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Asymptotics and oscillation of higher-order functional dynamic equations with Laplacian and deviating arguments
Advances in Difference Equations volume 2017, Article number: 14 (2017)
Abstract
In this paper, we deal with the asymptotics and oscillation of the solutions of higher-order nonlinear dynamic equations with Laplacian and mixed nonlinearities of the form
on an above-unbounded time scale. By using a generalized Riccati transformation and integral averaging technique we study asymptotic behavior and derive some new oscillation criteria for the cases without any restrictions on \(g(t)\) and \(\sigma(t)\) and when n is even and odd. Our results obtained here extend and improve the results of Chen and Qu (J. Appl. Math. Comput. 44(1-2):357-377, 2014) and Zhang et al. (Appl. Math. Comput. 275:324-334, 2016).
1 Introduction
We are concerned with the asymptotic and oscillatory behavior of the higher-order nonlinear functional dynamic equation
on an above-unbounded time scale \({\mathbb{T}}\), assuming without loss of generality that \(t_{0}\in{\mathbb{T}}\). For \(A\subset {\mathbb{T}}\) and \(B\subset{\mathbb{R}}\), we denote by \(C_{\mathrm{rd}}(A,B)\) the space of right-dense continuous functions from A to B and by \(C_{\mathrm{rd}}^{1}(A,B)\) the set of functions in \(C_{\mathrm{rd}}(A,B)\) with right-dense continuous Δ-derivatives. We refer the readers to the books by Bohner and Peterson [3, 4] for an excellent introduction of calculus of time scales. Throughout this paper, we suppose that:
-
(i)
\(n,N\in\mathbb{N}\), \(n\geq2\), and \(\phi_{\beta }(u):=\vert u\vert ^{\beta-1}u\), \(\beta>0\);
-
(ii)
\(r_{i}\in C_{\mathrm{rd}} ( [ {t}_{0},\infty ) _{\mathbb{T}},(0,\infty) ) \) for \(i=1,2,\ldots,n-1\) are such that
$$ \int_{{t}_{0}}^{\infty}r_{i}^{-1/\alpha_{i}}(\tau) \Delta\tau =\infty; $$(1.2) -
(iii)
\(\alpha_{i}>0\), \(i=1,2,\ldots,n-1\), and \(\gamma_{\nu}>0\), \(\nu =0,1,\ldots,N\), are constants such that
$$ \gamma_{\nu}>\gamma_{0},\quad \nu=1,2,\ldots,l\quad \text{and} \quad \gamma _{\nu }< \gamma_{0},\quad \nu=l+1,l+2, \ldots,N; $$(1.3) -
(iv)
\(p_{\nu}\in C_{\mathrm{rd}} ( [ t_{0},\infty ) _{\mathbb{T}},[0,\infty) ) \), \(\nu=0,1,\ldots,N\), are such that not all of the \(p_{\nu } ( t ) \) vanish in a neighborhood of infinity;
-
(v)
\(g_{\nu}:\mathbb{T\rightarrow T}\) are rd-continuous functions such that \(\lim_{t\rightarrow\infty}g_{\nu }(t)=\infty\), \(\nu=0,1,\ldots,N\).
By a solution of equation (1.1) we mean a function \(x\in C_{\mathrm{rd}}^{1}([T_{x},\infty)_{\mathbb{T}},{\mathbb{R}})\) for some \(T_{x}\geq{0}\) such that \(x^{[i]}\in C_{\mathrm{rd}}^{1}([T_{x},\infty)_{\mathbb {T}},{\mathbb{R}}), i=1,2,\ldots,n-1\), that satisfies equation (1.1) on \([T_{x},\infty)_{\mathbb{T}}\), where
A solution \(x(t)\) of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is nonoscillatory.
Oscillation criteria for higher-order dynamic equations on time scales have been studied by many authors. For instance, Grace et al. [5] obtained sufficient conditions for oscillation for the higher-order nonlinear dynamic equation
where γ is the quotient of positive odd integers, and where \(g(t)\leq t\). In [5], some comparison criteria have been studied when \(g(t)\leq t\), and some oscillation criteria are given when n is even and \(g ( t ) =t\). The results in [5] have been proved when
Wu et al. [6] established Kamanev-type oscillation criteria for the higher-order nonlinear dynamic equation
where α is the quotient of positive odd integers, \(g:\mathbb{T} \rightarrow\mathbb{T}\) with \(g(t)>t\) and \(\lim_{t\rightarrow\infty }g(t)=\infty\), and there exists a positive rd-continuous function \(p(t)\) such that \(\frac{f(t,u)}{u^{\alpha}}\geq p(t)\) for \(u\neq0\). Sun et al. [7] proved some criteria for oscillation and asymptotic behavior of the dynamic equation
where \(\alpha\geq1\) is the quotient of positive odd integers, \(g:\mathbb{T}\rightarrow\mathbb{T}\) is an increasing differentiable function with \(g(t)\leq t\), \(g\circ\sigma=\sigma\circ g\), and \(\lim_{t\rightarrow \infty}g(t)=\infty\), and there exists a positive rd-continuous function \(p(t)\) such that \(\frac{f(t,u)}{u^{\beta}}\geq p(t)\) for \(u\neq0\) and \(\beta \geq1\) is the quotient of positive odd integers. Sun et al. [8] studied quasilinear dynamic equations of the form
where α, β are the quotients of positive odd integers. Also, the results obtained in [6–8] are presented when
Hassan and Kong [9] obtained asymptotics and oscillation criteria for the nth-order half-linear dynamic equation
where \({\alpha}{[1,n-1]}:={\alpha}_{1}\cdots{\alpha}_{n-1}\), and Grace and Hassan [10] further studied the asymptotics and oscillation for the higher-order nonlinear dynamic equation
However, the establishment of the results in [10] requires the restriction on the time scale \(\mathbb{T}\) that \(g^{\ast}\circ\sigma =\sigma\circ g^{\ast}\) with \(g^{\ast}(t)=\min\{t,g(t)\}\), which is hardly satisfied. Hassan [11] improved the results in [9, 10] and established oscillation criteria for the higher-order quasilinear dynamic equation
when n is even or odd and when \(\alpha>\gamma\), \(\alpha=\gamma\), and \(\alpha<\gamma\) with \(\alpha=\alpha_{1}\cdots\alpha_{n-1}\). Chen and Qu [1] considered the even-order advanced type dynamic equation with mixed nonlinearities
where \(n\geq2\) is even, \(\gamma_{\nu}>0\), \(g_{\nu}(t)\geq t\), and \(\gamma _{1}>\cdots>\gamma_{l}>\gamma_{0}>\gamma_{l+1}>\cdots>\gamma_{N}>0\). Zhang et al. [2] studied the dynamic equation (1.7), where \(n\geq2\) is integer and \(g_{\nu}^{\Delta}(t)>0\), and obtained some of the results in [2] when \(\gamma_{0}\geq1\). Also, the results obtained in [1, 2] are given when
Huang [12] extended the work in [1] to the neutral advanced dynamic equation
where \(n\geq2\) is integer, \(y(t):=x(t)+p(t)x ( g(t) ) \), \(\gamma _{\nu}>0\), \(g(t)\leq t\), and \(g_{\nu}(t)\geq t\). For more results on dynamic equations, we refer the reader to the papers [13–29].
In this paper, we will discuss the higher-order nonlinear dynamic equation (1.1) with mixed nonlinearities on a general time scale without any restrictions on \(g(t)\) and \(\sigma(t)\) and also without conditions (1.5), (1.6), and (1.8). The results in this paper improve the results in [1, 2, 5–10] on the oscillation of various dynamic equations.
2 Main results
We introduce the following notations:
and
with \(\alpha=\gamma_{0}=\alpha[1,n-1]\) and \(\beta_{i}=\alpha [1,i]\). For any \(t,s\in{\mathbb{T}}\) and for a fixed \(m\in \{0,1,\dots,n-1\}\), define the functions \(R_{m,j}(t,s)\), \(j=0,1,\ldots,m\), and \(\hat{p}_{j}(t)\), \(j=0,1,\ldots,n-1\), by the following recurrence formulas:
and
For a fixed \(m\in\{0,\ldots,n-1\}\), define the functions \(\bar{p}_{m,j}(t,s)\), \(j=0,1,2,\ldots,n-1\), by the recurrence formula
with
and
such that
where
provided that the improper integrals involved are convergent.
In the sequel, we present conditions that guarantee the following conclusions:
- (C):
Theorem 2.1
Let conditions (i)-(v) hold. Furthermore, for each \(i\in \{1,2,\ldots,n-1\}\) and sufficiently large \(T,T_{1}\in[ t_{0},\infty){_{\mathbb{T}}}\), one of the following conditions is satisfied:
-
(a)
either \(\int_{T}^{\infty}\bar{p}_{i,n-i-1}(\tau ,T_{1})\Delta \tau=\infty\), or \(\int_{T}^{\infty}\bar{p}_{i,n-i-1}(\tau ,T_{1})\Delta \tau<\infty\) and either
$$ \limsup_{t\rightarrow\infty}R_{i,i}^{\beta_{i}}(t,T_{1}) \int _{t}^{\infty }\bar{p}_{i,n-i-1}( \tau,T_{1})\Delta\tau>1 $$or
$$ \limsup_{t\rightarrow\infty}R_{i,i}(t,T_{1}) \biggl( \int _{t}^{\infty}\bar{p}_{i,n-i-1}( \tau,T_{1})\Delta\tau \biggr) ^{1/\beta_{i}}>1; $$ -
(b)
there exists \(\rho_{i}\in C_{\mathrm{rd}}^{1}([t_{0},\infty )_{{\mathbb{T}}},(0,\infty))\) such that
$$ \limsup_{t\rightarrow\infty} \int_{T}^{t} \biggl[\rho_{i}(\tau) \bar {p}_{i,n-i-1}(\tau,T_{1})-\frac{ ( \rho_{i}^{\Delta}(\tau) ) _{+}}{R_{i,i}^{\beta_{i}}(\sigma(\tau),T_{1})} \biggr]\Delta\tau=\infty; $$(2.5) -
(c)
there exists \(\rho_{i}\in C_{\mathrm{rd}}^{1}([t_{0},\infty )_{{\mathbb{T}}},(0,\infty))\) such that
$$\begin{aligned}& \limsup_{t\rightarrow\infty} \int_{T}^{t} \biggl[\rho _{i}(\tau) \bar{p}_{i,n-i-1}(\tau,T_{1}) \\& \quad {}-\frac{1}{\rho_{i}^{\beta_{i}}(\tau)} \biggl[ \frac {(\rho _{i}^{\Delta}(\tau))_{+}}{1+\beta_{i}} \biggr] ^{1+\beta_{i}} \biggl[ \frac{r_{1}(\tau)}{R_{i,i-1}(\tau,T_{1})} \biggr] ^{\beta_{i}/\alpha_{1}} \biggr]\Delta\tau= \infty; \end{aligned}$$(2.6) -
(d)
there exist \(\rho_{i}\in C_{\mathrm{rd}}^{1}([t_{0},\infty )_{{\mathbb{T}}},(0,\infty))\) and \(H_{i},h_{i}\in C_{\mathrm{rd}} ( \mathbb {D},\mathbb{\mathbb{R}} ) \), where \(\mathbb{D}\equiv\{ ( t,\tau ) :t\geq \tau\geq t_{0}\}\), such that
$$ H_{i} ( t,t ) =0,\quad t\geq t_{0},\qquad H_{i} ( t,\tau ) >0, \quad t>\tau\geq t_{0}, $$(2.7)and \(H_{i}\) has a nonpositive continuous Δ-partial derivative \(H_{i}^{\Delta_{\tau}} ( t,\tau ) \) with respect to the second variable and satisfies
$$ H_{i}^{\Delta_{\tau}} ( t,\tau ) +H_{i} ( t,\tau ) \frac{\rho_{i}^{\Delta}(\tau)}{\rho_{i}^{\sigma} ( \tau ) }=-\frac{h_{i} ( t,\tau ) }{\rho_{i}^{\sigma} ( \tau ) }H_{i}^{\beta_{i}/ ( 1+\beta_{i} ) } ( t,\tau ) $$(2.8)and
$$\begin{aligned} \begin{aligned}[b] &\limsup_{t\rightarrow\infty}\frac{1}{H_{i} ( t,T ) }\int_{T}^{t} \biggl[\rho_{i}(\tau) \bar{p}_{i,n-i-1}(\tau ,T_{1})H_{i} ( t,\tau ) \\ &\quad {}-\frac{1}{\rho_{i}^{\beta_{i}}(\tau)} \biggl[ \frac { ( h_{i} ( t,\tau ) ) _{-}}{1+\beta_{i}} \biggr] ^{1+\beta_{i}} \biggl[ \frac{r_{1}(\tau)}{R_{i,i-1}(\tau,T_{1})} \biggr] ^{\beta _{i}/\alpha_{1}} \biggr] \Delta\tau=\infty; \end{aligned} \end{aligned}$$(2.9) -
(e)
there exists \(\rho_{i}\in C_{\mathrm{rd}}^{1}([t_{0},\infty )_{{\mathbb{T}}},(0,\infty))\) such that
$$\begin{aligned}& \limsup_{t\rightarrow\infty} \int_{T}^{t} \biggl[\rho_{i}(\tau) \bar {p}_{i,n-i-1}(\tau,T_{1}) \\& \quad {}-\frac{ ( \rho_{i}^{\Delta}(\tau) ) ^{2}}{4\beta_{i}\rho_{i}(\tau)\delta^{\sigma}(\tau,T_{1})} \biggl[ \frac {r_{1}(\tau)}{R_{i,i-1}(\tau,T_{1})} \biggr] ^{1/\alpha_{1}} \biggr]\Delta \tau=\infty; \end{aligned}$$(2.10) -
(f)
there exist \(\rho_{i}\in C_{\mathrm{rd}}^{1}([t_{0},\infty )_{{\mathbb{T}}},(0,\infty))\) and \(H_{i},h_{i}\in C_{\mathrm{rd}} ( \mathbb {D},\mathbb{\mathbb{R}} ) \), where \(\mathbb{D}\equiv\{ ( t,\tau ) :t\geq \tau\geq t_{0}\}\), such that (2.7) holds and \(H_{i}\) has a nonpositive continuous Δ-partial derivative \(H_{i}^{\Delta_{\tau}} ( t,\tau ) \) with respect to the second variable and satisfies
$$ H_{i}^{\Delta_{\tau}} ( t,\tau ) +H_{i} ( t,\tau ) \frac{\rho_{i}^{\Delta}(\tau)}{\rho_{i}^{\sigma} ( \tau ) }=-\frac{h_{i} ( t,\tau ) }{\rho_{i}^{\sigma} ( \tau ) }\sqrt{H_{i} ( t,\tau ) } $$(2.11)and
$$\begin{aligned}& \limsup_{t\rightarrow\infty}\frac{1}{H_{i} ( t,T ) }\int_{T}^{t} \biggl[\rho_{i}(\tau) \bar{p}_{i,n-i-1}(\tau ,T_{1})H_{i} ( t,\tau ) \\& \quad {}-\frac{ [ ( h_{i} ( t,\tau ) ) _{-} ] ^{2}}{4\beta_{i}\rho _{i}(\tau ) \delta^{\sigma}(\tau,T_{1})} \biggl[ \frac{r_{1}(\tau)}{R_{i,i-1}(\tau,T_{1})} \biggr] ^{1/\alpha_{1}} \biggr]\Delta\tau=\infty. \end{aligned}$$(2.12)
Moreover, for the case where n is odd, assume that, for an integer \(j\in \{0,1,\ldots,n-1 \} \),
Then conclusions (C) hold.
Example 2.1
Consider the higher-order nonlinear dynamic equation (1.1), where \(\beta _{i}=\alpha[1,i]\leq1\) and \(r_{1}(t):=\frac{t^{\xi}}{\beta_{1}}\) with
and where
Choose an n-tuple \(( \eta_{1},\eta_{2},\ldots,\eta_{n} ) \) with \(0<\eta_{j}<1\) satisfying (2.4). It is clear that conditions (1.2) hold since
by [3], Example 5.60. By the Pötzsche chain rule we get
Also, since (1.2) implies \(\lim_{t\rightarrow\infty}\frac {\varphi _{i,\nu} ( t,T_{1} ) }{\varphi_{i,\nu} ( t,t_{0} ) }=1\), we obtain
It is easy to see that
Therefore, we can find \(T_{\ast}\geq T\geq T_{1}\) such that \(R_{i,i-1}(t,T_{1})\geq1\) for \(t\geq T_{\ast}\). Let us take \(\rho _{i}(t)=t^{\beta_{i}}\). Then, by the Pötzsche chain rule,
Hence,
if
and hence (2.6) holds. Also,
If n is odd, then
so that condition (2.13) holds. Then, by Theorem 2.1(c) conclusions (C) hold if
3 Lemmas
In order to prove the main results, we need the following lemmas. The first two lemmas are extensions of Lemmas 1 and 2 in [9] to the nonlinear equation (1.1) with exactly the same proof.
Lemma 3.1
Let \(x(t)\in C_{\mathrm{rd}}^{n} ( \mathbb{T},[0,\infty ) ) \). Assume that \((x^{ [ n-1 ] })^{\Delta} ( t ) \) is of eventually one sign and not identically zero. Then there exists an integer \(m\in\{0,1,\ldots,n-1\}\) with \(m+n\) odd for \((x^{ [ n-1 ] })^{\Delta} ( t ) \leq0\) or with \(m+n\) even for \((x^{ [ n-1 ] })^{\Delta} ( t ) \geq0\) such that
and
eventually.
Lemma 3.2
Assume that equation (1.1) has an eventually positive solution \(x(t)\) and \(m\in\{0,1,\ldots,n-1\}\) is given in Lemma 3.1 such that (3.1) and (3.2) hold for \(t\in[ t_{1},\infty )_{\mathbb{T}}\) for some \(t_{1}\in[{t}_{0},\infty)_{\mathbb{T}}\). Then the following hold for \(t\in(t_{1},\infty)_{{\mathbb{T}}}\):
-
(a)
for \(i=0,1,\ldots,m\),
$$ \frac{x^{ [ m-i ] }(t)}{R_{m,i}(t,t_{1})}\quad \textit{is strictly decreasing}; $$(3.3) -
(b)
for \(i\in \{ 0,1,\ldots,m \} \) and \(j=0,1,\ldots,m-i\),
$$ x^{ [ j ] }(t)\geq\phi_{\alpha [ j+1,m-i ] }^{-1} \biggl[ \frac{x^{ [ m-i ] } ( t ) }{R_{m,i}(t,t_{1})} \biggr] R_{m,m-j}(t,t_{1}). $$(3.4)
Lemma 3.3
Assume that equation (1.1) has an eventually positive solution \(x(t)\) and m is given in Lemma 3.1 such that \(m\in\{1,2,\ldots,n-1\}\) and (3.1) and (3.2) hold for \(t\geq t_{1}\in[ t_{0},\infty)_{\mathbb{T}}\). Then, for \(t\in[ t_{2},\infty)_{{\mathbb{T}}}\), where \(g_{\nu}(t)>t_{1}\) for \(t\geq t_{2}\), and for \(j=m,m+1,\ldots,n-1\),
and
Proof
We show it by a backward induction. By Lemma 3.1 with \(m\geq1\) we see that \(x(t)\) is strictly increasing on \([t_{1},\infty)_{{\mathbb {T}}}\). As a result, (3.1) and (3.2) hold for \(t\in[ t_{1},\infty)_{{\mathbb{T}}}\). Let \(t\in [t_{1},\infty)_{{\mathbb{T}}}\) be fixed. Then, for \(\nu=0,1,\ldots ,N\), if \(g_{\nu}(t)\geq\sigma ( t ) \), then \(x(g_{\nu}(t))\geq x(t)\) by the fact that \(x(t)\) is strictly increasing. Now consider the case where \(g_{\nu}(t)\leq\sigma ( t ) \). In view of Lemma 3.2(a), we see that for \(i=m\), \(\frac{x(t)}{R_{m,m}(t,t_{1})}\) is decreasing on \((t_{1},\infty)_{{\mathbb{T}}}\) and that there exists \(t_{2}\geq t_{1}\) such that \(g_{\nu}(t)>t_{1}\) for \(t\geq t_{2}\), so that
In both cases, we have
Therefore,
Using the arithmetic-geometric mean inequality (see [30], p.17), we have
Then, for \(t\geq T_{1}\),
In view of (2.4), we have
Hence,
This, together with (1.1), shows that, for \(t\in[ t_{2},\infty)_{{\mathbb{T}}}\),
Replacing t by Ï„ in (3.6), integrating from \(t\in[ t_{2},\infty)_{{\mathbb{T}}}\) to \(v\in[ t,\infty)_{\mathbb{T}}\), and using (3.2), we have
Hence, by taking limits as \(v\rightarrow\infty\) we obtain that
This shows that \(\int_{t}^{\infty}\bar{p}_{m,0}(\tau,t_{1})\Delta \tau <\infty\) and (3.5) holds for \(j=n-1\). Assume that \(\int_{t}^{\infty}\bar {p}_{m,n-j-1}(\tau ,t_{1})\Delta\tau<\infty\) and (3.5) holds for some \(j\in \{m+1,m+2,\ldots,n-1\}\). Then, for (3.5),
Replacing t by Ï„ and then integrating it from \(t\in [ t_{2},\infty)_{{\mathbb{T}}}\) to \(v\in[ t,\infty )_{{\mathbb{T}}}\), we have
Taking limits as \(v\rightarrow\infty\), we obtain that
This shows that \(\int_{t}^{\infty}\bar{p}_{m,n-j}(\tau ,t_{1}) \Delta\tau <\infty\) and (3.5) holds for \(j-1\). Therefore, the conclusion holds. □
The following lemma improves [31], Lemma 1; also see [32–34].
Lemma 3.4
Let (1.3) hold. Then, there exists an N-tuple \((\eta_{1},\eta_{2},\ldots,\eta_{N})\) with \(\eta_{\nu}>0\) satisfying (2.4).
Lemma 3.5
see [35]
Let \(\omega(u)=au-bu^{1+1/\beta}\), where \(a,u\geq0\) and \(b,\beta>0\). Then
4 Proofs of main results
Proof of Theorem 2.1
Assume that equation (1.1) has a nonoscillatory solution \(x(t)\). Then, without loss of generality, assume that \(x ( t ) >0\) and \(x ( g_{\nu } ( t ) ) >0\) for \(t\in[{t}_{0},\infty){_{\mathbb {T}}}\). It follows from Lemma 3.1 that there exists an integer \(m\in\{ 0,1,\ldots ,n-1\}\) with \(m+n\) odd such that (3.1) and (3.2) hold for \(t\in [ t_{1},\infty)_{{\mathbb{T}}}\) for some \(t_{1}\in[{t}_{0},\infty)_{{\mathbb{T}}}\). Let \(t_{2}\geq t_{1}\) be such that \(g_{\nu }(t)>t_{1}\) for \(t\in[{t}_{2},\infty){_{\mathbb{T}}}\).
(i) Assume that \(m\geq1\).
Part I: Assume that (a) holds. By Lemma 3.3 we have that, for \(j=m\),
which contradicts \(\int_{t}^{\infty}\bar{p}_{m,n-m-1}(\tau ,t_{1})\Delta \tau=\infty\). If \(\int_{t}^{\infty}\bar{p}_{m,n-m-1}(\tau ,t_{1})\Delta \tau<\infty\), then by Lemma 3.3 we have that, for \(j=m\),
By Lemma 3.2(b) with \(i=0\) and \(j=0\) we get
Substituting (4.2) into (4.1), we obtain that
which contradicts \(\limsup_{t\rightarrow\infty}R_{m,m}^{\beta _{m}}(t,t_{1}) \int_{t}^{\infty}\bar{p}_{m,n-m-1}(\tau,t_{1})\Delta \tau >1 \). Substituting (4.1) into (4.2), we obtain that
which contradicts \(\limsup_{t\rightarrow\infty }R_{m,m}(t,t_{1}) ( \int_{t}^{\infty}\bar{p}_{m,n-m-1}(\tau,t_{1})\Delta\tau ) ^{1/\beta_{m}}>1\).
Part II: Assume that (b) holds. Define
By the product rule and the quotient rule we have
From Lemma 3.3 with \(j=m+1\) we have
which, together with (2.3), implies that, for \(t\in {}[ t_{1},\infty )_{{\mathbb{T}}}\),
Substituting (4.6) into (4.4), we obtain
When \(0<\beta _{m}\leq 1\), since \(x(t)\) is strictly increasing, by Pötzsche chain rule ([3], Thm. 1.90) we obtain
Hence,
When \(\beta _{m}\geq 1\), since \(x(t)\) is strictly increasing, again by Pötzsche chain rule we obtain
Therefore,
Then, for \(\beta _{m}>0\),
By using Lemma 3.2 (b) with \(i=0\) and \(j=0\) we see that
which implies
Substituting (4.10) into (4.9), we get
Integrating both sides from \(t_{2}\) to t we get
which contradicts (2.5).
Part III: Assume that (c) holds. When \(0<\beta _{m}\leq 1\), by the definition of \(w_{m}(t)\), since \(x(t)\) is strictly increasing, (4.7) can be written as
By using Lemma 3.2 (b) with \(i=0\) and \(j=1\) we see that
which implies
Substituting (4.13) into (4.11), we get, for \(0<\beta _{m}\leq 1\),
When \(\beta _{m}\geq 1\), by the definition of \(w_{m}(t)\), (4.8) can be written as
By using Lemma 3.2 (b) with \(i=0\) and \(j=1\) we see that
which implies
Substituting (4.15) into (4.14), we get, for \(\beta _{m}\geq 1\),
Hence, for \(\beta _{m}>0\) and \(t\in {}[ t_{2},\infty )_{{\mathbb{T}}}\),
Using Lemma 3.5 with
we obtain
From this and from (4.17) we have
Integrating both sides from \(t_{2}\) to t, we get
which contradicts (2.6).
Part IV: Assume that (d) holds. Multiplying both sides of (4.16), with t replaced by τ, by \(H_{m} ( t,\tau ) \) and integrating with respect to τfrom \(t_{2}\) to \(t\in[ t_{2},\infty)_{{\mathbb{T}}}\), we have
Integrating by parts and using (2.7) and (2.8), we obtain
Using Lemma 3.5 with
and
we get
From this last inequality and from (4.18) we have
which implies that
contradicting assumption (2.9).
Part V: Assume that (e) holds. From (4.16) we have
When \(0<\beta_{m}\leq1\), in view of the definition of w and (4.1), we get
When \(\beta_{m}\geq1\), in view of the definition of w and (4.2), we get
Thus, by (4.20), (4.21), and the definition of \(\delta(t,t_{1})\), (4.19) becomes
Now,
Therefore,
Integrating both sides from \(t_{2}\) to t, we get
which contradicts (2.10).
Part VI: Assume that (f) holds. Multiplying both sides of (4.22), with t replaced by τ, by \(H_{m} ( t,\tau ) \) and integrating with respect to τfrom \(t_{2}\) to \(t\in[ t_{2},\infty)_{{\mathbb{T}}}\), we have
Integrating by parts and using (2.7) and (2.11), we obtain
Now,
Consequently,
which contradicts assumption (2.12).
(ii) We show that if \(m=0\), then \(\lim_{t\rightarrow \infty }x(t)=0\). In fact, from Lemma 3.1 we see that it is only possible when n is odd. In this case,
Hence,
We claim that \(\lim_{t\rightarrow\infty}x(t)=l_{0}=0\). Assume that \(l_{0}>0\). Then, for sufficiently large \(t_{2}\in[ t_{1},\infty)_{{\mathbb {T}}}\), we have \(x(g_{\nu}(t))\geq l_{0}\) for \(t\geq t_{2}\). It follows that
where \(L:=\min_{\nu=0}^{N} \{ l_{0}^{\gamma_{\nu}} \} >0\). Then from (1.1) we obtain
Integrating this from t to \(v\in[ t,\infty)_{\mathbb{T}}\), we get
and by (4.23) we see that \(x^{[n-1]}(v)>0\). Hence, by taking limits as \(v\rightarrow \infty \) we have
If \(\int_{t}^{\infty}\hat{p}_{0} ( \tau ) \Delta\tau =\infty\), then we have reached a contradiction. Otherwise,
Integrating this from t to \(v\in[ t,\infty)_{\mathbb{T}}\) and letting \(v\rightarrow\infty\), by (4.23) we get
If \(\int_{t}^{\infty}\hat{p}_{1} ( \tau ) \Delta\tau =\infty\), then we have reached a contradiction. Otherwise,
Continuing this process, we get
If \(\int_{t}^{\infty}\hat{p}_{n-2} ( \tau ) \Delta\tau =\infty\), then we have reached a contradiction. Otherwise,
Again, integrating from \(t_{2}\) to \(t\in[ t_{2},\infty)_{\mathbb{T}}\), we get
If \(\int_{t}^{\infty}\hat{p}_{n-1} ( \tau ) \Delta\tau =\infty\), then we have \(\lim_{t\rightarrow\infty}x(t)=-\infty\), which contradicts the assumption that \(x(t)>0\) eventually. This shows that if \(m=0\), then \(\lim_{t\rightarrow\infty}x(t)=0\). This completes the proof. □
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This work was supported by Research Deanship of Hail University under grant No. 0150287.
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Hassan, T.S. Asymptotics and oscillation of higher-order functional dynamic equations with Laplacian and deviating arguments. Adv Differ Equ 2017, 14 (2017). https://doi.org/10.1186/s13662-016-1065-2
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DOI: https://doi.org/10.1186/s13662-016-1065-2