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

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).

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:

• (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

$$ x^{ [ i ] }:=r_{i} \phi_{\alpha_{i}} \bigl[ \bigl( x^{ [ i-1 ] } \bigr) ^{\Delta} \bigr] , \quad i=1,2,\ldots,n,\text{with }r_{n}=1, \alpha_{n}=1,\text{and }x^{ [ 0 ] }=x. $$

(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 [6–8] 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 [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:

$$ k_{+}:=\max\{k,0\},\qquad k_{-}:=\max\{-k,0\}\quad \text{for any }k\in {\mathbb{R}}, $$

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):

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

• (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:

• (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 $j\in \{0,1,\ldots,n-1 \}$,

$$ \int_{T}^{\infty}\hat{p}_{j}(\tau) \Delta \tau=\infty. $$

(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 $t\in(t_{1},\infty)_{{\mathbb{T}}}$:

• (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 $j=m,m+1,\ldots,n-1$,

$$ \int_{t}^{\infty}\bar{p}_{m,n-j-1}( \tau,t_{1})\Delta\tau< \infty $$

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 [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

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

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. □