoscillatory solution of (1.1), and \(y(t)>0\), \(y(g(t))>0\) for \(t\geq t_{1}\geq t_{0}\). Since \(L_{1}y(t)>0\), then
Thus
Now, integrating the above inequality from \(t_{1}\) to t, we have
This completes the proof. □
Lemma 2.3
Assume that \(\beta>0\) is the ratio of positive odd integers and \(x^{\beta}(t) \in C_{rd}^{1}(I,\mathbb{R})\). Then
Lemma 2.4
([14] (Theorem 1.14) (Mean value theorem))
Let f be a continuous function on \([a, b]\) that is differentiable on \([a, b)\). Then there exist \(\eta,\xi\in[a, b)\) such that
3 Oscillation results
Now we are prepared to provide our main oscillatory theorems. By using the Riccati transformation and the integral averaging technique due to Philos [15], we establish new oscillation results for (1.1). Firstly, let us introduce now the class of functions \(\mathcal{P}\) which will be used in this section. Let
A function \(H\in C_{rd}(D, \mathbb{R})\) is said to belong to the class \(\mathcal{P}\) if
and \(H(t, s)\) has a continuous and nonpositive partial derivative on \(D_{0}\) with respect to the second variable, and for a positive continuous function h̄,
When \(H(t,s) = (t-s)^{n}\), \(n\in N\), the Philos-type conditions reduce to the Kamenev-type ones.
Theorem 3.1
Assume that (1.2) (1.3) hold, \(\alpha\geq\beta\). If there exist a function \(m\in C_{rd}(I, \mathbb{R})\) such that \(m(t)>0\) and a function \(H(t, s)\in\mathcal{P}\) satisfying
where
and
Moreover, if every solution of the equation
is oscillatory, where
for all constants \(c,c^{*}> 0\). Then every solution \(y(t)\) of (1.1) or \(L_{2}y(t)\) is oscillatory.
Proof
If y is a nonoscillatory solution of (1.1) on \([t_{1},\infty)_{\mathbb{T}}\), \(t_{1}\geq t_{0}\). Assume that \(y(t)>0\) and \(y(g(t))>0\) for \(t\geq t_{1}\). By the proof of Lemma 2.1, we have that two cases of Lemma 2.1 hold. Now, we shall show that in each case we are led to a contradiction.
Case (1). Suppose that (2.1) of Lemma 2.1 holds. Define the following Riccati transformation:
Then \(w(t)>0\), and
By (H3) and \(y(g(\sigma(t)))\geq y(g(t))\), we have \(f(t, y(g(t)))\geq k_{2}y^{\beta}(g(\sigma(t)))\). From (1.1) and (2.4), then
Now, according to the method given in [16], and by Lemma 2.3, we have
Then, if \(\sigma(t) > t\), by Lemma 2.4, we get
where \(\xi\in[g(t),g(\sigma(t)))\). If \(\sigma(t) = t\), we obtain \(g(\sigma(t))=\sigma(g(t))=g(t)\) and
Moreover, since \(L_{2}y(t)>0\), which implies that \(r_{1}(t)(y^{\Delta}(t))^{\alpha}\) is increasing, then
that is,
thus
Then, for \(0<\beta\leq1\),
And for \(\beta\geq1\),
Altogether, for all \(\beta>0\), one has
Then (3.4) implies that
By (2.4), we have
Further,
Then (3.6) implies that
What is more,
holds for all \(t\geq t_{2}\), where \(c_{1}=L_{2}y(t_{1})\) and \(\tilde{c}_{1}=c_{1}+\frac{L_{1}y(t_{1})}{R_{2}(t_{2},t_{1})}\). And
holds for all \(t\geq t_{2}\geq t_{1}\), where \(c_{2}=\frac{y(t_{2})}{R^{*}(t_{2},t_{1})}+\tilde{c}_{1}^{1/\alpha}\). By (3.3) and (2.5), we have
Using (3.8) and (3.10) in (3.7), we obtain
where \(c^{*}=\beta c_{2}^{\beta-\alpha}\).
Next,
Therefore,
which contradicts with (3.1).
Case (2). Suppose that (2.2) of Lemma 2.1 holds. Now, for \(v\geq u\geq t_{2}\), we have
Letting \(u=g(t)\) and \(v=a(t)\),
for \(a(t)\geq g(t)\geq t_{2}\), where \(x(t)=(-L_{1}y(t))^{1/\alpha}>0\) for \(t\geq t_{2}\). By (H3) and \(y(g(t))\geq y(g(\sigma(t)))\), we have \(f(t, y(g(t)))\geq k_{2}y^{\beta}(g(t))\). Then from (1.1) and combined with the fact that \(x(t)\) is decreasing, we get
where \(z(t)= x^{\alpha}(t)>0\). Since \(z(t)\) is decreasing and \(\alpha\geq \beta\), there exists a constant \(c_{4}>0\) such that \(z^{\beta/\alpha-1}(t)\geq c_{4}\) for \(t\geq t_{2}\). Then
This gives
then
And \(z(t)\) is an eventually positive solution of inequation (3.14). Integrating \(y(t)=-z^{\Delta}(t)>0\) from \(t_{1}\) to \(t\geq t_{1}\), we obtain
then we have
and (3.14) can be written as
Integrating (3.15) from t to \(u\geq t\geq t_{1}\) and \(u\rightarrow\infty\), we obtain
Now define the sequence \(\{x_{j}(t)\}_{j\in N_{0}}\): \(x_{0}(t)=y(t)\):
Then by (3.16) we get
So we obtain that the sequence \(\{x_{j}(t)\}_{j\in N_{0}}\) is positive and nonincreasing on j. Then we define
By the Lebesgue control convergence theorem [17], from (3.17), we have
then
Let
and
From (3.18) we get
where
So v is a positive solution of (3.2), which contradicts with (3.2) is oscillatory. This completes the proof. □
Theorem 3.2
Assume that the hypotheses of Theorem 3.1 hold, except (3.1). Moreover, suppose that, for all \(t\in I\),
Then every solution \(y(t)\) of (1.1) or \(L_{2}y(t)\) is oscillatory.
Proof
If y is a nonoscillatory solution of (1.1) on \([t_{1},\infty )_{\mathbb{T}}\). Assume that \(y(t)>0\) and \(y(g(t))> 0\) for \(t\geq t_{1}\). By the proof of Lemma 2.1, we have that two cases of Lemma 2.1 hold.
Case (1). Suppose that (2.1) of Lemma 2.1 holds, then proceeding as in the proof of Theorem 3.1, we obtain (3.11), then
Integrating (3.22) from \(t_{2}\) to t, we get
which contradicts with (3.21).
Case (2). The proof of the case if (2.2) of Lemma 2.1 holds is similar to the proof of Theorem 3.1 and hence it is omitted. □
Theorem 3.3
Assume that the hypotheses of Theorem 3.1 hold, except (3.1). Moreover, suppose that, for every \(t_{1}>t_{0}\),
and there exists \(\psi\in C_{rd}(I)\) such that
Then every solution \(y(t)\) of (1.1) or \(L_{2}y(t)\) is oscillatory.
Proof
If y is a nonoscillatory solution of (1.1) on \([t_{1},\infty )_{\mathbb{T}}\). Assume that \(y(t)>0\) and \(y(g(t))> 0\) for \(t\geq t_{1}\). By the proof of Lemma 2.1, we have that two cases of Lemma 2.1 hold.
Case (1). Suppose that (2.1) of Lemma 2.1 holds, then proceeding as in the proof of Theorem 3.1, we obtain (3.12), then
By (IV), we get
which implies that
and
Now, define
It follows from (3.24) that
Suppose that
i.e.,
In fact, let l be an arbitrary positive number. By condition (I) we can take a constant δ with
Since (3.26), there exists \(T_{1}>t_{1}\) such that
Then, for every \(t>t_{1}\), we have
and consequently we have, for \(t\geq T_{1}>t_{1}\),
But
we can choose \(T'_{1}\geq T_{1}>t_{1}\) so that
for every \(t\geq T'_{1}\). Thus
which proves (3.27), since \(l>0\) is arbitrary.
Next, we consider a sequence \((\varphi_{v})_{v=1,2,3,\ldots}\) in the interval \((t_{1},\infty)_{\mathbb{T}}\) with \(\lim_{v\rightarrow\infty}\varphi_{v}=\infty\) and
Since (3.25), there exists a constant M so that
Furthermore, (3.27) guarantees that
Hence (3.29) implies
From (3.29) and (3.30), for sufficiently large v, we derive
Thus
from (3.31),
On the other hand, by Hölder’s inequality [18], for any positive integer v, we have
then
By (3.28), we obtain
therefore
Because of (3.32), we get
Thus
which contradicts with condition (II). We have thus proved that (3.26) fails. So, it holds that
By (3.23) and \(\psi_{+}(s)=\max\{\psi(s),0\}\), we get
which yields a contradiction to condition (III). This completes the proof.
Case (2). The proof of the case if (2.2) of Lemma 2.1 holds is similar to the proof of Theorem 3.1 and hence it is omitted. □
4 Examples
Example 4.1
As an illustrative example, we consider the following equation:
Here \(\mathbb{T}=\mathbb{R}^{+}\), and \(\alpha=\beta=1\), \(r_{1}(t)=1\), \(r_{2}(t)=1\), \(p(t)=\frac{7}{4t^{2}}\), \(\psi(t,x)=x\), \(a(t)=2t\), \(q(t)=t^{-3}\), \(f(t,x)=x\), \(g(t)=\frac {t}{2}\), \(t_{0}=2\). By taking \(m(t)=1\), \(c=k_{1}=k_{2}=1\), \(t_{1}=3\), \(H(t,s)=(t-s)^{4}\), we have \(Q(t)=-\frac{1}{4t^{2}} \). And \(R_{1}(t,2)=R_{2}(t,2)=\int_{2}^{t}1\,ds\rightarrow\infty\) as \(t\rightarrow \infty\), we see that (1.2) and (1.3) are clearly satisfied. By Corollary 1 of [3], we obtain that the equation
is oscillatory. It is easy to check that all hypotheses of Theorem 3.1 are satisfied, so we get that equation (4.1) is oscillatory.
5 Summary
We present some new theorems for the oscillation of (1.1) by using the Riccati transformation, the integral averaging technique, and a new comparison theorem. Our method essentially simplifies the examination of the third order equation and, what is more, our results here extend and complement some of results of Bohner et al. In addition, the next step that can be done is as follows:
-
1.
It would be of interest to consider (1.1) and try to obtain some oscillation criteria if \(p(t)<0\) or \(q(t)<0\).
-
2.
We can consider the dynamic equation with advanced nonlinear term, that is, when \(g(t)>t\) is considered.
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Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
Funding
This research is supported by the Natural Science Foundation of China (61703180), and supported by Shandong Provincial Natural Science Foundation (ZR2016AM17).
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Sui, Y., Sun, S. Oscillation of third order nonlinear damped dynamic equation with mixed arguments on time scales. Adv Differ Equ 2018, 233 (2018). https://doi.org/10.1186/s13662-018-1654-3
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DOI: https://doi.org/10.1186/s13662-018-1654-3
MSC
- 26E70
- 34C10
Keywords
- Time scales
- Oscillation
- Mixed arguments
- Damped