An Asymptotic Behavior Property of High-Order Nonlinear Dynamic Equations on Time Scales

Abstract

In this work, by using one dynamic Gronwall–Bihari-type integral inequality on time scales, an interesting asymptotic behavior property of high-order nonlinear dynamic equations on time scales was obtained, which also generalized two classical results belong to Máté and Nevai’s and Agarwal and Bohner’s, respectively.

1. Introduction

Since Stefan Hilger [1] introduced the theory of time scales, which unifies continuous and discrete analysis and extends these theories to intermediate cases, it has gained significant importance and attention in recent decades. This is due to its extensive applications across virtually all scientific disciplines, including statistics, biology, economics, finance, engineering, physics, and operations research. The literature on dynamic differential equations and their applications is extensive; see the monographs by Martin Bohner and Allan Peterson [2,3], as well as Martin Bohner and Svetlin G. Georgiev [4], along with the references cited therein.

The study of asymptotic behavior problems has a long history (Section 1) and has been extensively developed in the context of various types of differential equations, such as ordinary differential equations (ODEs), delay differential equations (DDEs), and dynamic equations on time scales (DEs). For an overview of results related to ODEs up to 2007, we refer readers to the excellent survey paper [5]. Recently, several noteworthy manuscripts [6,7,8,9,10] have emerged, focusing on this topic within the framework of DEs. On the other hand, it is well established that Gronwall-type integral inequalities and their discrete analogues play a crucial role in analyzing the quantitative properties of solutions for differential, integral, and difference equations. In recent years, many authors have investigated Gronwall-type integral inequalities on time scales and their applications; see, for example [11,12,13,14,15,16,17]. In this paper, inspired by the works in [10,18], we utilize a Gronwall–Bihari-type dynamic inequality to establish an interesting asymptotic behavior property of high-order dynamic equations on time scales.

For the convenience of readers, we provide a concise introduction to time-scale calculus in Section 2. For comprehensive details regarding definitions, notation, and theorems related to time scales, we refer readers to the authoritative monographs [2,3] and the references therein. Hereafter, R represents the set of real numbers, $R+=[0,+\infty );$ T denotes an arbitrary time scale, and $C_{rd}$ signifies the set of rd-continuous functions. Throughout this paper, it is assumed that $t_0\in T,T_0=[t_0,+\infty )\cap T.$

2. Some Basic Definitions of Time-Scale Calculus

A time scale T is an arbitrary nonempty closed subset of the real numbers R, which is assumed throughout this paper to be unbounded above since we will consider the asymptotic behavior of solutions near infinity.

On T, the forward operator and backward jump operator are defined by

$$\sigma \left(t\right):=inf\{s\in T:s>t\}and\rho \left(t\right):=sup\{s\in T:s<t\}fort\in T,$$

respectively.

A point $t\in T$ with $t>infT$ is said to be left-dense if $\rho \left(t\right)=t$ and right-dense if $\sigma \left(t\right)=t,$ left-scattered if $\rho \left(t\right)<t$ and right-scattered if $\sigma \left(t\right)>t$.

Next, the graininess function $\mu \left(t\right)$ is defined by $\mu \left(t\right):=\sigma \left(t\right)-t$ for $t\in T.$ For a function $f:T\to R,$ the $(\Delta )$ derivative $f^{\Delta }\left(t\right)$ at $t\in T$ is defined as the number (provided it exists) with a property such that for every $\epsilon >0$ there exists a neighbourhood U of t with

$$|f\left(\sigma \left(t\right)\right)-f\left(s\right)-f^{\Delta }\left(t\right)(\sigma \left(s\right)-s)|\le |\sigma \left(t\right)-s|foralls\in U.$$

For the $\Delta$ derivative, a simple and useful formula is

$$f^{\sigma }=f+\mu f^{\Delta }$$

where $f^{\sigma }:=f\circ \sigma$.

The function $f:T\to R$ is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points. The set of rd-continuous functions $f:T\to R$ usually is denoted by $C_{rd}=C_{rd}\left(T\right)=C_{rd}(T,R)$.

For $a,b\in T$ and a function $f:T\to R$, the Cauchy integral of f is defined by

$${\int }_a^{b}f\left(t\right)\Delta t=F\left(b\right)-F\left(a\right)$$

where $F^{\Delta }\left(t\right)=f\left(t\right)$; i.e., F is an antiderivative of f.

The Hilger complex numbers are defined by $C_h:=\{z\in C:z\ne -{\displaystyle \frac{1}{h}},h>0\}$; $Z_h:=\{z\in C:-{\displaystyle \frac{\pi }{h}}<Im\left(z\right)\le {\displaystyle \frac{\pi }{h}},h>0\}$; and for $h=0,Z_0:=C$.

By Hilger’s main existence theorem ([4], Theorem 1.74), rd-continuous functions possess antiderivatives. If $p:T\to R$ is rd-continuous and regressive (i.e., $1+\mu \left(t\right)p\left(t\right)\ne 0$ for all $t\in T)$, then another existence theorem says that the initial value problem $y^{\Delta }=p\left(t\right)y,$ $y\left(t_0\right)=1$ (where $t_0\in T)$, possesses a unique solution $e_p(·,t_0)$. The set of all rd-continuous and regressive functions is denoted by $C_{rd}$.

3. Some Lemmas and Main Results

Lemma 1

([15]). Let $\mathrm{T}$ be an unbounded time scale, $t,t_0\in \mathrm{T}$; and let $u\left(t\right),a\left(t\right),b\left(t\right)$ be nonnegative rd-continuous functions defined for $t\in \mathrm{T}$. Assume that $a\left(t\right)$ is nondecreasing for $t\in \mathrm{T}$ and $0<r\le 1$. If for $t\in \mathrm{T}$ we have

$$u\left(t\right)\le a\left(t\right)+{\int }_{t_0}^{t}b\left(s\right)u^r\left(s\right)\Delta s,$$

then

$$u\left(t\right)\le \left\{\begin{array}{ccccc}a\left(t\right)e_b(t,t_0),r=1,\hfill & & & & \\ & & & & \\ {\displaystyle a\left(t\right){\left[1+(1-r){\int }_{t_0}^{t}b\left(s\right)\Delta s\right]}^{\frac{1}{1-r}},0<r<1,}\hfill & & & & \end{array}\right.$$

where ${\displaystyle e_b(t,t_0)=exp\left\{{\int }_{t_0}^t{\xi }_{\mu \left(s\right)}\left(b\left(s\right)\right)\Delta s\right\},}$ and the cylinder transformation ${\xi }_h:C_h\to Z_h$ defined by

$${\xi }_h\left(z\right)={\displaystyle \frac{1}{h}}Log(1+zh),$$

where Log is the principal logarithm function.

Lemma 2.

For any rd-continuous nonnegative function $b\left(t\right)$, we have inequalitys

$$e_b(t,t_0)\le exp{\int }_{t_0}^{t}b\left(s\right)\Delta s.$$

Proof. 

By the representation ([2], (2.15)), we have

$$e_b(t,t_0)=exp\left\{{\int }_{t_0}^t{\xi }_{\mu \left(s\right)}\left(b\left(s\right)\right)\Delta s\right\}.$$

If $\mu \left(s\right)=0$, it follows that

$${\xi }_{\mu \left(s\right)}\left(b\left(s\right)\right)=b\left(s\right);$$

If $\mu \left(s\right)>0$, we have

$${\xi }_{\mu \left(s\right)}\left(b\left(s\right)\right)={\displaystyle \frac{Log(1+\mu (s\left)b\right(s\left)\right)}{\mu \left(s\right)}}={\displaystyle \frac{log(1+\mu (s\left)b\right(s\left)\right)}{\mu \left(s\right)}}$$

$$=b\left(s\right)-{\displaystyle \frac{\mu \left(s\right)b\left(s\right)-log(1+\mu (s\left)b\right(s\left)\right)}{\mu \left(s\right)}}.$$

(1)

Setting $f\left(x\right)=x-log(1+x)$ for $x>-1$, from (1) we obtain that

$${\xi }_{\mu \left(s\right)}\left(b\left(s\right)\right)=b\left(s\right)-{\displaystyle \frac{f\left(\mu \right(s\left)b\right(s\left)\right)}{\mu \left(s\right)}}\le b\left(s\right),$$

since $f\left(x\right)\ge 0$ for $x>-1$. The proof is completed. □

Theorem 1.

Let $I=[1,\infty ),T_1=T\cap I,$$n\in N^{+}$, rd-continuous functions $p_i:T_1\to R_{+},0\le i\le n-1$. If for  $t\in T_1$,  $r_i(i=0,1,2,...,n-1)$ are constants with $0<r_i\le 1$, a function y is n times differentiable on $T_1^{{\kappa }^n}$ and we assume that

$$|y^{{\Delta }^n}\left(t\right)|\le \sum _{i=0}^{n-1}{p}_i\left(t\right){|y^{{\Delta }^i}\left(t\right)|}^{r_i}$$

(2)

and

$${\displaystyle {\int }_1^{+\infty }{s}^{(n-i-1)r_i}{p}_i\left(s\right)\Delta s<+\infty .}$$

(3)

Then, there exists $\gamma >0$ such that

(i)

$|y^{{\Delta }^k}\left(t\right)|\le \gamma t^{n-k-1},k=0,1,2,\dots ,n-1;$

(ii)

${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{n-1}}\left(t\right)exists.}$

Proof. 

From (2), for any $t\ge t_0\ge 1$, we have

$$|y^{{\Delta }^{n-1}}\left(t\right)-y^{{\Delta }^{n-1}}\left(t_0\right)|\le \sum _{i=0}^{n-1}{\int }_{t_0}^t{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s,$$

(4)

from which it follows that for $t_0=1$,

$$|y^{{\Delta }^{n-1}}\left(t\right)|\le |y^{{\Delta }^{n-1}}\left(1\right)|+\sum _{i=0}^{n-1}{\int }_1^t{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s$$

$$=\left[|y^{{\Delta }^{n-1}}\left(1\right)|+\sum _{i=0}^{n-2}{\int }_1^t{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s\right]+{\int }_1^t{p}_{n-1}\left(s\right){|y^{{\Delta }^{n-1}}\left(s\right)|}^{r_{n-1}}\Delta s.$$

Without loss of generality, we assume that $|y^{{\Delta }^{n-1}}\left(1\right)|\ge 1$; from the last inequality and by Lemmas 1 and 2, we obtain that

$$|y^{{\Delta }^{n-1}}\left(t\right)|\le \left[|y^{{\Delta }^{n-1}}\left(1\right)|+\sum _{i=0}^{n-2}{\int }_1^t{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s\right]G_1\left(t\right),$$

(5)

where

$$G_1\left(t\right)=\left\{\begin{array}{ccccc}{\displaystyle exp{\int }_1^t{p}_{n-1}\left(s\right)\Delta s,r_{n-1}=1,}\hfill & & & & \\ & & & \\ {\displaystyle {\left[1+(1-r_{n-1}){\int }_1^t{p}_{n-1}\left(s\right)\Delta s\right]}^{\frac{1}{1-r_{n-1}}},0<r_{n-1}<1.}\hfill & & & & \end{array}\right.$$

(6)

From (5), (6), and condition (3), we have

$$|y^{{\Delta }^{n-1}}\left(t\right)|\le K_1+\sum _{i=0}^{n-2}{\int }_1^t{M}_1{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s,$$

(7)

where

$$K_1=|y^{{\Delta }^{n-1}}\left(1\right)|M_1,M_1=G_1(+\infty ).$$

Integrating (7) from 1 to $t,t\ge 1$ and using the change of order integration formula ([8], Lemma 2.1), we obtain that

$$|y^{{\Delta }^{n-2}}\left(t\right)|\le |y^{{\Delta }^{n-2}}\left(1\right)|+K_{1}t+t\sum _{i=0}^{n-2}{\int }_1^t{M}_1{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s,$$

from which it follows that

$${\displaystyle \frac{|y^{{\Delta }^{n-2}}\left(t\right)|}{t}}\le \left(|y^{{\Delta }^{n-2}}\left(1\right)|+K_1\right)+\sum _{i=0}^{n-3}{\int }_1^t{M}_1{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+$$

$$+{\int }_1^t{M}_1{s}^{r_{n-2}}{p}_{n-2}\left(s\right){\left({\displaystyle \frac{|y^{{\Delta }^{n-2}}\left(s\right)|}{s}}\right)}^{r_{n-2}}\Delta s.$$

Using Lemmas 1 and 2 to the last inequality again, we have

$${\displaystyle \frac{|y^{{\Delta }^{n-2}}\left(t\right)|}{t}}\le \left[|y^{{\Delta }^{n-2}}\left(1\right)|+K_1+\sum _{i=0}^{n-3}{\int }_1^t{M}_1{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s\right]G_2\left(t\right),$$

(8)

where

$$G_2\left(t\right)=\left\{\begin{array}{ccccc}{\displaystyle exp{\int }_1^t{M}_1{s}^{r_{n-2}}{p}_{n-2}\left(s\right)\Delta s,r_{n-2}=1,}\hfill & & & & \\ & & & & \\ {\displaystyle {\left[1+(1-r_{n-2}){\int }_1^t{M}_1{s}^{r_{n-2}}{p}_{n-2}\left(s\right)\Delta s\right]}^{\frac{1}{1-r_{n-2}}},0<r_{n-2}<1.}\hfill & & & & \end{array}\right.$$

Inequality (8) implies that

$$|y^{{\Delta }^{n-2}}\left(t\right)|\le K_{2}t+\sum _{i=0}^{n-3}t{\int }_1^t{M}_2{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s,$$

where

$$K_2=(|y^{{\Delta }^{n-2}}\left(1\right)|+K_1)G_2(+\infty ),M_2=M_1{G}_2(+\infty ).$$

By mathematical induction, we derive that

$$|y^{{\Delta }^{n-j}}\left(t\right)|\le K_j{t}^{j-1}+\sum _{i=0}^{n-j-1}{t}^{j-1}{\int }_1^t{M}_j{p}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s,$$

(9)

where $K_j$ and $M_j$ are constants, $j=1,2,\dots ,n-1$. Especially for $j=n-1$, we have

$$|y^{\Delta }\left(t\right)|\le K_{n-1}{t}^{n-2}+t^{n-2}{\int }_1^t{M}_{n-1}{p}_0\left(s\right){|y\left(s\right)|}^{r_0}\Delta s.$$

Integrating this inequality from 1 to $t,t\ge 1$, and using the change of order integration formula again, we can obtain that

$$|y\left(t\right)|\le K_0{t}^{n-1}+t^{n-1}{\int }_1^t{M}_{n-1}{p}_0\left(s\right){|y\left(s\right)|}^{r_0}\Delta s$$

where $K_0$ is a suitable constant. The last inequality can be re-written as

$${\displaystyle \frac{|y(t)|}{t^{n-1}}}\le K_0+{\int }_1^t{M}_{n-1}{p}_0\left(s\right)s^{(n-1)r_0}{\left({\displaystyle \frac{|y(s)|}{s^{n-1}}}\right)}^{r_0}\Delta s$$

Using Lemmas 1 and 2 to the last inequality, we have

$$|y\left(t\right)|\le M_0{t}^{n-1},$$

where

$$M_0=\left\{\begin{array}{ccccc}{\displaystyle K_{0}exp{\int }_1^{+\infty }{M}_1{s}^{(n-1)r_0}{p}_0\left(s\right)\Delta s,r_0=1,}\hfill & & & & \\ & & & \\ {\displaystyle K_0{\left[1+(1-r_0){\int }_1^{+\infty }{M}_1{s}^{(n-1)r_0}{p}_0\left(s\right)\Delta s\right]}^{\frac{1}{1-r_0}},0<r_0<1.}\hfill & & & & \end{array}\right.$$

(10)

From (9) and (10), we can derive that

$$|y^{{\Delta }^k}\left(t\right)|\le a_k{t}^{n-k-1},k=0,1,2,\dots ,n-1.$$

(11)

where $a_k,(k=0,1,2,\dots ,n-1)$ are some constants.

Set ${\displaystyle \gamma =\underset{0\le k\le n-1}{max}\left\{a_k\right\}}$; from (1), we have proved (i).

By condition (3), in combination with (4) and (11), we obtain that

$$\underset{t,t_0\to +\infty }{lim}|y^{{\Delta }^{n-1}}\left(t\right)-y^{{\Delta }^{n-1}}\left(t_0\right)|=0.$$

From the Cauchy criterion [4], it follows that ${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{n-1}}\left(t\right)}$ exists. □

By Theorem 1, we can easily obtain the following corollary.

Corollary 1.

Consider the initial value problem

$$y^{{\Delta }^n}=f(t,y,y^{\Delta },\dots ,y^{{\Delta }^{n-1}}),y^{{\Delta }^i}\left(1\right)={\alpha }_{i}for0\le i\le n-1$$

(12)

where ${\alpha }_i$ are some constants; $f:T\times R^n\to R$ is supposed to satisfy

$$|f(t,u_0,\dots ,u_{n-1})|\le \sum _{i=0}^{n-1}{p}_i\left(t\right){|u_i|}^{r_i}$$

(13)

for all $t\in T_1,$$\{u_i:0\le i\le n-1\}\subset R;$$0<r_i\le 1,p_i\left(t\right)(i=0,1,\dots ,n-1)$ are defined as in Theorem 1 and satisfy condition (6). Then, there exists $\gamma >0$ such that for every solution y of (12) it satisfies

(i)

$$|y^{{\Delta }^k}\left(t\right)|\le \gamma t^{n-k-1},k=0,1,2,\dots ,n-1;$$

(14)

and (ii)

$${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{n-1}}\left(t\right)exists.}$$

Remark 1.

When $T=R,$$r_i=1,i=0,1,\dots ,n-1$, from Theorem 1, we can obtain a main result of Máté and Nevai’s ([19], Lemma 2); when $T=Z,$$r_i=1,i=0,1,\dots ,n-1$, with some suitable conditions, we can obtain another main result of Máté and Nevai’s ([19], Lemma 6).

Remark 2.

When $r_i=1,i=0,1,\dots ,n-1$, from (14), we can obtain a similar result to that of Agarwal and Bohner ([18], Theorem 7) under some simpler conditions on $p_i\left(t\right)(i=0,1,\dots ,n-1)$.

Lemma 3.

Let T be an unbounded time scale, $t,t_0\in T$; and let $u\left(t\right),b\left(t\right)$ be nonnegative rd-continuous functions defined for $t\in T$. Assume that k and r are positive constants with $0<r\le 1$. If for $t\in T$ we have

$$u\left(t\right)\le k+{\int }_t^{+\infty }b\left(t\right)u^r\left(t\right)\Delta t<+\infty ,$$

then

$$u\left(t\right)\le \left\{\begin{array}{ccccc}kexp{\displaystyle {\int }_t^{+\infty }b\left(t\right)\Delta t},r=1,\hfill & & & & \\ & & & & \\ {\displaystyle {\left\{k^{1-r}+(1-r){\int }_t^{+\infty }b\left(t\right)\Delta t\right\}}^{\frac{1}{1-r}},0<r<1.}\hfill & & & & \end{array}\right.$$

Proof. 

The proof is similar to the result of Anderson ([12], Lemma 2.2) in one-variable. We omit the detail here. □

Theorem 2.

Assume that $y^{{\Delta }^k}\left(t\right)$ are rd-continuous on $T_1$ for $k\ge 1$, $y^{{\Delta }^k}\left(t\right)$ are bounded on $T_1$, and $y\left(t\right)$ satisfies

$$|y^{{\Delta }^k}\left(t\right)|\le {\int }_t^{+\infty }\sum _{i=0}^k{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+{\int }_t^{+\infty }p\left(s\right)\Delta s,t\ge 1,$$

(15)

where rd-continuous functions $q_i,p:T_1\to R^{+},0\le i\le k,0<r_i\le 1(i=0,1,\dots ,k)$,

$${\int }_1^{+\infty }{q}_i\left(s\right)s^{k-i}\Delta s<\infty$$

and

$$p\left(t\right)\equiv 0or{\int }_1^{+\infty }s(p\left(s\right)+q_k\left(s\right))\Delta s<\infty ;$$

(16)

then, ${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{k-1}}\left(t\right)=y^{{\Delta }^{k-1}}(+\infty )}$ exists and

$$|y^{{\Delta }^{k-1}}\left(t\right)-y^{{\Delta }^{k-1}}(+\infty )|\le K_1{\int }_t^{+\infty }\sum _{i=0}^{k-1}s{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+$$

$$+K_1{\int }_t^{+\infty }s(p\left(s\right)+q_k\left(s\right))\Delta s.$$

(17)

Proof. 

(I) When $0<r_k<1$, by condition (16), ${\displaystyle {\int }_1^{+\infty }{q}_k\left(s\right)\Delta s<+\infty }$. We can choose $t_0$ to be large enough such that

$${\int }_{t_0}^{+\infty }{q}_k\left(s\right)\Delta s<1.$$

Now, for any $t\ge t_0>1$, from (15), we have

$$|y^{{\Delta }^k}\left(t\right)|\le {\int }_{t_0}^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+{\int }_{t_0}^{+\infty }p\left(s\right)\Delta s+$$

$$+{\int }_t^{+\infty }{q}_k\left(s\right){|y^{{\Delta }^k}\left(s\right)|}^{r_k}\Delta s$$

(15)′

The infinite integrals on the right of the last inequality are convergent by condition (16), since $y^{{\Delta }^k}\left(t\right)$ was assumed to be bounded on $T_1$ and so $y^{{\Delta }^i}\left(t\right)=O\left(t^{k-i}\right)$.

Using Lemma 3 in ${\left(15\right)}^{\prime }$, we obtain that

$$|y^{{\Delta }^k}\left(t\right)|\le {\left\{{\left[{\int }_{t_0}^{+\infty }(\sum _{i=0}^{k-1}{q}_i\left(s\right)|y^{{\Delta }^i}\left(s\right){|}^{r_i}+p\left(s\right))\Delta s\right]}^{1-r_k}+(1-r_k){\int }_t^{+\infty }{q}_k\left(s\right)\Delta s\right\}}^{{\displaystyle \frac{1}{1-r_k}}}$$

$$\le 2^{{\displaystyle \frac{r_k}{1-r_k}}}\left[{\int }_{t_0}^{+\infty }(\sum _{i=0}^{k-1}{q}_i\left(s\right)|y^{{\Delta }^i}\left(s\right){|}^{r_i}+p\left(s\right))\Delta s+{\int }_t^{+\infty }{q}_k\left(s\right)\Delta s\right].$$

Here, we have used the elementary inequality ${(X+Y)}^{\gamma }\le 2^{\gamma -1}(X^{\gamma }+Y^{\gamma }),X,Y\ge 0,\gamma \ge 1$ in the last inequality.

Setting $t=t_0$ in the last inequality, we obtain

$$|y^{{\Delta }^k}\left(t\right)|\le K_1{\int }_t^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+K_1{\int }_t^{+\infty }(p\left(s\right)+q_k\left(s\right))\Delta s$$

where $K_1=2^{{\displaystyle \frac{r_k}{1-r_k}}}$, from which it follows by integrating that

$$|y^{{\Delta }^{k-1}}\left(t\right)-y^{{\Delta }^{k-1}}\left(t_0\right)|\le K_1{\int }_{t_0}^t{\int }_s^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(x\right){|y^{{\Delta }^i}\left(x\right)|}^{r_i}\Delta x\Delta s+$$

$$+K_1{\int }_{t_0}^t{\int }_s^{+\infty }(p\left(x\right)+q_k\left(x\right))\Delta x\Delta s.$$

(18)

Using the time-scale change of order integration formula [8], we obtain

$${\int }_{t_0}^t{\int }_s^{+\infty }(p\left(x\right)+q_k\left(x\right))\Delta x\Delta s\le {\int }_{t_0}^{+\infty }{\int }_s^{+\infty }(p\left(x\right)+q_k\left(x\right))\Delta x\Delta s$$

$$\le {\int }_{t_0}^{+\infty }x(p\left(x\right)+q_k\left(x\right))\Delta x<+\infty ,$$

and using this in (18), it follows that for $t\ge t_0\ge 1$,

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le K^{*}+K_1{\int }_{t_0}^t{\int }_s^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(x\right){|y^{{\Delta }^i}\left(x\right)|}^{r_i}\Delta x\Delta s,$$

(19)

where ${\displaystyle K^{*}=|y^{{\Delta }^{k-1}}\left(t_0\right)|+K_1{\int }_{t_0}^{+\infty }x(p\left(x\right)+q_k\left(x\right))\Delta x}$.

As $|y^{{\Delta }^k}\left(t\right)|$ is bounded, we have

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le A_{0}t+A_1$$

(20)

for some constants $A_0$ and $A_1$ (we may as well set $A_1>1)$. Integrating (20) $0,1,2,\dots ,k-1$ times, we can obtain

$$|y^{{\Delta }^i}\left(t\right)|\le A_0{t}^{k-i}+A_1{t}^{k-i-1}+A_2{t}^{k-i-2}(0\le i\le k-1,t\ge t_0)$$

where $A_2$ is a nonnegative constant and depends on $y^{{\Delta }^j}\left(t_0\right)(j=0,1,2,\dots ,k-2$). And so we have (note that $A_1>1$ and $t_0\ge 1$)

$$|y^{{\Delta }^i}{\left(t\right)|}^{r_i}\le {(A_0{t}^{k-i}+A_1{t}^{k-i-1}+A_2{t}^{k-i-2})}^{r_i}\le A_0{t}^{k-i}+A_1{t}^{k-i-1}+A_2{t}^{k-i-2}.$$

Substituting this into (19) and making an extension of the domains of integration, we obtain

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le K^{*}+K_1{\int }_{t_0}^t{\int }_s^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(x\right)(A_0{x}^{k-i}+A_1{x}^{k-i-1}+A_2{x}^{k-i-2})\Delta x\Delta s$$

$$\le K^{*}+K_1{\int }_{t_0}^t{\int }_{t_0}^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(x\right)A_0{x}^{k-i}\Delta x\Delta s+$$

$$+K_1{\int }_{t_0}^{+\infty }{\int }_s^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(x\right)(A_1{x}^{k-i-1}+A_2{x}^{k-i-2})\Delta x\Delta s.$$

(21)

We set

$$I\left(t_0\right)=K_1{\int }_{t_0}^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(x\right)x^{k-i}\Delta x;$$

the integral is finite by condition (16). Interchanging the order of integrations in (21), (21) can be re-written as

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le K^{*}+A_{0}I\left(t_0\right)t+(A_1+A_2)I\left(t_0\right).$$

(22)

We should note that (22) is the same type inequality as (20). In the same way as we obtain (22) from (20), we can use (22) to obtain a new inequality of the same kind. Let $t_0$ be large enough such that $I\left(t_0\right)<1$, and iterate this procedure n times; then, we can obtain

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le A_0{\left[I\left(t_0\right)\right]}^{n}t+A_1{\left[I\left(t_0\right)\right]}^n+A_2\left[\sum _{i=0}^n{I}^i\left(t_0\right)\right]+K^{*}\left[\sum _{i=0}^{n-1}{I}^i\left(t_0\right)\right]$$

$$\le A_0{\left[I\left(t_0\right)\right]}^{n}t+A_1{\left[I\left(t_0\right)\right]}^n+\left(A_2+K^{*}\right)\sum _{i=0}^n{I}^i\left(t_0\right)$$

Letting $n\to +\infty$ in the last inequality, we obtain

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le {\displaystyle \frac{A_1+K^{*}}{1-I\left(t_0\right)}},$$

which implies that $y^{{\Delta }^{k-1}}\left(t\right)$ is bounded, and so we have

$$|y^{{\Delta }^{k-1}}\left(t\right)|\le H(t\ge t_0>1)$$

where $H>1$ is a constant. Integrating this $0,1,2,\dots ,k-1$ times, we obtain that

$$|y^{{\Delta }^i}\left(t\right)|\le H{t}^{k-i-1}+H^{\prime }{t}^{k-i-2}(0\le i\le k-1)$$

(23)

where $H^{\prime }$ is a nonnegative constant. From (23), we have

$$|y^{{\Delta }^i}{\left(t\right)|}^{r_i}\le {\left(H{t}^{k-i-1}+H^{\prime }{t}^{k-i-2}\right)}^{r_i}\le H{t}^{k-i-1}+H^{\prime }{t}^{k-i-2},(0\le i\le k-1).$$

(24)

Substituting (24) into (18), then extending the upper limit of the outer integrals on the right side to $+\infty$, and then interchanging the order of integrations, it follows from condition (16) that

$$\underset{t,t_0\to +\infty }{lim}|y^{{\Delta }^{k-1}}\left(t\right)-y^{{\Delta }^{k-1}}\left(t_0\right)|=0,$$

which shows that ${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{k-1}}\left(t\right)}$ exists by the Cauchy criterion [4].

Because ${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{k-1}}\left(t\right)}$ exists, (17) follows from (18) by letting $t\to +\infty$ and then interchanging the order of integrations.

(II) When $r_k=1$, the inequality ${\left(15\right)}^{\prime }$ becomes

$$|y^{{\Delta }^k}\left(t\right)|\le {\int }_{t_0}^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+{\int }_{t_0}^{+\infty }p\left(s\right)\Delta s+$$

$$+{\int }_t^{+\infty }{q}_k\left(s\right)|y^{{\Delta }^k}\left(s\right)|\Delta s$$

(15)″

Now, using Lemma 2 in ${\left(15\right)}^{″}$ again, we obtain that

$$|y^{{\Delta }^k}\left(t\right)|\le {\overline{K}}_1\left({\int }_{t_0}^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+{\int }_{t_0}^{+\infty }p\left(s\right)\Delta s\right),$$

(25)

where ${\displaystyle {\overline{K}}_1={\int }_1^{+\infty }{q}_k\left(t\right)\Delta t}$.

Setting $t=t_0$ in (25), we can obtain

$$|y^{{\Delta }^k}\left(t\right)|\le {\overline{K}}_1\left({\int }_t^{+\infty }\sum _{i=0}^{k-1}{q}_i\left(s\right){|y^{{\Delta }^i}\left(s\right)|}^{r_i}\Delta s+{\int }_t^{+\infty }p\left(s\right)\Delta s\right).$$

(26)

Based on (26), following a similar procedure from (18) to (24), we will derive the desired result as (I); and the condition ${\displaystyle {\int }_1^{+\infty }s(p\left(s\right)+q_k\left(s\right))\Delta s<\infty }$ is also to be simply replaced by ${\displaystyle {\int }_1^{+\infty }sp\left(s\right)\Delta s<\infty }$ in the process. We omit the details here. □

Remark 3.

When $T=\mathrm{R},r_i=1,i=0,1,\dots ,n-1,p\left(t\right)\equiv 0$, from Theorem 2, we can obtain a main result of Máté and Nevai’s ([19], Lemma 3); when $\mathrm{T}=\mathrm{N},r_i=1,i=0,1,\dots ,n-1,p\left(t\right)\equiv 0$ we can obtain one another main result of Máté and Nevai’s ([19], Lemma 7) under some suitable conditions.

4. Some Examples

Example 1.

Consider the equation 

$$y^{{\Delta }^n}\left(t\right)=\sum _{i=0}^{n-1}{\displaystyle \frac{i+1}{t^{\frac{n-i-1}{i+2}}+i+2}}sin\left(y^{{\Delta }^i}+i\right){\left(y^{{\Delta }^i}\left(t\right)\right)}^{{\displaystyle \frac{1}{i+2}}},y^{{\Delta }^{i-1}}\left(1\right)={\alpha }_i,$$

(27)

for $t\ge 1,0\le i\le n-1$, where  ${\alpha }_i(i=0,1,\dots ,n-1)$ are some constants. Note that 

$$f(t,u_0,\dots ,u_{n-1})=\sum _{i=0}^{n-1}{\displaystyle \frac{i+1}{t^{\frac{n-i-1}{i+1}+2}+i+2}}sin\left(u_i+i\right){\left(u_i\right)}^{{\displaystyle \frac{1}{i+2}}},$$

and 

$$|f(t,u_0,\dots ,u_{n-1})|=|\sum _{i=0}^{n-1}{\displaystyle \frac{i+1}{t^{\frac{n-i-1}{i+2}+2}+i+2}}sin\left(u_i+i\right){\left(u_i\right)}^{{\displaystyle \frac{1}{i+2}}}|$$

$$\le \sum _{i=0}^{n-1}|{\displaystyle \frac{i+1}{t^{\frac{n-i-1}{i+2}+2}+i+2}}sin\left(u_i+i\right){\left(u_i\right)}^{{\displaystyle \frac{1}{i+2}}}|$$

$$\le \sum _{i=0}^{n-1}{\displaystyle \frac{1}{t^{\frac{n-i-1}{i+2}+2}}}{|u_i|}^{{\displaystyle \frac{1}{i+2}}};$$

i.e., we have

$$|y^{{\Delta }^n}\left(t\right)|\le \sum _{i=0}^{n-1}{\displaystyle \frac{1}{t^{\frac{n-i-1}{i+2}+2}}}{|u_i|}^{{\displaystyle \frac{1}{i+2}}}.$$

Because of

$${\int }_1^{+\infty }{s}^{(n-i-1)r_i}{p}_i\left(s\right)\Delta s={\int }_1^{+\infty }{s}^{(n-i-1){\displaystyle \frac{1}{i+2}}}{\displaystyle \frac{1}{s^{\frac{n-i-1}{i+2}+2}}}\Delta s={\int }_1^{+\infty }{\displaystyle \frac{1}{s^2}}\Delta s.$$

From ([3], Theorems 5.64 and 5.65)), ${\displaystyle {\int }_1^{+\infty }\frac{1}{s^2}\Delta s<+\infty }$ for many time scales. Thus, all assumptions of Corollary 1 are satisfied, and the solution $y\left(t\right)$ of (27) satisfies

(i)

$$|y^{{\Delta }^k}\left(t\right)|\le \gamma t^{n-k-1},k=0,1,2,\dots ,n-1;$$

and (ii)

$${\displaystyle \underset{t\to +\infty }{lim}{y}^{{\Delta }^{n-1}}\left(t\right)exists,}$$

where $\gamma >0$ is a constant.

Example 2.

As an application of Theorem 2, consider an integro-differential equation

$$y^{\prime }\left(t\right)={\int }_t^{+\infty }{s}^{-\beta }{y}^{1/3}\left(s\right)ds+t^{2-\beta },\beta >4,t\ge 1.$$

From this, we can obtain that

$$|y^{\prime }\left(t\right)|\le {\int }_t^{+\infty }{s}^{-\beta }{|y\left(s\right)|}^{1/3}ds+t^{2-\beta }.$$

(28)

By (28), we observe that the solution $y\left(t\right)$ satisfies all conditions of Theorem 2; it follows that

$${\displaystyle \underset{t\to +\infty }{lim}y\left(t\right)=y(+\infty )exists,}$$

and

$$|y\left(t\right)-y(+\infty )|\le K_1{\int }_t^{+\infty }{s}^{1-\beta }{|y\left(s\right)|}^{1/3}ds+K_1{\int }_t^{+\infty }s\left(s^{-\beta }+s^{2-\beta }\right)ds.$$

By a simple computation, the last inequality can be re-written as

$$|y\left(t\right)|\le |y(+\infty )|+K_1\left({\displaystyle \frac{1}{\beta -2}}{t}^{2-\beta }+{\displaystyle \frac{1}{\beta -3}}{t}^{3-\beta }\right)+K_1{\int }_t^{+\infty }{s}^{1-\beta }{|y\left(s\right)|}^{1/3}ds$$

$$\le |y(+\infty )|+K_1\left({\displaystyle \frac{1}{\beta -2}}+{\displaystyle \frac{1}{\beta -3}}\right)+K_1{\int }_t^{+\infty }{s}^{1-\beta }{|y\left(s\right)|}^{1/3}ds.$$

Using Lemma 3 to the last inequality, we can obtain a $prior$ bound of $y\left(t\right)$ as follows

$$|y\left(t\right)|\le {\left\{{\left({\overline{K}}^{*}\right)}^{2/3}+{\displaystyle \frac{K_1}{\beta -2}}{t}^{2-\beta }\right\}}^{3/2},$$

where ${\overline{K}}^{*}=|y(+\infty )|+K_1\left({\displaystyle \frac{1}{\beta -2}}+{\displaystyle \frac{1}{\beta -3}}\right)$.

5. Conclusions

In this manuscript, we employed several useful nonlinear integral dynamic inequalities on time scales and combined them with advanced analysis techniques in time-scale calculus to investigate the asymptotic behavior of a class of high-order nonlinear dynamic equations. Our main theorems not only generalize some well-known previous results but also demonstrate the effectiveness of nonlinear integral inequalities as a powerful tool for qualitative analysis in various types of differential equations.