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Asymptotics and oscillation of higher-order functional dynamic equations with Laplacian and deviating arguments

Advances in Difference Equations20172017:14

https://doi.org/10.1186/s13662-016-1065-2

  • Received: 29 August 2016
  • Accepted: 17 December 2016
  • Published:

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
$$\begin{aligned}& \bigl\{ r_{n-1}(t) \phi_{\alpha_{n-1}} \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)\phi _{\alpha_{1}} \bigl[x^{\Delta}(t) \bigr] \bigr)^{\Delta}\cdots \bigr)^{\Delta} \bigr)^{\Delta } \bigr] \bigr\} ^{\Delta} \\& \quad {}+\sum_{\nu=0}^{N}p_{\nu} ( t ) \phi _{\gamma _{\nu}} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) =0 \end{aligned}$$
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).

Keywords

  • asymptotic behavior
  • oscillation
  • higher order
  • dynamic equations
  • dynamic inequality
  • time scales

MSC

  • 34K11
  • 34N05
  • 39A10
  • 39A13
  • 39A21
  • 39A99

1 Introduction

We are concerned with the asymptotic and oscillatory behavior of the higher-order nonlinear functional dynamic equation
$$\begin{aligned}& \bigl\{ r_{n-1}(t) \phi_{\alpha_{n-1}} \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)\phi _{\alpha_{1}} \bigl[x^{\Delta}(t) \bigr] \bigr)^{\Delta}\cdots \bigr)^{\Delta} \bigr)^{\Delta } \bigr] \bigr\} ^{\Delta} \\& \quad {}+\sum_{\nu=0}^{N}p_{\nu} ( t ) \phi _{\gamma _{\nu}} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) =0 \end{aligned}$$
(1.1)
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:
  1. (i)

    \(n,N\in\mathbb{N}\), \(n\geq2\), and \(\phi_{\beta }(u):=\vert u\vert ^{\beta-1}u\), \(\beta>0\);

     
  2. (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)
     
  3. (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)
     
  4. (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;

     
  5. (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
$$ x^{ [ i ] }:=r_{i} \phi_{\alpha_{i}} \bigl[ \bigl( x^{ [ i-1 ] } \bigr) ^{\Delta} \bigr] , \quad i=1,2,\ldots,n,\text{with }r_{n}=1, \alpha_{n}=1,\text{and }x^{ [ 0 ] }=x. $$
(1.4)
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
$$ x^{\Delta^{n}} ( t ) +p ( t ) \bigl( x^{\sigma } \bigl( g ( t ) \bigr) \bigr) ^{\gamma}=0, $$
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
$$ \int_{{t}_{0}}^{\infty} \int_{t}^{\infty} \int_{s}^{\infty }p(u)\Delta u\Delta s\Delta t=\infty. $$
(1.5)
Wu et al. [6] established Kamanev-type oscillation criteria for the higher-order nonlinear dynamic equation
$$ \bigl\{ r_{n-1}(t) \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)x^{\Delta }(t) \bigr)^{\Delta }\cdots \bigr)^{\Delta} \bigr)^{\Delta} \bigr] ^{\alpha} \bigr\} ^{\Delta }+f \bigl( t,x \bigl( g(t) \bigr) \bigr) =0, $$
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
$$ \bigl\{ r_{n-1}(t) \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)x^{\Delta }(t) \bigr)^{\Delta }\cdots \bigr)^{\Delta} \bigr)^{\Delta} \bigr] ^{\alpha} \bigr\} ^{\Delta }+f \bigl( t,x \bigl( g(t) \bigr) \bigr) =0, $$
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
$$ \bigl\{ r_{n-1}(t) \bigl[ \bigl(r_{n-2}(t) \bigl(\cdots \bigl(r_{1}(t)x^{\Delta }(t) \bigr)^{\Delta }\cdots \bigr)^{\Delta} \bigr)^{\Delta} \bigr] ^{\alpha} \bigr\} ^{\Delta }+p ( t ) x^{\beta} ( t ) =0, $$
where α, β are the quotients of positive odd integers. Also, the results obtained in [68] are presented when
$$ \int_{{t}_{0}}^{\infty}\frac{1}{r_{n-2}(t)} \biggl\{ \int _{t}^{\infty} \biggl[ \frac{1}{r_{n-1}(s)} \int_{s}^{\infty}p(u)\Delta u \biggr] ^{1/\alpha } \Delta s \biggr\} \Delta t=\infty. $$
(1.6)
Hassan and Kong [9] obtained asymptotics and oscillation criteria for the nth-order half-linear dynamic equation
$$ \bigl(x^{ [ n-1 ] } \bigr)^{\Delta} ( t ) +p ( t ) \phi_{{\alpha}[1,n-1]} \bigl( x \bigl( g ( t ) \bigr) \bigr) =0, $$
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
$$ \bigl(x^{ [ n-1 ] } \bigr)^{\Delta} ( t ) +p ( t ) \phi _{\gamma} \bigl( x^{\sigma} \bigl( g ( t ) \bigr) \bigr) =0. $$
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
$$ \bigl(x^{ [ n-1 ] } \bigr)^{\Delta} ( t ) +p ( t ) \phi_{{\gamma}} \bigl( x \bigl( g ( t ) \bigr) \bigr) =0 $$
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
$$ \bigl\{ r(t) \phi_{\gamma_{0}} \bigl( x^{\Delta^{n-1}}(t) \bigr) \bigr\} ^{\Delta}+\sum_{\nu=0}^{N}p_{\nu} ( t ) \phi_{\gamma _{\nu }} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) =0, $$
(1.7)
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
$$ \int_{t_{0}}^{\infty} \Biggl[ \int_{v}^{\infty} \Biggl( r^{-1} ( s ) \int_{s}^{\infty}\sum_{\nu=0}^{N}p_{\nu} ( \tau ) \Delta\tau \Biggr) ^{1/\gamma_{0}}\Delta s \Biggr] \Delta v=\infty. $$
(1.8)
Huang [12] extended the work in [1] to the neutral advanced dynamic equation
$$ \bigl\{ r(t) \phi_{\alpha} \bigl( y^{\Delta^{n-1}}(t) \bigr) \bigr\} ^{\Delta}+\sum_{\nu=0}^{N}p_{\nu} ( t ) \phi_{\gamma _{\nu }} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) =0, $$
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 [1329].

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, 510] on the oscillation of various dynamic equations.

2 Main results

We introduce the following notations:
$$ k_{+}:=\max\{k,0\},\qquad k_{-}:=\max\{-k,0\}\quad \text{for any }k\in {\mathbb{R}}, $$
and
$$ {\alpha} {[h,k]}:=\left \{ \textstyle\begin{array}{l@{\quad}l} {\alpha}_{h}\cdots{\alpha}_{k}, & h\leq k, \\ 1, & h>k, \end{array}\displaystyle \right . $$
(2.1)
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:
$$ R_{m,j} ( t,s ) :=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 , & j=0 , \\ \int_{s}^{t} [ \frac{R_{m,j-1}(\tau,s)}{r_{m-j+1}(\tau )} ] ^{1/\alpha_{m-j+1}}\Delta\tau , & j=1,2,\ldots,m ,\end{array}\displaystyle \right . $$
(2.2)
and
$$ \hat{p}_{j}(t):=\left \{ \textstyle\begin{array}{l@{\quad}l} \sum_{\nu=0}^{N}p_{\nu} ( t ) , & j=0 , \\ {[ \frac{1}{r_{n-j}(t)}\int_{t}^{\infty}\hat{p}_{j-1}(\tau )\Delta \tau ]} ^{1/\alpha_{n-j}} , & j=1,2,\ldots,n-1 .\end{array}\displaystyle \right . $$
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
$$ \bar{p}_{m,j}(t,s):=\left \{ \textstyle\begin{array}{l@{\quad}l} p_{m}(t,s) , & j=0 , \\ {[ \frac{1}{r_{n-j}(t)}\int_{t}^{\infty}\bar{p}_{m,j-1}(\tau ,s)\Delta\tau ]} ^{1/\alpha_{n-j}} , & j=1,2,\ldots ,n-1 , \end{array}\displaystyle \right . $$
(2.3)
with
$$ \varphi_{m,\nu}(t,t_{1}):=\left \{ \textstyle\begin{array}{l@{\quad}l} 1 , & g_{\nu} ( t ) \geq\sigma(t) , \\ \frac{R_{m,m}(g_{\nu} ( t ) ,t_{1})}{R_{m,m}(\sigma(t),t_{1})} , & g_{\nu} ( t ) \leq\sigma(t) ,\end{array}\displaystyle \right . $$
and
$$ p_{m}(t,s)=p_{0} ( t ) \phi_{\alpha} \bigl( \varphi _{m,0} ( t,s ) \bigr) +\prod_{\nu=1}^{N} \biggl[ \frac{p_{\nu } ( t ) \phi_{\gamma_{\nu}} ( \varphi_{m,\nu} ( t,s ) ) }{\eta_{\nu}} \biggr] ^{\eta_{\nu}} $$
such that
$$ \sum_{\nu=1}^{N}\gamma_{\nu} \eta_{\nu}=\alpha \quad \text{and}\quad \sum _{\nu=1}^{N}\eta_{\nu}=1, $$
(2.4)
where
$$ \delta(t,s):=\left \{ \textstyle\begin{array}{l@{\quad}l} {[ \int_{t}^{\infty}\bar{p}_{m,n-m-1}(\tau,s)\Delta\tau]} ^{1/\beta_{m}-1} , & 0< \beta_{m}\leq1 , \\ R_{m,m}^{\beta_{m}-1} ( t,s ) , & \beta_{m}\geq1 ,\end{array}\displaystyle \right . $$
provided that the improper integrals involved are convergent.
In the sequel, we present conditions that guarantee the following conclusions:
(C): 
  1. (i)

    every solution of equation (1.1) is oscillatory if n is even;

     
  2. (ii)

    every solution of equation (1.1) either is oscillatory or tends to zero eventually if n is odd.

     

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:
  1. (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
    $$ \limsup_{t\rightarrow\infty}R_{i,i}(t,T_{1}) \biggl( \int _{t}^{\infty}\bar{p}_{i,n-i-1}( \tau,T_{1})\Delta\tau \biggr) ^{1/\beta_{i}}>1; $$
     
  2. (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)
     
  3. (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)
     
  4. (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)
     
  5. (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)
     
  6. (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 \(j\in \{0,1,\ldots,n-1 \} \),
$$ \int_{T}^{\infty}\hat{p}_{j}(\tau) \Delta \tau=\infty. $$
(2.13)
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
$$ \xi=\left \{ \textstyle\begin{array}{l@{\quad}l} {>}0 & \mbox{if } n \mbox{ is even}, \\ {\leq}0 & \mbox{if } n \mbox{ is odd},\end{array}\displaystyle \right . $$
and where
$$ r_{i}(t):=\frac{t^{\alpha_{i}}}{\beta_{i}},\quad i=2,\ldots ,n-1\quad \text{and} \quad p_{0} ( t ) :=\frac{\zeta}{t^{\alpha+1}\phi_{\alpha } ( \varphi_{i,0} ( t,t_{0} ) ) }\quad \text{with }\zeta>0. $$
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
$$ \int_{{t}_{0}}^{\infty}r_{1}^{-1/\alpha_{1}}(\tau) \Delta\tau =\beta _{1}^{1/\beta_{1}} \int_{{t}_{0}}^{\infty}\frac{\Delta\tau}{\tau ^{\xi /\alpha_{1}}}=\infty\quad \text{and}\quad \int_{{t}_{0}}^{\infty }r_{i}^{-1/\alpha _{i}}(\tau) \Delta\tau=\beta_{i}^{1/\alpha_{i}} \int _{{t}_{0}}^{\infty}\frac{\Delta\tau}{\tau}=\infty $$
by [3], Example 5.60. By the Pötzsche chain rule we get
$$\begin{aligned} \hat{p}_{1}(t) =& \biggl[ \frac{1}{r_{n-1}(t)} \int_{t}^{\infty }\hat{p}_{0}( \tau)\Delta\tau \biggr] ^{1/\alpha_{n-1}} \\ \geq&\zeta^{1/\alpha_{n-1}} \biggl[ \frac{\beta_{n-1}}{t^{\alpha _{n-1}}}\int_{t}^{\infty}\frac{1}{\tau^{\beta_{n-1}+1}}\Delta\tau \biggr] ^{1/\alpha_{n-1}} \\ \geq&\zeta^{1/\alpha_{n-1}} \biggl[ \frac{1}{t^{\alpha_{n-1}}}\int_{t}^{\infty} \biggl( \frac{-1}{\tau^{\beta_{n-1}}} \biggr) ^{\Delta }\Delta\tau \biggr] ^{1/\alpha_{n-1}} \\ =&\frac{\zeta^{1/\alpha_{n-1}}}{t^{\beta_{n-2}+1}}=\frac{\zeta ^{1/\alpha[ n-1,n-1]}}{t^{\beta_{n-2}+1}}. \end{aligned}$$
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
$$\begin{aligned} \bar{p}_{i,1}(t,T_{1}) =& \biggl[ \frac{1}{r_{n-1}(t)} \int _{t}^{\infty}\bar{p}_{i,0}( \tau,T_{1})\Delta\tau \biggr] ^{1/\alpha_{n-1}} \\ \geq&\zeta^{1/\alpha_{n-1}} \biggl[ \frac{\beta_{n-1}}{t^{\alpha _{n-1}}}\int_{t}^{\infty}\frac{1}{\tau^{\beta_{n-1}+1}}\Delta\tau \biggr] ^{1/\alpha_{n-1}} \\ \geq&\frac{\zeta^{1/\alpha[ n-1,n-1]}}{t^{\beta_{n-2}+1}}. \end{aligned}$$
It is easy to see that
$$ \hat{p}_{j}(t), \bar{p}_{i,j}(t,T_{1})\geq \frac{\zeta ^{1/\alpha [ n-j,n-1]}}{t^{\beta_{n-j-1}+1}},\quad j=0,1,\ldots,n-2. $$
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,
$$ \rho_{i}^{\Delta}(t)= \bigl( t^{\beta_{i}} \bigr) ^{\Delta}=\beta _{i} \int_{0}^{1} \bigl(t+h\mu(t) \bigr)^{\beta_{i}-1} \, dh\leq\beta_{i}t^{\beta_{i}-1}. $$
Hence,
$$\begin{aligned}& \limsup_{t\rightarrow\infty} \int_{T}^{t} \biggl[\rho_{i}(\tau ) \bar{p}_{i,n-i-1}(\tau,T_{1}) \\& \qquad {}-\frac{1}{\rho_{i}^{\beta_{i}}(\tau)} \biggl[ \frac{(\rho _{i}^{\Delta }(\tau))_{+}}{1+\beta_{i}} \biggr] ^{1+\beta_{i}} \biggl[ \frac {r_{1}(\tau)}{R_{i,i-1}(\tau,T_{1})} \biggr] ^{\beta_{i}/\alpha_{1}} \biggr]\Delta \tau \\& \quad \geq \biggl[\zeta^{1/\alpha[ i+1,n-1]}- \biggl[ \frac {1}{\alpha_{1}} \biggr] ^{\beta_{i}/\alpha_{1}} \biggl[ \frac{\beta_{i}}{1+\beta _{i}} \biggr] ^{1+\beta_{i}} \biggr]\limsup_{t\rightarrow\infty} \int _{T^{\ast }}^{t}\frac{1}{\tau}\Delta\tau \\& \quad = \infty \end{aligned}$$
if
$$ \zeta^{1/\alpha[ i+1,n-1]}> \biggl[ \frac{1}{\alpha_{1}} \biggr] ^{\beta_{i}/\alpha_{1}} \biggl[ \frac{\beta_{i}}{1+\beta _{i}} \biggr] ^{1+\beta_{i}}, $$
and hence (2.6) holds. Also,
$$\begin{aligned} \hat{p}_{n-1}(t) =& \biggl[ \frac{1}{r_{1}(t)} \int_{t}^{\infty }\hat{p}_{n-2}( \tau)\Delta\tau \biggr] ^{1/\alpha_{1}} \\ \geq&\zeta^{1/\alpha} \biggl[ \frac{\alpha_{1}}{t^{\xi}} \int _{t}^{\infty }\frac{1}{\tau^{\alpha_{1}+1}}\Delta\tau \biggr] ^{1/\alpha_{1}} \\ \geq&\zeta^{1/\alpha} \biggl[ \frac{1}{t^{\xi}} \int_{t}^{\infty } \biggl( \frac{-1}{\tau^{\alpha_{1}}} \biggr) ^{\Delta}\Delta\tau \biggr] ^{1/\alpha_{1}}=\frac{\zeta^{1/\alpha}}{t^{1+\xi/\alpha_{1}}}. \end{aligned}$$
If n is odd, then
$$ \int_{T}^{\infty}\hat{p}_{n-1}(\tau) \Delta \tau=\zeta^{1/\alpha } \int_{T}^{\infty}\frac{\Delta\tau}{\tau^{1+\xi/\alpha _{1}}}=\infty, $$
so that condition (2.13) holds. Then, by Theorem 2.1(c) conclusions (C) hold if
$$ \zeta^{1/\alpha[ i+1,n-1]}> \biggl[ \frac{1}{\alpha_{1}} \biggr] ^{\beta_{i}/\alpha_{1}} \biggl[ \frac{\beta_{i}}{1+\beta _{i}} \biggr] ^{1+\beta_{i}}. $$

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
$$ x^{ [ k ] }(t)>0\quad \textit{for }k=0,1,\ldots,m $$
(3.1)
and
$$ ( -1 ) ^{m+k}x^{ [ k ] }(t)>0\quad \textit{for }k=m,m+1,\ldots,n-1 $$
(3.2)
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 \(t\in(t_{1},\infty)_{{\mathbb{T}}}\):
  1. (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)
     
  2. (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 \(j=m,m+1,\ldots,n-1\),
$$ \int_{t}^{\infty}\bar{p}_{m,n-j-1}( \tau,t_{1})\Delta\tau< \infty $$
and
$$ (-1)^{m+j}x^{ [ j ] }(t)\geq\phi_{\alpha[ 1,j]} \bigl( x^{\sigma} ( t ) \bigr) \int_{t}^{\infty}\bar {p}_{m,n-j-1}(\tau ,t_{1})\Delta\tau. $$
(3.5)

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
$$ x \bigl(g_{\nu}(t) \bigr)\geq\frac{R_{m,m}(g_{\nu}(t),t_{1})}{R_{m,m}(\sigma (t),t_{1})}x^{\sigma}(t) \quad \text{for }t\in[ t_{2},\infty )_{{\mathbb{T}}}. $$
In both cases, we have
$$ x \bigl(g_{\nu} ( t ) \bigr)\geq\varphi_{m,\nu} ( t,t_{1} ) x^{\sigma} ( t ) \quad \text{for }t\in[ t_{2},\infty)_{{\mathbb{T}}}. $$
Therefore,
$$\begin{aligned} \sum_{\nu=0}^{N}p_{\nu} ( t ) \phi_{\gamma_{\nu}} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) \geq&\sum _{\nu=0}^{N}p_{\nu} ( t ) \phi _{\gamma_{\nu}} \bigl( \varphi_{m,\nu} ( t,t_{1} ) \bigr) \bigl[ x^{\sigma}(t) \bigr] ^{\gamma_{\nu}} \\ =&\phi_{\alpha} \bigl( x^{\sigma} ( t ) \bigr) \sum _{\nu =0}^{N}p_{\nu} ( t ) \phi_{\gamma_{\nu}} \bigl( \varphi _{m,\nu } ( t,t_{1} ) \bigr) \bigl[ x^{\sigma}(t) \bigr] ^{\gamma_{\nu }-\alpha}. \end{aligned}$$
Using the arithmetic-geometric mean inequality (see [30], p.17), we have
$$ \sum_{\nu=1}^{N}\eta_{\nu}v_{\nu} \geq \prod_{\nu =1}^{N}v_{\nu}^{\eta_{\nu}} \quad \text{for any }v_{\nu}\geq0, \nu =1,\ldots,N. $$
Then, for \(t\geq T_{1}\),
$$\begin{aligned}& \sum_{\nu=0}^{N}p_{\nu} ( t ) \phi_{\gamma_{\nu }} \bigl( \varphi_{m,\nu} ( t,t_{1} ) \bigr) \bigl[ x^{\sigma }(t) \bigr] ^{\gamma_{\nu}-\alpha} \\& \quad = p_{0} ( t ) \phi_{\alpha} \bigl( \varphi_{m,0} ( t,t_{1} ) \bigr) +\sum_{\nu=1}^{N} \eta_{\nu}\frac{p_{\nu } ( t ) \phi_{\gamma_{\nu}} ( \varphi_{m,\nu} ( t,t_{1} ) ) }{\eta_{\nu}} \bigl[ x^{\sigma}(t) \bigr] ^{\gamma_{\nu }-\alpha} \\& \quad \geq p_{0} ( t ) \phi_{\alpha} \bigl( \varphi _{m,0} ( t,t_{1} ) \bigr) +\prod _{\nu=1}^{N} \biggl[ \frac {p_{\nu} ( t ) \phi_{\gamma_{\nu}} ( \varphi_{m,\nu} ( t,t_{1} ) ) }{\eta_{\nu}} \biggr] ^{\eta_{\nu}} \bigl[ x^{\sigma }(t) \bigr] ^{\eta_{\nu} ( \gamma_{\nu}-\alpha ) }. \end{aligned}$$
In view of (2.4), we have
$$ \sum_{\nu=1}^{N}\gamma_{\nu} \eta_{\nu}-\alpha\sum_{\nu =1}^{N} \eta _{\nu}=0. $$
Hence,
$$\begin{aligned}& \sum_{\nu=0}^{N}p_{\nu} ( t ) \phi_{\gamma_{\nu }} \bigl( \varphi_{m,\nu} ( t,t_{1} ) \bigr) \bigl[ x^{\sigma }(t) \bigr] ^{\gamma_{\nu}-\alpha} \\& \quad \geq p_{0} ( t ) \phi_{\alpha} \bigl( \varphi _{m,0} ( t,t_{1} ) \bigr) +\prod _{\nu=1}^{N} \biggl[ \frac {p_{\nu} ( t ) \phi_{\gamma_{\nu}} ( \varphi_{m,\nu} ( t,t_{1} ) ) }{\eta_{\nu}} \biggr] ^{\eta_{\nu}}=p(t,t_{1}). \end{aligned}$$
This, together with (1.1), shows that, for \(t\in[ t_{2},\infty)_{{\mathbb{T}}}\),
$$ - \bigl( x^{[n-1]}(t) \bigr) ^{\Delta}\geq p(t,t_{1}) \phi_{\alpha } \bigl( x^{\sigma} ( t ) \bigr) =\bar{p}_{m,0}(t,t_{1}) \phi _{\alpha } \bigl( x^{\sigma} ( t ) \bigr) . $$
(3.6)
Replacing t by τ 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
$$\begin{aligned} x^{[n-1]}(t) >&-x^{[n-1]}(v)+x^{[n-1]}(t)\geq \int_{t}^{v}\bar{p}_{m,0}( \tau,t_{1}) \phi_{\alpha} \bigl( x^{\sigma} ( \tau ) \bigr) \Delta\tau \\ \geq&\phi_{\alpha} \bigl( x^{\sigma} ( t ) \bigr) \int_{t}^{v}\bar{p}_{m,0}( \tau,t_{1})\Delta\tau. \end{aligned}$$
Hence, by taking limits as \(v\rightarrow\infty\) we obtain that
$$ x^{[n-1]}(t)\geq\phi_{\alpha} \bigl( x^{\sigma} ( t ) \bigr) \int_{t}^{\infty}\bar{p}_{m,0}( \tau,t_{1})\Delta\tau. $$
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),
$$\begin{aligned} (-1)^{m+j} \bigl[ x^{[j-1]}(t) \bigr] ^{\Delta} =&(-1)^{m+j}\phi _{\alpha _{j}}^{-1} \biggl[ \frac{x^{[j]}(t)}{r_{j}(t)} \biggr] \\ \geq&\phi_{\alpha_{j}}^{-1} \bigl\{ \phi_{\alpha[ 1,j]} \bigl( x^{\sigma} ( t ) \bigr) \bigr\} \biggl[ \frac {1}{r_{j}(t)}\int_{t}^{\infty}\bar{p}_{m,n-j-1}( \tau,t_{1})\Delta\tau \biggr] ^{1/\alpha_{j}} \\ =&\phi_{\alpha[1,j-1]} \bigl( x^{\sigma} ( t ) \bigr) \bar{p}_{m,n-j}(t,t_{1}). \end{aligned}$$
Replacing t by τ and then integrating it from \(t\in [ t_{2},\infty)_{{\mathbb{T}}}\) to \(v\in[ t,\infty )_{{\mathbb{T}}}\), we have
$$\begin{aligned} (-1)^{m+j-1}x^{[j-1]}(t) >& (-1)^{m+j} \bigl(x^{[j-1]}(v)-x^{[j-1]}(t) \bigr) \\ \geq& \int_{t}^{v}\phi_{\alpha[1,j-1]} \bigl( x^{\sigma } ( \tau ) \bigr) \bar{p}_{m,n-j}(\tau,t_{1}) \Delta\tau \\ \geq&\phi_{\alpha[1,j-1]} \bigl( x^{\sigma} ( t ) \bigr) \int_{t}^{v}\bar{p}_{m,n-j}( \tau,t_{1}) \Delta\tau. \end{aligned}$$
Taking limits as \(v\rightarrow\infty\), we obtain that
$$ (-1)^{m+j-1}x^{[j-1]}(t)\geq\phi_{\alpha[1,j-1]} \bigl( x^{\sigma } ( t ) \bigr) \int_{t}^{\infty}\bar{p}_{m,n-j}(\tau ,t_{1}) \Delta\tau. $$
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 [3234].

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
$$ \omega(u)\leq \biggl( \frac{\beta}{b} \biggr) ^{\beta} \biggl( \frac {a}{1+\beta} \biggr) ^{1+\beta}. $$

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 \(j=m\),
$$ \int_{t}^{\infty}\bar{p}_{m,n-m-1}( \tau,t_{1})\Delta\tau< \infty, $$
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 \(j=m\),
$$\begin{aligned} x^{ [ m ] }(t) \geq&\phi_{\alpha[1,m]} \bigl( x^{\sigma } ( t ) \bigr) \int_{t}^{\infty}\bar{p}_{m,n-m-1}(\tau ,t_{1})\Delta\tau \\ \geq&\phi_{\beta_{m}} \bigl( x ( t ) \bigr) \int _{t}^{\infty}\bar{p}_{m,n-m-1}( \tau,t_{1})\Delta\tau. \end{aligned}$$
(4.1)
By Lemma 3.2(b) with \(i=0\) and \(j=0\) we get
$$\begin{aligned} x(t) \geq&\phi_{\alpha [ 1,m ] }^{-1} \bigl( x^{ [ m ] } ( t ) \bigr) R_{m,m}(t,t_{1}) \\ =&\phi_{\beta_{m}}^{-1} \bigl( x^{ [ m ] } ( t ) \bigr) R_{m,m}(t,t_{1}). \end{aligned}$$
(4.2)
Substituting (4.2) into (4.1), we obtain that
$$ 1\geq R_{m,m}^{\beta_{m}}(t,t_{1}) \int_{t}^{\infty}\bar {p}_{m,n-m-1}(\tau ,t_{1})\Delta\tau, $$
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
$$ 1\geq R_{m,m}(t,t_{1}) \biggl( \int_{t}^{\infty}\bar {p}_{m,n-m-1}(\tau ,t_{1})\Delta\tau \biggr) ^{1/\beta_{m}}, $$
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
$$ w_{m}(t):=\rho_{m}(t)\frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) }. $$
(4.3)
By the product rule and the quotient rule we have
$$\begin{aligned} w_{m}^{\Delta }(t) =&\rho _{m}(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\Delta }+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } \\ =&\rho _{m}(t) \biggl( \frac{x^{\beta _{m}} ( t ) ( x^{ [ m ] }(t) ) ^{\Delta }- ( x^{\beta _{m}} ( t ) ) ^{\Delta }x^{ [ m ] }(t)}{ ( x^{\beta _{m}} ( t ) ) ^{\sigma }x^{\beta _{m}} ( t ) } \biggr) +\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } \\ =&\rho _{m}(t)\frac{ ( x^{ [ m ] }(t) ) ^{\Delta }}{ ( x^{\beta _{m}} ( t ) ) ^{\sigma }}-\rho _{m}(t) \frac{ ( x^{\beta _{m}} ( t ) ) ^{\Delta }}{ ( x^{\beta _{m}} ( t ) ) ^{\sigma }}\frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) }+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma }. \end{aligned}$$
(4.4)
From Lemma 3.3 with \(j=m+1\) we have
$$ -x^{ [ m+1 ] }(t)\geq \phi _{\alpha {}[ 1,m+1]} \bigl( x^{\sigma } ( t ) \bigr) ~ \int_{t}^{\infty }\bar{p}_{m,n-m-2}(\tau ,t_{1})~\Delta \tau , $$
(4.5)
which, together with (2.3), implies that, for \(t\in {}[ t_{1},\infty )_{{\mathbb{T}}}\),
$$\begin{aligned} - \bigl( x^{ [ m ] }(t) \bigr) ^{\Delta } \geq &\phi _{\alpha {}[ 1,m]} \bigl( x^{\sigma } ( t ) \bigr) \biggl[ \frac{1}{r_{m+1}(t)} \int_{t}^{\infty }\bar{p}_{m,n-m-2}(\tau ,t_{1})~\Delta \tau \biggr] ^{1/\alpha _{m+1}} \\ =&\phi _{\beta _{m}} \bigl( x^{\sigma } ( {t} ) \bigr) ~ \bar{p}_{m,n-m-1}(t,t_{1}). \end{aligned}$$
(4.6)
Substituting (4.6) into (4.4), we obtain
$$ w_{m}^{\Delta }(t)\leq -\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma }-\rho _{m}(t) \frac{ ( x^{\beta _{m}} ( t ) ) ^{\Delta }}{ ( x^{\beta _{m}} ( t ) ) ^{\sigma }}\frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) }. $$
When \(0<\beta _{m}\leq 1\), since \(x(t)\) is strictly increasing, by Pötzsche chain rule ([3], Thm. 1.90) we obtain
$$\begin{aligned} \bigl( x^{\beta _{m}} ( t ) \bigr) ^{\Delta } =&\beta _{m} \int_{0}^{1} \bigl[ x ( t ) +h~\mu (t)x^{\Delta } ( t ) \bigr] ^{\beta _{m}-1}\, dh~x^{\Delta } ( t ) \\ =&\beta _{m} \int_{0}^{1} \bigl[ ( 1-h ) x ( t ) +h~x^{\sigma } ( t ) \bigr] ^{\beta _{m}-1}\, dh~x^{\Delta } ( t ) \\ \geq &\beta _{m}~ \bigl[ x^{\sigma } ( t ) \bigr] ^{\beta _{m}-1}x^{\Delta } ( t ) . \end{aligned}$$
Hence,
$$\begin{aligned} \begin{aligned}[b] w_{m}^{\Delta } ( t ) &\leq {-}\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } -\beta _{m}\rho _{m}(t)\frac{x^{\Delta } ( t ) }{x^{\sigma } ( t ) } \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } \\ &\leq{-}\rho _{m}(t)~\bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma }. \end{aligned} \end{aligned}$$
(4.7)
When \(\beta _{m}\geq 1\), since \(x(t)\) is strictly increasing, again by Pötzsche chain rule we obtain
$$\begin{aligned} \bigl( x^{\beta _{m}} ( t ) \bigr) ^{\Delta } =&\beta _{m} \int_{0}^{1} \bigl[ x ( t ) +h~\mu (t)x^{\Delta } ( t ) \bigr] ^{\beta _{m}-1}\, dh~x^{\Delta } ( t ) \\ =&\beta _{m} \int_{0}^{1} \bigl[ ( 1-h ) x ( t ) +h~x^{\sigma } ( t ) \bigr] ^{\beta _{m}-1}\, dh~x^{\Delta } ( t ) \\ \geq &\beta _{m}~ \bigl[ x ( t ) \bigr] ^{\beta _{m}-1}x^{\Delta } ( t ) . \end{aligned}$$
Therefore,
$$\begin{aligned} w_{m}^{\Delta }(t) \leq &-\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } -\beta _{m}\rho _{m}(t)\frac{x^{\Delta } ( t ) }{x ( t ) } \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } \\ \leq &-\rho _{m}(t)~\bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma }. \end{aligned}$$
(4.8)
Then, for \(\beta _{m}>0\),
$$ w_{m}^{\Delta } ( t ) \leq -\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma }. $$
(4.9)
By using Lemma 3.2 (b) with \(i=0\) and \(j=0\) we see that
$$ x(t)\geq \phi _{\alpha [ 1,m ] }^{-1} \bigl( x^{ [ m ] } ( t ) \bigr) R_{m,m}(t,t_{1}), $$
which implies
$$ \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) }\leq \frac{1}{R_{m,m}^{\beta _{m}}(t,t_{1})}. $$
(4.10)
Substituting (4.10) into (4.9), we get
$$\begin{aligned} w_{m}^{\Delta } ( t ) \leq &-\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\frac{\rho _{m}^{\Delta }(t)}{R_{m,m}^{\beta _{m}}(\sigma (t),t_{1})} \\ \leq &-\rho _{m}(t)~\bar{p}_{m,n-m-1}(t,t_{1})+ \frac{ ( \rho _{m}^{\Delta }(t) ) _{+}}{R_{m,m}^{\beta _{m}}(\sigma (t),t_{1})}\quad \text{ for }t\in {}[ t_{2},\infty )_{{\mathbb{T}}}. \end{aligned}$$
Integrating both sides from \(t_{2}\) to t we get
$$ \int_{t_{2}}^{t} \biggl[\rho _{m}(\tau )~ \bar{p}_{m,n-m-1}(\tau ,t_{1})-\frac{ ( \rho _{m}^{\Delta }(\tau ) ) _{+}}{R_{m,m}^{\beta _{m}}(\sigma (\tau ),t_{1})} \biggr]\Delta \tau \leq w_{m}(t_{2})-w_{m}(t)\leq w_{m}(t_{2}), $$
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
$$ w_{m}^{\Delta } ( t ) \leq -\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma }-\beta _{m}\rho _{m}(t)\frac{x^{\Delta } ( t ) }{x^{\sigma } ( t ) } \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma }. $$
(4.11)
By using Lemma 3.2 (b) with \(i=0\) and \(j=1\) we see that
$$ x^{[1]}(t)\geq \phi _{\alpha [ 2,m ] }^{-1} \bigl( x^{ [ m ] } ( t ) \bigr) R_{m,m-1}(t,t_{1}), $$
(4.12)
which implies
$$\begin{aligned} \frac{x^{\Delta } ( t ) }{x^{\sigma } ( t ) } \geq &\frac{\phi _{\alpha [ 1,m ] }^{-1} ( x^{ [ m ] } ( t ) ) }{x^{\sigma } ( t ) } \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \\ \geq &\frac{\phi _{\alpha [ 1,m ] }^{-1} ( x^{ [ m ] } ( t ) ) }{x^{\sigma } ( t ) } \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \\ \geq & \biggl[ \biggl( \frac{x^{ [ m ] } ( t ) }{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } \biggr] ^{1/\beta _{m}} \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \\ =& \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1/\beta _{m}} \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}}. \end{aligned}$$
(4.13)
Substituting (4.13) into (4.11), we get, for \(0<\beta _{m}\leq 1\),
$$\begin{aligned} w_{m}^{\Delta } ( t ) \leq &-\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \\ &{}-\beta _{m}\rho _{m}(t) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta _{m}}. \end{aligned}$$
When \(\beta _{m}\geq 1\), by the definition of \(w_{m}(t)\), (4.8) can be written as
$$ w_{m}^{\Delta } ( t ) \leq -\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma }-\beta _{m}\rho _{m}(t)\frac{x^{\Delta } ( t ) }{x ( t ) } \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma }. $$
(4.14)
By using Lemma 3.2 (b) with \(i=0\) and \(j=1\) we see that
$$ x^{[1]}(t)\geq \phi _{\alpha [ 2,m ] }^{-1} \bigl( x^{ [ m ] } ( t ) \bigr) R_{m,m-1}(t,t_{1}), $$
which implies
$$\begin{aligned} \frac{x^{\Delta } ( t ) }{x ( t ) } =&\frac{x^{\Delta } ( t ) }{x ( t ) } \geq \frac{\phi _{\alpha [ 1,m ] }^{-1} ( x^{ [ m ] } ( t ) ) }{x ( t ) } \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \\ \geq &\frac{\phi _{\alpha [ 1,m ] }^{-1} ( x^{ [ m ] } ( t ) ) }{x ( t ) } \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \\ =& \biggl[ \biggl( \frac{x^{ [ m ] } ( t ) }{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma } \biggr] ^{1/\beta _{m}} \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \\ =& \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1/\beta _{m}} \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}}. \end{aligned}$$
(4.15)
Substituting (4.15) into (4.14), we get, for \(\beta _{m}\geq 1\),
$$\begin{aligned} w_{m}^{\Delta } ( t ) \leq &-\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \\ &{}-\beta _{m}\rho _{m}(t) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta _{m}}. \end{aligned}$$
Hence, for \(\beta _{m}>0\) and \(t\in {}[ t_{2},\infty )_{{\mathbb{T}}}\),
$$\begin{aligned} w_{m}^{\Delta } ( t ) \leq &-\rho _{m}(t)~ \bar{p}_{m,n-m-1}(t,t_{1})+\rho _{m}^{\Delta }(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \\ &{}-\beta _{m}\rho _{m}(t) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta _{m}} \end{aligned}$$
(4.16)
$$\begin{aligned} \leq &-\rho _{m}(t)~\bar{p}_{m,n-m-1}(t,t_{1})+ \bigl( \rho _{m}^{\Delta }(t) \bigr) _{+} \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \\ &{}-\beta _{m}\rho _{m}(t) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta _{m}}. \end{aligned}$$
(4.17)
Using Lemma 3.5 with
$$ a:=\bigl(\rho _{m}^{\Delta }(t)\bigr)_{+},\qquad b:=\beta _{m}\rho _{m}(t) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}},\qquad \beta :=\beta _{m}\quad \text{and} \quad u:= \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma }, $$
we obtain
$$\begin{aligned}& \bigl( \rho _{m}^{\Delta }(t) \bigr) _{+} \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma }-\beta _{m}\rho _{m}(t) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta _{m}} \\& \quad \leq \biggl( \frac{\beta _{m}}{\beta _{m}\rho _{m}(t)} \biggl[ \dfrac{r_{1}(t)}{R_{m,m-1}(t,t_{1})} \biggr] ^{1/\alpha _{1}} \biggr) ^{\beta _{m}} \biggl[ \dfrac{(\rho _{m}^{\Delta }(t))_{+}}{1+\beta _{m}} \biggr] ^{1+\beta _{m}} \\& \quad =\frac{1}{\rho _{m}^{\beta _{m}}(t)} \biggl[ \dfrac{(\rho _{m}^{\Delta }(t))_{+}}{1+\beta _{m}} \biggr] ^{1+\beta _{m}} \biggl[ \frac{r_{1}(t)}{R_{m,m-1}(t,t_{1})} \biggr] ^{\beta _{m}/\alpha _{1}}. \end{aligned}$$
From this and from (4.17) we have
$$ w_{m}^{\Delta} ( t ) \leq-\rho_{m}(t) \bar {p}_{m,n-m-1}(t,t_{1})+\frac{1}{\rho_{m}^{\beta_{m}}(t)} \biggl[ \frac{(\rho_{m}^{\Delta }(t))_{+}}{1+\beta_{m}} \biggr] ^{1+\beta_{m}} \biggl[ \frac{r_{1}(t)}{R_{m,m-1}(t,t_{1})} \biggr] ^{\beta_{m}/\alpha_{1}}. $$
Integrating both sides from \(t_{2}\) to t, we get
$$\begin{aligned}& \int_{t_{2}}^{t} \biggl[\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1}) \\& \quad {}-\frac{1}{\rho_{m}^{\beta_{m}}(\tau)} \biggl[ \frac{(\rho _{m}^{\Delta }(\tau))_{+}}{1+\beta_{m}} \biggr] ^{1+\beta_{m}} \biggl[ \frac {r_{1}(\tau)}{R_{m,m-1}(\tau,t_{1})} \biggr] ^{\beta_{m}/\alpha_{1}} \biggr] \Delta \tau \leq w_{m}(t_{2})-w_{m}(t)\leq w_{m}(t_{2}), \end{aligned}$$
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
$$\begin{aligned}& \int_{t_{2}}^{t}\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1})H_{m} ( t,\tau ) \Delta \tau \\& \quad \leq - \int_{t_{2}}^{t}H_{m} ( t,\tau ) w_{m}^{\Delta } ( \tau ) \Delta\tau \\& \qquad {}+ \int_{t_{2}}^{t}H_{m} ( t,\tau ) \rho_{m}^{\Delta }(\tau ) \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma }\Delta\tau \\& \qquad {}-\beta_{m} \int_{t_{2}}^{t}\rho_{m}( \tau)H_{m} ( t,\tau ) \biggl[ \frac{R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta_{m}}\Delta\tau. \end{aligned}$$
Integrating by parts and using (2.7) and (2.8), we obtain
$$\begin{aligned}& \int_{t_{2}}^{t}\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1})H_{m} ( t,\tau ) \Delta \tau \\& \quad \leq H_{m} ( t,t_{2} ) w_{m} ( t_{2} ) + \int_{t_{2}}^{t}H_{m}^{\Delta_{\tau}} ( t, \tau ) w_{m}^{\sigma } ( \tau ) \Delta\tau \\& \qquad {}+ \int_{t_{2}}^{t}H_{m} ( t,\tau ) \rho_{m}^{\Delta }(\tau ) \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma }\Delta\tau \\& \qquad {}-\beta_{m} \int_{t_{2}}^{t}\rho_{m}( \tau)H_{m} ( t,\tau ) \biggl[ \frac{R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha _{1}} \biggl[ \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma } \biggr] ^{1+1/\beta_{m}}\Delta\tau \\& \quad \leq H_{m} ( t,t_{2} ) w ( t_{2} ) + \int_{t_{2}}^{t} \biggl[ \bigl( h_{m} ( t, \tau ) \bigr) _{-} \bigl( H_{m} ( t,\tau ) \bigr) ^{\frac{\beta _{m}}{1+\beta _{m}}} \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma } \\& \qquad {}-\beta_{m}\rho_{m}(\tau)H_{m} ( t, \tau ) \biggl[ \frac{R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma} \biggr] ^{1+1/\beta _{m}} \biggr] \Delta \tau. \end{aligned}$$
(4.18)
Using Lemma 3.5 with
$$ a:= \bigl( h_{m} ( t,\tau ) \bigr) _{-} \bigl( H_{m} ( t,\tau ) \bigr) ^{\frac{\beta_{m}}{1+\beta_{m}}},\qquad b:=\beta _{m}\rho _{m}(\tau)H_{m} ( t,\tau ) \biggl[ \frac{R_{m,m-1}(\tau ,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}}, $$
and
$$ \beta:=\beta_{m}, \qquad u:= \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma}, $$
we get
$$\begin{aligned} \begin{aligned} &\bigl( h_{m} ( t,\tau ) \bigr) _{-} \bigl( H_{m} ( t,\tau ) \bigr) ^{\frac{\beta_{m}}{1+\beta_{m}}} \biggl( \frac {w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma} \\ &\qquad {}-\beta_{m}\rho_{m}(\tau)H_{m} ( t, \tau ) \biggl[ \frac {R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma} \biggr] ^{1+1/\beta _{m}} \\ &\quad \leq \frac{1}{ ( 1+\beta_{m} ) ^{1+\beta_{m}}}\frac { [ ( h_{m} ( t,\tau ) ) _{-} ] ^{1+\beta _{m}}}{\rho _{m}^{\beta_{m}}(\tau)} \biggl[ \frac{r_{1}(\tau)}{R_{m,m-1}(\tau ,t_{1})} \biggr] ^{\beta_{m}/\alpha_{1}} \\ &\quad = \frac{1}{\rho_{m}^{\beta_{m}}(\tau)} \biggl[ \frac{ ( h_{m} ( t,\tau ) ) _{-}}{1+\beta_{m}} \biggr] ^{1+\beta _{m}} \biggl[ \frac{r_{1}(\tau)}{R_{m,m-1}(\tau,t_{1})} \biggr] ^{\beta _{m}/\alpha_{1}}. \end{aligned} \end{aligned}$$
From this last inequality and from (4.18) we have
$$\begin{aligned}& \int_{t_{2}}^{t} \biggl[\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1})H_{m} ( t,\tau ) \\& \quad {}-\frac{1}{\rho_{m}^{\beta_{m}}(\tau)} \biggl[ \frac { ( h_{m} ( t,\tau ) ) _{-}}{1+\beta_{m}} \biggr] ^{1+\beta_{m}} \biggl[ \frac{r_{1}(\tau)}{R_{m,m-1}(\tau,t_{1})} \biggr] ^{\beta _{m}/\alpha_{1}} \biggr]\Delta\tau\leq H_{m} ( t,t_{2} ) w_{m} ( t_{2} ) , \end{aligned}$$
which implies that
$$\begin{aligned}& \frac{1}{H_{m} ( t,t_{2} ) } \int_{t_{2}}^{t} \biggl[\rho _{m}(\tau) \bar{p}_{m,n-m-1}(\tau,t_{1})H_{m} ( t, \tau ) \\& \quad {}-\frac{1}{\rho_{m}^{\beta_{m}}(\tau)} \biggl[ \frac { ( h_{m} ( t,\tau ) ) _{-}}{1+\beta_{m}} \biggr] ^{1+\beta_{m}} \biggl[ \frac{r_{1}(\tau)}{R_{m,m-1}(\tau,t_{1})} \biggr] ^{\beta _{m}/\alpha_{1}} \biggr]\Delta\tau\leq w_{m} ( t_{2} ) , \end{aligned}$$
contradicting assumption (2.9).
Part V: Assume that (e) holds. From (4.16) we have
$$\begin{aligned} w_{m}^{\Delta} ( t ) \leq&-\rho_{m}(t) \bar{p}_{m,n-m-1}(t,t_{1})+\rho_{m}^{\Delta}(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma} \\ &{}-\beta_{m}\rho_{m}(t) \biggl[ \frac {R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] ^{1+1/\beta_{m}} \\ \leq&-\rho_{m}(t) \bar{p}_{m,n-m-1}(t,t_{1})+ \rho_{m}^{\Delta }(t) \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \\ &{}-\beta_{m}\rho_{m}(t) \biggl[ \frac {R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] ^{1/\beta_{m}-1} \biggl[ \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma} \biggr] ^{2}. \end{aligned}$$
(4.19)
When \(0<\beta_{m}\leq1\), in view of the definition of w and (4.1), we get
$$ \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] ^{1/\beta_{m}-1}= \biggl[ \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma} \biggr] ^{1/\beta _{m}-1}\geq \biggl[ \int_{\sigma(t)}^{\infty}\bar{p}_{m,n-m-1}( \tau,t_{1})\Delta\tau \biggr] ^{1/\beta_{m}-1}. $$
(4.20)
When \(\beta_{m}\geq1\), in view of the definition of w and (4.2), we get
$$ \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] ^{1/\beta_{m}-1}= \biggl[ \biggl( \frac{x^{ [ m ] }(t)}{x^{\beta _{m}} ( t ) } \biggr) ^{\sigma} \biggr] ^{1/\beta _{m}-1}\geq \bigl[ R_{m,m}^{\sigma} ( t,t_{1} ) \bigr] ^{\beta_{m}-1}. $$
(4.21)
Thus, by (4.20), (4.21), and the definition of \(\delta(t,t_{1})\), (4.19) becomes
$$\begin{aligned} w_{m}^{\Delta} ( t ) \leq&-\rho_{m}(t) \bar{p}_{m,n-m-1}(t,t_{1})+\rho_{m}^{\Delta}(t) \biggl( \frac{w_{m}(t)}{\rho _{m}(t)} \biggr) ^{\sigma} \\ &{}-\beta_{m}\rho_{m}(t)\delta^{\sigma}(t,t_{1}) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] ^{2}. \end{aligned}$$
(4.22)
Now,
$$\begin{aligned}& \rho_{m}^{\Delta}(t) \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma }-\beta_{m}\rho_{m}(t)\delta^{\sigma}(t,t_{1}) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] ^{2} \\& \quad = \frac{ ( \rho_{m}^{\Delta}(t) ) ^{2}}{4\beta_{m}\rho _{m}(t)\delta^{\sigma}(t,t_{1})} \biggl[ \frac{r_{1}(t )}{R_{m,m-1}(t ,t_{1})} \biggr] ^{1/\alpha_{1}} \\& \qquad {}- \biggl[\sqrt{\beta_{m}\rho_{m}(t) \delta^{\sigma}(t,t_{1}) \biggl[ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} \biggr] ^{1/\alpha_{1}}} \biggl[ \biggl( \frac{w_{m}(t)}{\rho_{m}(t)} \biggr) ^{\sigma} \biggr] \\& \qquad {}-\frac{\rho_{m}^{\Delta}(t)}{2\sqrt{\beta_{m}\rho_{m}(t)\delta ^{\sigma}(t,t_{1}) [ \frac{R_{m,m-1}(t,t_{1})}{r_{1}(t)} ] ^{1/\alpha_{1}}}} \biggr]^{2} \\& \quad \leq \frac{ ( \rho_{m}^{\Delta}(t) ) ^{2}}{4\beta _{m}\rho _{m}(t)\delta^{\sigma}(t,t_{1})} \biggl[ \frac{r_{1}(t )}{R_{m,m-1}(t ,t_{1})} \biggr] ^{1/\alpha_{1}}. \end{aligned}$$
Therefore,
$$ w_{m}^{\Delta} ( t ) \leq-\rho_{m}(t) \bar {p}_{m,n-m-1}(t,t_{1})+\frac{ ( \rho_{m}^{\Delta}(t) ) ^{2}}{4\beta_{m}\rho _{m}(t)\delta^{\sigma}(t,t_{1})} \biggl[ \frac{r_{1}(t )}{R_{m,m-1}(t ,t_{1})} \biggr] ^{1/\alpha_{1}}. $$
Integrating both sides from \(t_{2}\) to t, we get
$$\begin{aligned}& \int_{t_{2}}^{t} \biggl[\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1}) \\& \quad {}-\frac{ ( \rho_{m}^{\Delta}(\tau) ) ^{2}}{4\beta _{m}\rho_{m}(\tau)\delta^{\sigma}(\tau,t_{1})} \biggl[ \frac{r_{1}(\tau)}{ R_{m,m-1}(\tau,t_{1})} \biggr] ^{1/\alpha_{1}} \biggr]\Delta\tau\leq w_{m}(t_{2})-w_{m}(t) \leq w_{m}(t_{2}), \end{aligned}$$
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
$$\begin{aligned} \begin{aligned} & \int_{t_{2}}^{t}\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1})H_{m} ( t,\tau ) \Delta \tau \\ &\quad \leq - \int_{t_{2}}^{t}H_{m} ( t,\tau ) w_{m}^{\Delta } ( \tau ) \Delta\tau+ \int_{t_{2}}^{t}H_{m} ( t,\tau ) \rho _{m}^{\Delta}(\tau) \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau )} \biggr) ^{\sigma}\Delta\tau \\ &\qquad {}-\beta_{m} \int_{t_{2}}^{t}\rho_{m}( \tau)H_{m} ( t,\tau ) \delta^{\sigma}(\tau,t_{1}) \biggl[ \frac{R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(\tau )}{\rho _{m}(\tau)} \biggr) ^{\sigma} \biggr] ^{2}\Delta\tau. \end{aligned} \end{aligned}$$
Integrating by parts and using (2.7) and (2.11), we obtain
$$\begin{aligned}& \int_{t_{2}}^{t}\rho_{m}(\tau) \bar{p}_{m,n-m-1}(\tau ,t_{1})H_{m} ( t,\tau ) \Delta \tau \\& \quad \leq H_{m} ( t,t_{2} ) w_{m} ( t_{2} ) + \int_{t_{2}}^{t}H_{m}^{\Delta_{\tau}} ( t, \tau ) w_{m}^{\sigma } ( \tau ) \Delta\tau + \int_{t_{2}}^{t}H_{m} ( t,\tau ) \rho_{m}^{\Delta }(\tau ) \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma }\Delta\tau \\& \qquad {} -\beta_{m} \int_{t_{2}}^{t}\rho_{m}( \tau)H_{m} ( t,\tau ) \delta^{\sigma}(\tau,t_{1}) \biggl[ \frac{R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(\tau )}{\rho _{m}(\tau)} \biggr) ^{\sigma} \biggr] ^{2}\Delta\tau \\& \quad \leq H_{m} ( t,t_{2} ) w ( t_{2} ) \\& \qquad {} - \int_{t_{2}}^{t} \biggl[ \beta_{m} \rho_{m}(\tau)H_{m} ( t,\tau ) \delta^{\sigma}( \tau,t_{1}) \biggl[ \frac{R_{m,m-1}(\tau ,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(\tau )}{\rho _{m}(\tau)} \biggr) ^{\sigma} \biggr] ^{2} \\& \qquad {} - \bigl( h_{m} ( t,\tau ) \bigr) _{-}\sqrt {H_{m} ( t,\tau ) } \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma } \biggr] \Delta\tau. \end{aligned}$$
Now,
$$\begin{aligned}& \beta_{m}\rho_{m}(\tau)H_{m} ( t,\tau ) \delta ^{\sigma }(\tau,t_{1}) \biggl[ \frac{R_{m,m-1}(\tau,t_{1})}{r_{1}(\tau )} \biggr] ^{1/\alpha_{1}} \biggl[ \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau )} \biggr) ^{\sigma} \biggr] ^{2} \\& \qquad {}- \bigl( h_{m} ( t,\tau ) \bigr) _{-} \sqrt{H_{m} ( t,\tau ) } \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau)} \biggr) ^{\sigma} \\& \quad = \biggl[ \sqrt{\beta_{m}\rho_{m}( \tau)H_{m} ( t,\tau ) \delta ^{\sigma}(\tau,t_{1}) \biggl[ \frac{R_{m,m-1}(\tau ,t_{1})}{r_{1}(\tau)} \biggr] ^{1/\alpha_{1}}} \biggl( \frac{w_{m}(\tau)}{\rho_{m}(\tau )} \biggr) ^{\sigma} \\& \qquad {}- \frac{ ( h_{m} ( t,\tau ) ) _{-}}{2\sqrt{\beta_{m}\rho_{m}(\tau ) \delta^{\sigma}(\tau,t_{1}) [ \frac {R_{m,m-1}(\tau ,t_{1})}{r_{1}(\tau)} ] ^{1/\alpha_{1}}}} \biggr] ^{2} \\& \qquad {}-\frac{ [ ( h_{m} ( t,\tau ) ) _{-} ] ^{2}}{4\beta_{m}\rho_{m}(\tau) \delta^{\sigma}(\tau,t_{1})} \biggl[ \frac{r_{1}(\tau )}{R_{m,m-1}(\tau ,t_{1})} \biggr] ^{1/\alpha_{1}} \\& \quad \geq {}-\frac{ [ ( h_{m} ( t,\tau ) ) _{-} ] ^{2}}{4\beta_{m}\rho_{m}(\tau ) \delta^{\sigma}(\tau,t_{1})} \biggl[ \frac {r_{1}(\tau)}{R_{m,m-1}(\tau,t_{1})} \biggr] ^{1/\alpha_{1}}. \end{aligned}$$
Consequently,
$$\begin{aligned}& \frac{1}{H_{m} ( t,t_{2} ) } \int_{t_{2}}^{t} \biggl[\rho _{m}(\tau) \bar{p}_{m,n-m-1}(\tau,t_{1})H_{m} ( t, \tau ) \\& \quad {}-\frac{ [ ( h_{m} ( t,\tau ) ) _{-} ] ^{2}}{4\beta_{m}\rho_{m}(\tau ) \delta^{\sigma}(\tau,t_{1})} \biggl[ \frac{r_{1}(\tau)}{R_{m,m-1}(\tau,t_{1})} \biggr] ^{1/\alpha_{1}} \biggr] \Delta\tau\leq w_{m} ( t_{2} ) , \end{aligned}$$
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,
$$ \begin{aligned} &(-1)^{k}x^{ [ k ] }(t)>0\quad \text{and} \\ & \bigl( (-1)^{k}x^{ [ k ] }(t) \bigr) ^{\Delta}< 0 \quad \text{for }t\in[ t_{1},\infty )_{\mathbb{T}}\text{ and }k=0,1, \ldots,n-1. \end{aligned} $$
(4.23)
Hence,
$$ \lim_{t\rightarrow\infty}(-1)^{k}x^{ [ k ] }(t)=l_{k} \geq 0 \quad \text{for }k=0,1,\ldots,n-1. $$
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
$$ \phi_{\gamma_{\nu}} \bigl( x \bigl(g_{\nu}(t) \bigr) \bigr) \geq l_{0}^{\gamma _{\nu }}\geq L \quad \text{for }t\in[ t_{2}, \infty)_{\mathbb{T}}, $$
where \(L:=\min_{\nu=0}^{N} \{ l_{0}^{\gamma_{\nu}} \} >0\). Then from (1.1) we obtain
$$ - \bigl( x^{ [ n-1 ] } ( t ) \bigr) ^{\Delta }\geq L\sum _{\nu=0}^{N}p_{\nu} ( t ) =L \hat{p}_{0}(t). $$
Integrating this from t to \(v\in[ t,\infty)_{\mathbb{T}}\), we get
$$ -x^{[n-1]}(v)+x^{[n-1]}(t)\geq L \int_{t}^{v}\hat{p}_{0} ( \tau ) \Delta \tau, $$
and by (4.23) we see that \(x^{[n-1]}(v)>0\). Hence, by taking limits as \(v\rightarrow \infty \) we have
$$ x^{[n-1]}(t)\geq L \int_{t}^{\infty}\hat{p}_{0} ( \tau ) \Delta \tau. $$
If \(\int_{t}^{\infty}\hat{p}_{0} ( \tau ) \Delta\tau =\infty\), then we have reached a contradiction. Otherwise,
$$ \bigl( x^{[n-2]}(t) \bigr) ^{\Delta}\geq L^{1/\alpha_{n-1}} \biggl[ \frac{1}{r_{n-1}(t)} \int_{t}^{\infty}\hat{p}_{0}(\tau)\Delta\tau \biggr] ^{1/\alpha_{n-1}}=L^{1/\alpha_{n-1}} \hat{p}_{1}(t). $$
Integrating this from t to \(v\in[ t,\infty)_{\mathbb{T}}\) and letting \(v\rightarrow\infty\), by (4.23) we get
$$ -x^{[n-2]}(t)\geq L^{1/\alpha_{n-1}} \int_{t}^{\infty}\hat {p}_{1}(\tau ) \Delta \tau. $$
If \(\int_{t}^{\infty}\hat{p}_{1} ( \tau ) \Delta\tau =\infty\), then we have reached a contradiction. Otherwise,
$$ - \bigl( x^{[n-3]}(t) \bigr) ^{\Delta}\geq L^{1/\alpha[ n-2,n-1]} \biggl[ \frac{1}{r_{n-2}(t)} \int_{t}^{\infty}\hat{p}_{1}(\tau )\Delta \tau \biggr] ^{1/\alpha_{n-2}}=L^{1/\alpha[ n-2,n-1]} \hat{p}_{2}(t). $$
Continuing this process, we get
$$ -x^{[1]}(t)\geq L^{1/\alpha[2,n-1]} \int_{t}^{\infty}\hat{p}_{n-2}(\tau) \Delta\tau. $$
If \(\int_{t}^{\infty}\hat{p}_{n-2} ( \tau ) \Delta\tau =\infty\), then we have reached a contradiction. Otherwise,
$$ -x^{\Delta}(t)\geq L^{1/\alpha[1,n-1]} \biggl[ \frac {1}{r_{1}(t)}\int_{t}^{\infty}\hat{p}_{n-2}(\tau)\Delta\tau \biggr] ^{1/\alpha _{1}}=L^{1/\alpha} \hat{p}_{n-1}(t). $$
Again, integrating from \(t_{2}\) to \(t\in[ t_{2},\infty)_{\mathbb{T}}\), we get
$$ -x(t)+x(t_{2})\geq L^{1/\alpha} \int_{t_{2}}^{t}\hat{p}_{n-1}(\tau ) \Delta \tau. $$
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. □

Declarations

Acknowledgements

This work was supported by Research Deanship of Hail University under grant No. 0150287.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, University of Hail, Hail, 2440, Saudi Arabia
(2)
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

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