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Nonoscillation for higher-order nonlinear delay dynamic equations on time scales
Advances in Difference Equations volume 2016, Article number: 58 (2016)
Abstract
In this paper, we investigate the nonoscillation of the higher-order nonlinear delay dynamic equation
where \(\mathbb{T}\) is a scale with \(\sup\mathbb{T}=\infty\), \(t_{0}\in\mathbb{T}\), and \([t_{0},\infty)_{\mathbb{T}}= \{t\in\mathbb{T}:t\geq t_{0}\}\). We obtain some sufficient conditions for all solutions of this equation to be nonoscillatory.
1 Introduction
A time scale \(\mathbb{ T}\) is an arbitrary nonempty closed subset of the real numbers. Thus, the set \(\mathbb{R}\) of all real numbers, the set \(\mathbb{N}\) of all natural numbers, and the set \(\mathbb{Z}\) of all integers are examples of time scales. On a time scale \(\mathbb{ T}\), the forward jump operator, the backward jump operator, and the graininess function are defined as
respectively.
In this paper, we investigate the nonoscillation of the higher-order nonlinear delay dynamic equation
where \(t_{0}\in{\mathbb { T}}\), the time scale interval \([t_{0},\infty)_{\mathbb{ T}}\equiv\{t\in\mathbb{ T}:t\geq t_{0}\}\), \(a_{i}\in C_{rd}([t_{0},\infty)_{\mathbb{ T}}, (0,\infty)) \) (\(1\leq i\leq n-1\)), \(u,R\in C_{rd}([t_{0},\infty)_{\mathbb{ T}}, {\mathbb{ R}})\), \(\delta\in C_{rd}([t_{0},\infty)_{\mathbb{T}},\mathbb{T})\) is surjective with \(\delta(t)\leq t\) and \(\delta(t)\rightarrow\infty \) as \(t\rightarrow\infty\), and \(g\in C([t_{0},\infty)_{\mathbb{ T}}\times{\mathbb{R}}, {\mathbb{ R}})\). Our goal is to obtain sufficient conditions for all solutions of (1.1) to be nonoscillatory.
We define
Then (1.1) reduces to the equation
We can suppose the \(\sup{\mathbb{ T}}=\infty\) since we are interested in the oscillatory behavior of solutions near infinity. By a solution of (1.1) we mean a nontrivial real-valued function \(x\in C_{rd}([T_{x},\infty)_{\mathbb{ T}},{\mathbb{ R}})\), \(T_{x}\geq t_{0}\), such that \(R_{n-1}(t,x(t))\in C^{1}_{rd}([T_{x},\infty)_{\mathbb{ T}},{\mathbb{ R}})\) and satisfies (1.1) on \([T_{x},\infty)\). Since we are working on a time scale, the notion of oscillation takes the form of what is known as a generalized zero of a function. We say that \(x(t)\) has a generalized zero at a point T if \(x(T)x(\sigma(T))\leq0\). A function is said to be oscillatory if it has arbitrarily large generalized zeros and nonoscillatory otherwise.
In order to create a theory that can unify discrete and continuous analysis, the theory of time scale was initiated by Hilger’s landmark paper [1], which has received a lot of attention. There exist a variety of interesting time scales, and they give rise to many applications (see [2]). We refer the reader to [3, 4] for further results on time-scale calculus. In the thousands of papers in the literature, finding sufficient conditions for all solutions of an equation to be oscillatory have been a major focus of study (see [5–28]), but finding necessary and sufficient conditions for the existence of a nonoscillatory bounded solution of an equation are more rare (see [29]).
Zhu and Wang [21] studied the existence of nonoscillatory solutions to neutral dynamic equation
Karpuz and Öcalan [22] studied the asymptotic behavior of delay dynamic equations of the form
Wu et al. [25] investigated the oscillation of the higher-order dynamic equation
Sun et al. [26] obtained some necessary and sufficient conditions for the existence of nonoscillatory solution for the higher-order equation
2 Auxiliary results
We state the following conditions, which are needed in the sequel.
- (H1):
-
There exist constants \(\alpha,\beta\geq0\) and \(\gamma\geq0\) such that \(|g(u)|\leq \alpha|u|^{\gamma}+\beta\).
- (H2):
-
\(\int_{t_{0}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})}\int_{t_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots\int_{t_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{t_{0}}^{s_{n-1}}|R(s_{n})|\Delta s_{n}<\infty\).
- (H3):
-
\(\int_{t_{0}}^{\infty}\frac{\Delta s_{1}}{a_{1}(s_{1})}\int_{t_{0}}^{s_{1}}\frac{\Delta s_{2}}{a_{2}(s_{2})}\cdots\int_{t_{0}}^{s_{n-2}}\frac{\Delta s_{n-1}}{a_{n-1}(s_{n-1})} \int_{t_{0}}^{s_{n-1}}|u(s_{n})|\Delta s_{n}<\infty\).
We shall employ the following lemma.
Lemma 2.1
Let \(\mathbb{R}_{+}\equiv[0,\infty)\) and \(H=\{(t,s_{1},s_{2},\ldots,s_{n-1}):0\leq s_{n-1}\leq s_{n-2}\leq\cdots\leq s_{1}\leq t<\infty\}\). Suppose that \({r}\in C_{rd}([t_{0},\infty)_{\mathbb{T}},\mathbb{R}_{+})\), \(h\in C_{rd}(H,\mathbb{R}_{+})\), and that \(p\in C(\mathbb{R}_{+},\mathbb{R}_{+})\) is nondecreasing with \(p(r)>0\) for \(r>0\). If there exists a constant \(c>0 \) such that
then
where
\(P^{-1}\) is the inverse function of P, and
Proof
Let \(z(t)\) denote the right side of inequality (2.1). Then \(z(t_{0})=c\), \({r}(t)\leq z(t)\), and
Since \(z^{\Delta} (t)\geq0\) and p is nondecreasing, we obtain
Noting that
we have
Since \(P(w) \) is increasing, we have
The proof is complete. □
Notice that taking \(p(\nu)=\nu^{\xi}\) and \(\xi>1 \) in Lemma 2.1, we have
So
that is,
We have
provided that
3 Main results
Now, we state and prove our main results.
Theorem 3.1
Assume that conditions (H1)-(H3) hold and for some \(k\geq0\),
If \(x(t)\) is an oscillatory solution of (1.1) such that
then \(x(t)\rightarrow0\) as \(t\rightarrow\infty\).
Proof
We will show \(\limsup_{t\longrightarrow\infty}x(t)=0 \) and \(\liminf_{t\longrightarrow\infty}x(t)=0\). Suppose that \(\limsup_{t\longrightarrow\infty}x(t) =L>0\). Then for any \(t_{1}\geq t_{0}\), there exists \(t_{2}\geq t_{1}\) such that \(x(t_{2})>\frac{L}{2}\). In view of conditions (H2), (H3), (3.1), and (3.2), there exist \(T_{0}\geq t_{0}\) and \(K>0\) such that \(|x(t)|\leq Kt^{k}\) (\(t\geq T_{0}\)) and
Since \(x(t)\) is an oscillatory solution of (1.1), every \(R_{i}{(t,x(t))}\) is oscillatory for \(i=1,2,\ldots,n-1\). Choose \(T_{0}< T_{1}\leq T_{2}\leq\cdots\leq T_{n-1}\) such that
and
Integrating (1.1) from \(T_{i}\) to t, \(i=1,2,\ldots,n-1\), successively \(n-1\) times with \(t>T_{n-1}\), we obtain
Choose \(T_{n}>T_{n-1}\) so that
Take \(T_{n+1}\geq T_{n}\) such that
Note that such \(T_{n+1}\) exists since \(\limsup_{t\longrightarrow\infty}x(t)>\frac{L}{2}\). Dividing (3.6) by \(a_{1}(t)\) and integrating once more from \(T_{n}\) to \(T_{n+1}\), we have
It follows from (H1) that
In view of (3.3), we have a contradiction.
In a similar fashion, we can show that \(\liminf_{t\longrightarrow\infty }x(t)=0\). The proof is complete. □
Theorem 3.2
Assume that conditions (H1)-(H3) hold with \(\gamma\geq1\). Then every oscillatory solution of (1.1) is bounded.
Proof
Let \(x(t)\) be an oscillatory solution of (1.1), and \(d>0\) be a constant.
If \(\gamma>1\), then it follows from conditions (H2) and (H3) that there exists \(T^{*}\geq t_{0}\) such that
and
We will show that eventually for any interval on which \(x(t)\) is positive, we have that \(x(t)\) is bounded by a constant independent of \(x(t)\). Choose \(T^{*}< T_{1}\leq T_{2}\leq\cdots\leq T_{n-1}\leq T_{n}\) so that (3.4)-(3.5) are satisfied, \(\delta(t)>T_{n-1}\) for \(t\geq T_{n}\), and \(x(\delta(T_{n}))x(\delta(\sigma(T_{n})))\leq0\) with \(x(\delta(T_{n}))\leq0\). As in the proof of Theorem 3.1, using (3.8), we have
We can apply Lemma 2.1 with \(c=d\), \(h(s_{1},s_{2},\ldots,s_{n})=\frac{\alpha |u(s_{n})|}{a_{1}(s_{1})a_{2}(s_{2})\cdots a_{n-1}(s_{n-1})}\), \(\xi=\gamma\), and \(p(s)=s^{\gamma}\). From condition (3.9) we have
Thus, (2.4) holds. It follows from Lemma 2.1 that
So \(x(\delta(t))\) is bounded. A similar argument holds for intervals where \(x(t)\) is negative.
If \(\gamma=1\), then choose \(\hat{T}\geq t_{0}\) so that (3.8) holds with \(T^{*}\) replaced by T̂ and
Choose \(T^{*}< T'_{1}\leq T'_{2}\leq\cdots\leq T'_{n-1}\leq T'_{n}\) so that \(R_{n-i}(T'_{i},x(T'_{i}))R_{n-i}(\sigma(T'_{i}),x(\sigma(T'_{i})))\leq 0\) with \(R_{n-i}(T'_{i},x(T'_{i}))\geq0\) for \(1\leq i\leq n\) and \(\delta(t)>T'_{n-1}\) for \(t\geq T'_{n}\). As in the proof of Theorem 3.1, using (3.8), we have
Combining (3.10) with this inequality, we obtain
where \(L=\min\{T_{1},T'_{1}\}\). Denoting by \(z(t)\) the right side of inequality (3.11), we see that \(|x(\delta(t))|\leq z(t)\), \(z(\delta (t))\leq z(t)\), and
which implies \(x(\delta(t))\leq d(\alpha+1)\). The proof is complete. □
After seeing the proof of Theorem 3.2, the proof of the following Theorem 3.3 becomes obvious.
Theorem 3.3
Assume that conditions (H1)-(H3) hold with \(\gamma\geq1\). If (3.1) holds, then every oscillatory solution of (1.1) converges to zero as \(t\rightarrow\infty\).
In a similar fashion as before, we can show the following theorem.
Theorem 3.4
Assume that conditions (H1)-(H3) hold with \(0<\gamma< 1\). If (3.1) holds, then every oscillatory solution of (1.1) is bounded and converges to zero as \(t\rightarrow\infty\).
Proof
Notice that taking \(p(\nu)=\nu^{\xi}\) and \(0<\xi<1\) in Lemma 2.1, we have
So
that is,
We have
Further, the proof is similar to that of Theorem 3.2, so we have
So we can conclude that every oscillatory solution of (1.1) is bounded, and by Theorem 3.1 \(x(t)\) converges to zero as \(t\rightarrow\infty\). The proof is complete. □
Theorem 3.5
Assume that conditions (H1)-(H3) hold with \(g(0)=0\). If there exists \(N>0\) such that for all large T, either
or
then all solutions of (1.1) are nonoscillatory.
Proof
For contradiction, let \(x(t)\) be an oscillatory solution of (1.1). By Theorem 3.3 and Theorem 3.4, \(x(t)\) converges to 0 as \(t\rightarrow\infty\). Hence, there exists \(T_{0}\geq t_{0}\) such that \(|g(x(\delta(t)))|\leq N\) for \(t\geq T_{0}\). From (1.3) we have
If (3.12) holds, then we choose \(T\geq T_{0}\) such that \(\delta(t)\geq T_{0}\) for \(t\geq T\),
and integrating the left inequality in (3.14) from T to t, we obtain
This is a contradiction since if \(x(t)\) is oscillatory, then \(R_{n-2}(t,x(t))\) is also oscillatory.
If (3.13) holds, then we choose T so that the second inequality in (3.15) is reversed. This completes the proof of the theorem. □
4 Example
In this section, we give an example to illustrate our main results.
Lemma 4.1
Assume that \(s,t\in {\mathbb{ T}}\) and \(g\in C_{rd}({\mathbb{ T}}\times{\mathbb{ T}},{\mathbb{ R}})\). Then
Example 4.1
Let \(\mathbb{ T}=\{q^{n}:n\in\mathbb{ Z}\}\cup\{0\}\) with \(q>1\). Consider the higher-order dynamic equation
where \(t\in[q,\infty)_{\mathbb{ T}}\), \(\gamma >0\), \(k\geq0\), \(a_{1}(t)=t^{2+\frac{1}{\gamma}}\), \(a_{i}(t)=t\) (\(2\leq i\leq n-1\)), \(u(t)=\frac{1}{t^{1+k\gamma}+\frac{1}{\gamma}}\), \(\delta(t)=\frac{t}{q}\), \(R(t)=\frac{1}{t^{1+\frac{1}{\gamma}}}\), and \(g(u)=|u|^{\gamma}\operatorname{sgn}(u)\).
It is easy to verify that \(R(t)\) and \(u(t)\) satisfy the condition (3.12). We will use the following inequality: if \(s>t\geq q\), then
Applying Lemma 4.1 and the last inequality, we have
Thus, conditions (H1)-(H3) and (3.1) hold. Then it follows from Theorem 3.5 that every solution \(x(t)\) of (4.1) is nonoscillatory.
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This project is supported by NNSF of China (11461003).
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Tao, C., Sun, T. & He, Q. Nonoscillation for higher-order nonlinear delay dynamic equations on time scales. Adv Differ Equ 2016, 58 (2016). https://doi.org/10.1186/s13662-016-0786-6
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DOI: https://doi.org/10.1186/s13662-016-0786-6
MSC
- 34K11
- 39A10
- 39A99
Keywords
- nonoscillation
- dynamic equation
- time scale