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Properties of thirdorder nonlinear delay dynamic equations with positive and negative coefficients
Advances in Difference Equations volume 2019, Article number: 292 (2019)
Abstract
We study asymptotic properties of a class of thirdorder nonlinear delay dynamic equations with positive and negative coefficients on time scales. In this paper, we establish new results on asymptotic behavior of nonoscillatory and oscillatory solutions for the corresponding equations by making a comparison with suitable firstorder dynamic equations. Finally, we offer an example to illustrate the feasibility of our results.
Introduction
We are interested in studying oscillation and nonoscillation of thirdorder nonlinear delay dynamic equations with positive and negative coefficients
Throughout this paper, we impose the following conditions:

(A1)
\(\mathbb{T}\) is an arbitrary time scale with \(\sup \mathbb{T}=+\infty \), \(b,a,p,q\in C_{rd}(\mathbb{T},\mathbb{R^{+}})\);

(A2)
\(\int _{t_{0}}^{\infty }\frac{1}{b(s)}\Delta s=\int _{t_{0}} ^{\infty }\frac{1}{a(s)}\Delta s=\infty \);

(A3)
\(f,h:\mathbb{R}\to \mathbb{R}\) are continuous functions with \(uf(u)>0\), \(uh(u)>0\) for \(u\neq 0\);

(A4)
\(\tau ,\delta :\mathbb{T}\to \mathbb{T}\) are nondecreasing, \(\tau (t)\leq t\), \(\delta (t)\geq t\), \(\lim_{t\to \infty } \tau (t)=\infty \).
Definition 1.1
A function x is said to be a solution of (1.1) if a nontrivial realvalued function \(x\in C_{rd}([t_{0},\infty )_{\mathbb{T}}, \mathbb{R})\) with \(ax^{\Delta }\in C^{1}_{rd}([t_{0},\infty )_{ \mathbb{T}},\mathbb{R})\) and \(b(ax^{\Delta })^{\Delta }\in C^{1}_{rd}([t _{0},\infty )_{\mathbb{T}},\mathbb{R})\), and such that (1.1) is satisfied on the interval \([t_{0},\infty )_{\mathbb{T}}\).
About the existence and uniqueness of solutions to delay dynamic equations on time scales, we refer to [16]. As usual, a solution of (1.1) is called nonoscillatory if it is either eventually negative or eventually positive; otherwise it is called oscillatory. Equation (1.1) is said to be oscillatory provided that every solution of (1.1) is oscillatory.
The theory of time scales introduced by Hilger (see [15]) unifies continuous and discrete analysis. Oscillations of delay dynamic equations are common in applications, for example, in economics, where the demand depends on current price and the supply depends on the price at an earlier time, and in the study of population dynamic models (see [3]). Compared to dynamic equations with positive coefficients, the research on oscillation and nonoscillation of dynamic equations with positive and negative coefficients has received less attention. What is more, most of the papers focus on the oscillation of secondorder dynamic equations with positive and negative coefficients (see [5, 6, 8, 17, 19,20,21]), much fewer do on thirdorder dynamic equations with positive and negative coefficients. However, for thirdorder delay dynamic equations with positive coefficients, the oscillation theory has been developed well on the binomial equations. For example, when \(q(t)=0\), \(\tau (t)=t\), (1.1) becomes the nonlinear dynamic equation
If \(q(t)=0\), \(b(t)=a(t)=1\), \(\tau (t)=t\), \(f(x(t))=x(t)\), then (1.1) is simplified to the linear dynamic equation
And it is easy to see that (1.1) can be turned into the thirdorder nonlinear delay dynamic equation with positive coefficients
In 2005, Erbe et al. [10] considered (1.2). With the help of the generalized Riccati transformation techniques, they established oscillation criteria which ensure that every solution of (1.2) is oscillatory or converges to zero. In 2006 and 2007, Erbe et al. [11, 12] studied (1.3), and in [12] they obtained Hille and Nehari type oscillation criteria. In 2011, Han et al. [14] extended and improved the results of [10,11,12]. Han et al. offered new oscillatory criteria of (1.4) and demonstrated that every solution of (1.4) oscillates or is convergent to zero.
However, when using the Riccati transformation, we meet an unexpected obstacle: the nonoscillatory solution space of (1.1) is unclear from Lemma 2.1. Agarwal et al. [1] studied oscillation of secondorder dynamic equations and turned to the nonexistence of eventually positive solutions of a firstorder dynamic inequality. Grace et al. [13] and Erbe et al. [9] investigated the oscillation of higher order dynamic equations to make a comparison with secondorder dynamic equations and dynamic inequalities respectively. Jozef et al. [7] also used the comparison method to study the oscillatory properties of (1.1) when \(\mathbb{T}= \mathbb{R}\), and then, in 2017, they [2] continued the investigation to study nonoscillation properties of (1.1).
Motivated by the works [1, 2, 7, 9, 13], we obtain some sufficient conditions which guarantee that Equation (1.1) is oscillatory, and we give certain new results for asymptotic behavior of nonoscillatory solutions of (1.1). Our work generalizes and improves the main results of [2, 7] and those related research works in the literature for oscillation of thirdorder nonlinear delay dynamic equations. Moreover, our results are new for the corresponding thirdorder nonlinear difference equations and differential equations.
For convenience, we denote
The paper is organized as follows. In Sect. 2, we present some basic definitions and useful results from the theory of calculus on time scales and several lemmas. In Sect. 3, we investigate the oscillation of (1.1) by making a comparison with firstorder dynamic equations and establish some sufficient conditions for (1.1) to be oscillatory. In Sect. 4, we give new results for asymptotic behavior of nonoscillatory solutions of (1.1). In Sect. 5, we offer an example to illustrate the feasibility of our primary results.
Preliminaries
We define the graininess function μ:
and for any function \(f:\mathbb{T}\rightarrow \mathbb{R}\) the notation \(f^{\sigma }(t)\) denotes \(f(\sigma (t))\).
The set of functions \(f:\mathbb{T}\to \mathbb{R}\) that are delta differentiable on \([c,d]_{\mathbb{T}}\) and whose delta derivative function is rdcontinuous on \([c,d]_{\mathbb{T}}\) is denoted by \(C^{1}_{rd}\).
A function \(p:\mathbb{T}\to \mathbb{R}\) is said to be regressive if
We denote the set of all \(F:\mathbb{T}\rightarrow \mathbb{R}\) which are rightdense continuous and regressive by ℜ. The exponential function is defined by
where \(p\in \Re \), the cylinder transformation \(\zeta _{h}(z)\) is given by
For more details, see [3].
Equation (1.1) covers the couple of threeorder binomial dynamic equations:
and
We point out that the solution spaces of (2.2) are different from the solution spaces of (2.3). Let S be the set of all positive solutions of the considered equations. Then for (2.2) the set S is the union of \(S_{0}\) and \(S_{2}\):
where the positive solution sets \(S_{0}\) and \(S_{2}\) for (2.2) satisfy
Similarly, for (2.3), the set S is also the union of \(S_{1}\) and \(S_{3}\):
where the positive solution sets \(S_{1}\) and \(S_{3}\) for (2.3) satisfy
Lemma 2.1
Assume that x is an eventually positive solution of (2.2) or (2.3). Then, for (2.2), there exists \(t_{1}\in [t_{0},\infty )_{\mathbb{T}}\) such that, for \(t\in [t_{1},\infty )_{\mathbb{T}}\), either
And, for (2.3), we have also either
The proof is similar to that of [10, Lemma 1].
Lemma 2.2
Assume that \(g\in C_{rd}(\mathbb{T},[0,\infty ))\), \(m:\mathbb{T} \rightarrow \mathbb{R}\) is nondecreasing, \(n:\mathbb{T}\rightarrow \mathbb{T}\) is nonincreasing, and \(j:\mathbb{T}\rightarrow \mathbb{T}\) is nondecreasing. If \(s< t\), then
Proof
Since \(s< t\), we can divide the integral into two parts:
Using the fact that m is nondecreasing, n is nonincreasing, and j is nondecreasing, the first part gives
Since \(g\circ m \circ n \circ j\) is rdcontinuous, by \(\int _{t}^{ \sigma (t)} y(t) \Delta t=\mu (t) y(t)\), the second part yields
Combining (2.4) and (2.5), we get
This completes the proof. □
The oscillation of Equation (1.1)
When we use generalized Riccati transformation techniques, we meet an unexpected obstacle: the nonoscillatory solution space of (1.1) with positive and negative coefficients is unclear from Lemma 2.1. Fortunately, by reducing (1.1) to oscillation of an appropriate firstorder delay dynamic equation, we obtain several new oscillatory properties of Equation (1.1) by comparison method.
In this section, we always assume that

(A5)
h is bounded, f is nondecreasing, and \(f(uv)\geq f(u)f(v)\) for \(uv>0\);

(A6)
\(\int _{t_{0}}^{\infty } \frac{1}{a(t)} \int _{t}^{\infty } \frac{1}{b(s)} \int _{s}^{\infty }q(u)\Delta u\Delta s \Delta t<\infty \).
Hypothesis (A6) reveals that the positive term of (1.1) is dominating. So the structure of the positive solution space of (1.1) is similar to that of (2.2).
Theorem 3.1
Assume that either
or
If the firstorder nonlinear delay dynamic equation
oscillates, then every solution x of (1.1) either is oscillatory or \(\lim_{t\rightarrow \infty }x(t)=0\).
Proof
Suppose to the contrary that \(x(t)\) is an eventually positive solution of (1.1). Then there exists \(t_{1}\in \mathbb{T}\) large enough and \(t_{1}\geq t_{0}\) such that
Let
Conditions (A5) and (A6) imply that the function \(k(t)\) exists for all \(t\geq t_{1}\) and is well defined. Moreover,
By (3.4) and (1.1), we find that
which implies that
Hence, a suitable modification of Lemma 2.1 together with condition (A2) implies that either
or
for all \(t\geq t_{1}\). Firstly, we consider the case \(k(t)\in S_{0}\). Since \(k(t)>0\) and \(k^{\Delta }(t)<0\), by (3.4), we find
We claim that \(l=0\). Otherwise, \(l>0\), then \(x(\tau (t))\geq l>0\) holds for all \(t\geq t_{1}\).
If (3.1) holds, noting that f is nondecreasing, integrate (3.5) from \(t_{1}\) to t (\(t\in [t_{1},\infty )_{\mathbb{T}}\))
which contradicts \(b(t) ( a(t) k^{\Delta }(t) )^{\Delta }>0\).
On the other hand, if (3.2) holds, similar to the proof of [7, Theorem 2.1], we find the limit \(\lim_{t\to \infty }k(t)=\infty \), which contradicts the fact \(k(t)>0\). So,
Now we consider \(k(t)\in S_{2}\). By \([b(t) (a(t)k^{\Delta }(t) ) ^{\Delta }]^{\Delta }<0\), we get two inequalities:
and
The last inequality together with \(x(t)>0\) and \(k^{\Delta }(t)< x^{ \Delta }(t)\) shows that
Let
By (3.6) and \(z_{1}(t)=b(t) (a(t)k^{\Delta }(t) ) ^{\Delta }>0\), we have
According to (3.5), (A5), and the definition of \(A(t)\), we see that \(z_{1}>0\) is a solution of the delay dynamic inequality
By [4, Theorem 3.1], it follows that the corresponding firstorder delay dynamic equation (3.3) has a positive solution, too. This contradiction shows that \(k(t)\notin S_{2}\). This completes the proof. □
Remark 3.2
Note that (A2) implies that
Then there exists \(t_{2}\geq t_{1}\) such that, for \(t \geq t_{2} > t _{1}\),
Set it into (3.5),
Thus, we can remove the condition \(f(uv) \geq f(u)f(v)\) for \(uv>0\) in (A5) and (3.3) can be replaced by
to make sure that the conclusion of Theorem 3.1 is also true. However, [7, Theorem 2.1] did not give the corresponding result. When we use oscillation criterion (3.9) of Corollary 3.3, we also need not the condition \(f(uv) \geq f(u)f(v)\) for \(uv>0\) in (A5).
Corollary 3.3
or
then every solution x of (1.1) either oscillates or \(\lim_{t\to \infty }x(t)=0\).
Proof
Due to Theorem 3.1 and 3.2, we only need to show that (3.3) and (3.7) oscillate.
First, assume that Equation (3.3) has a positive solution \(z(t)\). Since \(\lim_{t\to \infty }\tau (t)=\infty \), there is a positive number \(t_{1}\geq t_{0}\) such that \(z(\tau (t))>0\) for \(t\geq t_{1}\). By (3.3),
which implies the limit \(\lim_{t\to \infty }z(t)=r\geq 0\).
Integrating (3.10) from \(\tau (t)\) to \(\sigma (t)\) yields
Since z is Δdifferentiable and rdcontinuous and the function τ is nondecreasing, Lemma 2.2 is applicable for the integral term in the previous equation. By Lemma 2.2, we get
so
Note that \(z(t)>0\), \(p(t)>0\), \(zf(z)>0\) for all \(t\geq t_{1}\), we have
Taking upper limits on both sides brings an obvious contradiction with (3.8).
Second, assume that Equation (3.7) has a positive solution \(z_{2}(t)\). The proof is the same as the previous case and we omit it. □
Let \(f(u)= u^{\beta 1}u\). We have the following result.
Corollary 3.4
Let \(0<\beta <1\). Assume that (3.1) or (3.2) holds and
then every solution x of the nonlinear dynamic equation
either oscillates or \(\lim_{t\to \infty }x(t)=0\).
Remark 3.5
When (3.2) holds, if (3.11) is replaced by
the conclusion of Corollary 3.4 is also true. What is more, if \(\beta =1\)
then the conclusion of Corollary 3.4 still holds.
Compared with the following Corollary 3.6, we can see that (3.15) is better than (3.14). But it is easier to compute (3.14) than (3.15). When \(\mathbb{T}=\mathbb{R}\), [22, Example 2] shows that [7, Corollary 2.2] is a special case of Corollary 3.6.
Corollary 3.6
where \(E=\{\lambda \lambda >0,1\lambda p(t)A(\tau (t))\mu (t)>0, t \in \mathbb{T}\}\), then every solution x of the delay dynamic equation with positive and negative coefficients
either oscillates or \(\lim_{t\to \infty }x(t)=0\).
Proof
A modification of [22, Corollary 2] guarantees (3.3) to be oscillatory with \(f(u)=u\). The conclusion of Corollary 3.6 is correct owing to Theorem 3.1. □
On the other hand, using the supplementary condition, we will establish another oscillation criterion for (1.1) to make sure that (1.1) is oscillatory.
Theorem 3.7
Assume that there exists a function \(\xi \in C_{rd}^{1}([t_{0}, \infty )_{\mathbb{T}}, \mathbb{R})\) such that
If both the firstorder delay dynamic equation (3.3) and
are oscillatory, then (1.1) is oscillatory.
Proof
Assume not. Let x be an eventually positive solution of (1.1). The function \(k(t)\) is defined by (3.4). As in the proof of Theorem 3.1, we see that \(k(t)\in S_{0}\), and there exists \(\lim_{t\to \infty }k(t)=2l\geq 0\). Assumption (A2) implies
In view of (3.4), it follows that
Let
Conditions (A5) and (A6) imply that \(w(t)\) is well defined. By (A1), (A3), (1.1), and (3.20), we have
and
Then, hypothesis (A2) combined with a proper modification of Lemma 2.1 implies that either
or
eventually. However, \(w(t)\in S_{2}\) will not happen. In fact, if \(w(t)\in S_{2}\), then by \(0< a(t)w^{\Delta }(t)< a(t)x^{\Delta }(t)\) and \(\lim_{t\rightarrow \infty }a(t)x^{\Delta }(t)=0\), we get \(\lim_{t\rightarrow \infty }a(t)w^{\Delta }(t)=0\). We have a contradiction with \(a(t)w^{\Delta }(t)>0\) and \((a(t)w^{\Delta }(t) ) ^{\Delta }>0\), eventually. So, we claim that \(w(t)\in S_{0}\).
Let \(\psi (t)= b(t) (a(t)w^{\Delta }(t) )^{\Delta } \), then \(\psi ^{\Delta }(t)= ( b(t) (a(t)w^{\Delta }(t) )^{ \Delta } )^{\Delta }<0\). By (3.17) and (3.19), integrate \((a(t)w^{\Delta }(t) )^{\Delta }< (a(t)x ^{\Delta }(t) )^{\Delta }\) from t to ∞,
which implies \(x^{\Delta }(t) \geq \psi ( \xi (t) ) \frac{1}{a(t)} \int _{t}^{\xi (t)}\frac{1}{b(s)} \Delta s\). We carry out the calculation
Substituting it into (3.21), we find that the delay dynamic inequality
has a solution \(\psi >0\). Owing to [4, Theorem 3.1], the corresponding firstorder dynamic equation (3.18) has a positive solution as well. The contradiction shows that \(w(t)\notin S_{0}\) and we reach an agreement that (1.1) oscillates. □
It is evident that Corollaries 3.8, 3.9, 3.10 are true.
Corollary 3.8
Assume that (3.8) and (3.17) hold. If
then (1.1) oscillates.
Corollary 3.9
Assume that (3.15) and (3.17) hold. If
where \(E^{\prime }=\{\lambda \lambda >0,1\lambda p(t)B(t)\mu (t)>0, t \in \mathbb{T}\}\), then (3.16) oscillates.
Corollary 3.10
Assume that (3.11) and (3.17) hold. If
then (3.12) oscillates.
Remark 3.11
When \(\mathbb{T}=\mathbb{R}\), Theorem 3.7 and Corollaries 3.8, 3.9, 3.10 reduce to Theorem 2.6 and Corollaries 2.8–2.10 in [7].
The asymptotic properties of Equation (1.1)
Now, we explore the asymptotic behavior of solutions for (1.1) by reducing the exploration of (1.1) to oscillation of a proper firstorder dynamic equation.
In this section, we need the following set of hypotheses.

(A7)
f is bounded, h is nondecreasing;

(A8)
\(\int _{t_{0}}^{\infty } \frac{1}{a(t)} \int _{t}^{\infty } \frac{1}{b(s)} \int _{s}^{\infty }p(u) \Delta u \Delta s \Delta t < \infty \).
Theorem 4.1
Suppose that
holds for every \(c>0\). If the firstorder nonlinear delay dynamic equation
is oscillatory, then every nonoscillatory solution x of (1.1) either
or, for some \(M>0\),
Proof
Suppose to contrary that (1.1) has a nonoscillatory solution \(x>0\). We may assume that there exists \(t_{1}\geq t_{0}\) such that
Let
According to (A7) and (A8), \(r(t)\) exists for all \(t\in \mathbb{T}\) and is well defined. Furthermore, \(r(t)< x(t)\), and
Hence, by (A2), a proper modification of Lemma 2.1 implies that, for all \(t\geq t_{1}\), either
or
or
The rest of the proof is similar to that of [2, Theorem 1]. □
The function \(r\in S_{3}\) implies
By direct calculation, we have
Hence, if we remove condition (4.1), we conclude that every nonoscillatory solution x of (1.1) either is unbounded or converges to zero. The following result improves [2, Theorem 1].
Remark 4.2
If (4.2) is oscillatory, then every nonoscillatory solution x of (1.1) either, for sufficiently large \(T \in \mathbb{T}\) and some \(c>0\), is such that
or satisfies (4.4).
Furthermore, applying extra condition, we obtain the other new criterion that every nonoscillatory solution of (1.1) is asymptotic stable.
Theorem 4.3
Assume that there exists a function \(\chi \in C_{rd}^{1}([t_{0}, \infty )_{\mathbb{T}}, \mathbb{R})\) such that
If both the firstorder delay dynamic equation (4.2) and
are oscillatory, then every nonoscillatory solution of (1.1) satisfies (4.4).
Proof
The proof is similar to [2, Theorem 2]. □
Remark 4.4
When \(\mathbb{T}=\mathbb{R}\), Theorems 4.1 and 4.3 reduce to Theorems 1–2 in [2].
Application
In this section, we present an example to illustrate the feasibility of our results.
Example 5.1
Consider the following thirdorder delay dynamic equation with positive and negative coefficients:
for \(t\in [t_{0},\infty )_{\mathbb{T}}\), where \(b, a, p, q\in C_{rd}([1, \infty )_{\mathbb{T}},\mathbb{R^{+}})\), \(\tau (t)\leq t\), \(\delta (t) \geq t\), \(\lim_{t\to \infty }\tau (t) =\infty \). Now \(f(u)=u\), \(h(u)=\frac{x(\delta (t))}{2+x(\delta (t))^{3}}\) is bounded, so conditions (A1), (A3)–(A4) are satisfied. Then we can choose a, b satisfying (A2) and choose p, q satisfying (A6) and (3.1).
(I) \(\mathbb{T}=\mathbb{R}\). Taking account of Corollary 3.6, (3.15) can be written as follows:
where \(E=\mathbb{R^{+}}\). Let
It follows from Lagrange theorem that
Therefore, if
then the differential equation
is oscillatory. That is just a modification of [18, Theorem 1].
(II) \(\mathbb{T}=h\mathbb{Z}=\{hkk\in \mathbb{Z}\}\) for \(h>0\). Assume that \(p(t)A(\tau (t))=1\), \(\tau (t)=tlh\) with \(l\in Z^{+}\), and from (3.15) we get
where \(E=\{\lambda 0<\lambda <\frac{1}{h}\}\). By simply calculating, (5.4) turns out to be
So, if we pick up some \(h>0\) such that (5.5) holds, then the equation
oscillates.
If we apply Remark 3.5,
implies that \(\frac{ 1 }{ k+1 }< h\) guarantees Equation (5.6) to oscillate. Note that
this reveals that (3.15) is better than (3.14). However, it is convenient to compute (3.14), but not to compute (3.15).
(III) \(\mathbb{T}=\{t_{n}=\sum_{k=1}^{n}\frac{1}{k}: n\in \mathbb{N}\}\). Assume that \(p(t)A(\tau (t))=\alpha \), where α is an integer. By (3.15), we see that, if
where \(E=\{\lambda 0<\lambda <\frac{n}{\alpha }\}\), then the equation
is oscillatory.
By Corollary 3.6, on time scales \(\mathbb{T}=\mathbb{R}\), \(\mathbb{T}=h\mathbb{Z}=\{hkk\in \mathbb{Z}\}\) for \(h>0\), and \(\mathbb{T}=\{\sum_{k=1}^{n}\frac{1}{k}, k\in \mathbb{N}\}\), then every solution x of (5.1) oscillates or
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Acknowledgements
The authors would like to thank Professors Yu Huang and QiRu Wang for their helpful discussions.
Authors’ information
Xianyong Huang, Associate Professor, the major field of interest is in the area of ordinary differential equations and dynamical systems.
Funding
This project is partially supported by the NSFC of China (Grant Nos. 11271379, 11671406, 11801092, 11801093), Construction Project of Teaching Quality and Teaching Reform in Guangdong Undergraduate Colleges and Universities in 2018 (Speciality of Financial Mathematics), the Distinctive Innovation of Ordinary University of Guangdong Province (Grant No. 2016KTSCX093), the Funds of Guangzhou Science and Technology (Grant No. 201804010088), and the NSF of Guangdong University of Education (Grant Nos. 2016sjjxjd04, 2018sfzy01).
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Correspondence to XunHuan Deng.
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MSC
 34N05
 34C10
Keywords
 Nonlinear delay dynamic equations
 Comparison method
 Positive and negative coefficients
 Oscillation
 Asymptotic behavior