Partial Variable Stability for a Class of Nonlinear Systems with Time Delay

Abstract

This article first gives a new class of integral inequalities. Then, as an application, the nonlinear neutral differential system with multiple delays is considered, and the trivial solution of the nonlinear neutral system with multiple delays is obtained. Uniform asymptotic Lipschitz stability. Obviously, the above system is a generalization of the traditional differential system. The purpose of this paper is to study the dual stability of neutral differential equations with delays, including equal asymptotically Lipschitz stability and uniformly asymptotic Lipschitz stability. The author uses the method of integral inequality to establish a double stability criterion. As a result, the local stability of differential equations is widely used in theory and practice, such as dynamic systems and control systems.

Share and Cite:

Huo, R. and Wang, X. (2020) Partial Variable Stability for a Class of Nonlinear Systems with Time Delay. Applied Mathematics, 11, 377-388. doi: 10.4236/am.2020.115027.

1. Introduction

In 1892, Lyapunov, a Russian mathematician, mechanician and physicist, proposed the notion of the stability of motion. He gave the general research methods in his doctoral dissertation “The general problem of the stability of motion” , in which he established the foundation of the stability theory. When studying nonlinear systems, especially studying dynamic systems or control systems, we cannot study the stability of all variables because of the technology difficulties, the limitation of practical conditions, or it is not necessary to study all variables considering the actual need. As a result, studying the partial stability of differential equations becomes more important. In addition, the partial stability is widely used in science and technology. For instance the absolute stability of famous Lurie adjusting systems can be changed into a problem of partial stability. In a word, it is of practical significance to study the partial stability of differential equations.

Since Bellman created a class of integral inequalities in 1958, integral inequalities have been greatly developed. The main results are:

In 1960, Li Yuesheng gave the following inequality in :

$u\left(t\right)\le {u}_{0}+{\int }_{0}^{t}g\left(s\right)u\left(s\right)\text{d}s+{\int }_{0}^{t}f\left(s\right){u}^{\alpha }\left(s\right)\text{d}s$

In 2005, Sligeng discussed the following inequality in :

$\begin{array}{c}{u}_{1}\left(t\right)\le {k}_{1}+{\int }_{0}^{t}{h}_{1}\left(s\right){u}_{1}\left(s\right)\text{d}s+{\int }_{0}^{t}{h}_{2}\left(s\right){u}_{2}\left(s\right){\text{e}}^{\mu s}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{t}{\stackrel{¯}{h}}_{1}\left(s\right){u}_{1}^{\alpha }\left(s\right){\text{e}}^{-\left(\alpha -1\right)\mu s}\text{d}s+{\int }_{0}^{t}{\stackrel{¯}{h}}_{2}\left(s\right){u}_{2}^{\alpha }\left(s\right){\text{e}}^{\mu s}\text{d}s\end{array}$

$\begin{array}{c}{u}_{2}\left(t\right)\le {k}_{2}+{\int }_{0}^{t}{h}_{3}\left(s\right){u}_{1}\left(s\right){\text{e}}^{-\mu s}\text{d}s+{\int }_{0}^{t}{h}_{4}\left(s\right){u}_{2}\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{0}^{t}{\stackrel{¯}{h}}_{3}\left(s\right){u}_{1}^{\alpha }\left(s\right){\text{e}}^{-\alpha \mu s}\text{d}s+{\int }_{0}^{t}{\stackrel{¯}{h}}_{4}\left(s\right){u}_{2}^{\alpha }\left(s\right)\text{d}s\end{array}$

In 2009, the author discussed a new class of inequalities in .

Vorotnikov, V.I.   considered the following system

$\left\{\begin{array}{l}\frac{\text{d}y}{\text{d}t}=A\left(t\right)y+B\left(t\right)z+Y\left(t,y,z\right)\\ \frac{\text{d}z}{\text{d}t}=C\left(t\right)y+D\left(t\right)z+Z\left(t,y,z\right)\end{array}$

and studied the double stability as $‖y‖+‖z‖\to 0$ and $\frac{‖Y\left(t,y,z\right)‖+‖Z\left(t,y,z\right)‖}{‖y‖+‖z‖}\to 0$.

In 2002, Wang Feng used the differential inequality of delay in article  to study the following delay system:

$\left\{\begin{array}{l}\frac{\text{d}y}{\text{d}t}=A\left(t\right)y+B\left(t\right)z+Y\left(t,y,z,y\left(t-\tau \right),z\left(t-\tau \right)\right)\\ \frac{\text{d}z}{\text{d}t}=C\left(t\right)y+D\left(t\right)z+Z\left(t,y,z,y\left(t-\tau \right),z\left(t-\tau \right)\right)\end{array}$

In 2006, Siligeng used the integral inequality extended in  in  to discuss the double stability of the following system to some variables:

$\left\{\begin{array}{l}\frac{\text{d}y}{\text{d}t}=A\left(t\right)y+{f}_{1}\left(t,y,z\right)\\ \frac{\text{d}z}{\text{d}t}=B\left(t\right)z+{f}_{2}\left(t,y,z\right)\end{array}$

In this paper the author consider a new class of the nonlinearly perturbed differential systems with time-delay

$\left\{\begin{array}{l}\frac{\text{d}y}{\text{d}t}=B\left(t\right)y+C\left(t\right)z+Y\left(s,y\left(s\right),z\left(s\right),{\int }_{0}^{t}{h}_{1}\left(s,y\left(s\right),z\left(s\right),y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)\hfill \\ \frac{\text{d}z}{\text{d}t}=D\left(t\right)y+E\left(t\right)z+Z\left(s,y\left(s\right),z\left(s\right),{\int }_{0}^{t}{h}_{2}\left(s,y\left(s\right),z\left(s\right),y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)\hfill \end{array}$

It is obvious that the above system is a generalization of the systems in    .

The aim of this paper is to investigate the double stability of neutural differential equations, including Uniform stability and Uniform Lipschitz stability. The author uses the method of differential inequalities with time-delay and integral inequalities to establish double stability criteria.

2. Preliminaries

Consider the following system:

$\frac{\text{d}x}{\text{d}t}=f\left(t,x\left(t\right),x\left(t-\tau \right),\stackrel{˙}{x}\left(t-\tau \right)\right)$ (1)

where $x\in {R}^{n}$, $y=col\left({x}_{1},{x}_{2},\cdots ,{x}_{m}\right)$, $z=col\left({x}_{m+1},{x}_{m+2},\cdots ,{x}_{n}\right)$, $x=col\left(y,z\right)$, $f\left(t,0,0\right)\equiv 0$, $\tau$ is a nonnegative constant. Let $\varphi \left(t\right)$ be a continuous function, for $\forall t\in {E}_{{t}_{0}}=\left[{t}_{0}-\tau ,{t}_{0}\right]$.

Definition 1    The trivial solution of system (1) has uniform stability and exponential asymptotic stability with respect to y if, for $\forall \epsilon >0$, $\forall {t}_{0}\in I$, $\exists \delta \left(\epsilon \right)>0$, and $\lambda >0$, when $‖\varphi ‖<\delta$ (for $\forall t\in {E}_{{t}_{0}}$ ), such that

$‖y\left(t;{t}_{0},\varphi \right)‖+‖\stackrel{˙}{y}\left(t;{t}_{0},\varphi \right)‖<\epsilon \mathrm{exp}\left(-\lambda \left(t-{t}_{0}\right)\right),\text{\hspace{0.17em}}\forall t\ge {t}_{0}.$

Definition 2    The trivial solution of system (1) has Lipschitz stability with respect to y if, there exists constants $M\left({t}_{0}\right)>0$ and $\delta \left({t}_{0}\right)>0$, when $‖\phi ‖+‖\stackrel{˙}{\phi }‖<\delta$ (for $\forall t\in {E}_{{t}_{0}}$ ), such that

$‖y\left(t;{t}_{0},\varphi \right)‖+‖\stackrel{˙}{y}\left(t;{t}_{0},\varphi \right)‖\le M\left({t}_{0}\right)\left(‖\varphi ‖+‖\stackrel{˙}{\varphi }‖\right),\text{\hspace{0.17em}}\forall t\ge {t}_{0}\ge 0.$

Definition 3    The trivial solution of system (1) has equi-exponential Lipschitz asymptotic stability with respect to y if, there exists $\lambda >0$, $K\left({t}_{0}\right)>0$ and $\delta \left({t}_{0}\right)>0$, when $‖\phi ‖+‖\stackrel{˙}{\phi }‖<\delta$ (for $\forall t\in {E}_{{t}_{0}}$ ), such that

$‖y\left(t;{t}_{0},\varphi \right)‖+‖\stackrel{˙}{y}\left(t;{t}_{0},\varphi \right)‖\le K\left({t}_{0}\right)\left(‖\varphi ‖+‖\stackrel{˙}{\varphi }‖\right)\mathrm{exp}\left(-\lambda \left(t-{t}_{0}\right)\right),\text{\hspace{0.17em}}\forall t\ge {t}_{0}\ge 0.$

Definition 4    The trivial solution of system (1) has uniform exponential Lipschitz asymptotic stability with respect to y if, K and $\delta >0$ in definition 3 are ndependent of ${t}_{0}$.

Lemma 1  The following conditions are established on $t\ge {t}_{0}$ :

i) ${k}_{1},{k}_{2},\mu$ are non-negative constants;

ii) 1)

$\begin{array}{c}{u}_{1}\left(t\right)\le {k}_{1}+{\int }_{{t}_{0}}^{t}a\left(s\right){u}_{1}\left(s\right)\text{d}s+{\int }_{{t}_{0}}^{t}b\left(s\right){u}_{2}\left(s\right){\text{e}}^{\mu \left(s-{t}_{0}\right)}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(s\right){u}_{1}^{{\alpha }_{i}+1}\left(s\right){\text{e}}^{-{\alpha }_{i}\mu \left(s-{t}_{0}\right)}\text{d}s+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(s\right){u}_{2}^{{\alpha }_{i}+1}\left(s\right){\text{e}}^{\mu \left(s-{t}_{0}\right)}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{{t}_{0}}^{t}\underset{j=1}{\overset{J}{\sum }}{e}_{j}\left(s\right){\int }_{{t}_{0}}^{s}{f}_{j}\left(\tau \right){u}_{1}\left(\tau \right)\text{d}\tau \text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{{t}_{0}}^{t}\underset{j=1}{\overset{J}{\sum }}{g}_{j}\left(s\right)\left[{\int }_{{t}_{0}}^{s}{h}_{j}\left(\tau \right){u}_{2}\left(\tau \right){\text{e}}^{\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \right]\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{{t}_{0}}^{t}\underset{k=1}{\overset{K}{\sum }}{o}_{k}\left(s\right){\int }_{{t}_{0}}^{s}{p}_{k}\left(\tau \right){u}_{1}^{{\beta }_{k}+1}\left(\tau \right)\text{ }\text{ }{\text{e}}^{-{\beta }_{k}\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}s\end{array}$

$\begin{array}{l}\text{ }+{\int }_{{t}_{0}}^{t}\underset{k=1}{\overset{K}{\sum }}{q}_{k}\left(s\right){\int }_{{t}_{0}}^{s}{r}_{k}\left(\tau \right){u}_{2}^{{\beta }_{k}+1}\left(\tau \right)\text{ }\text{ }{\text{e}}^{\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}s\\ \text{ }+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(s\right){u}_{1}^{{\alpha }_{i}}\left(s\right){\text{e}}^{-{\alpha }_{i}\mu \left(s-{t}_{0}\right)}{\int }_{{t}_{0}}^{s}\underset{l=1}{\overset{L}{\sum }}{v}_{l}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{w}_{l}\left(\tau \right){u}_{1}\left(\tau \right)\text{d}\tau \text{d}\theta \text{d}s\\ +{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(s\right){u}_{2}^{{\alpha }_{i}}\left(s\right){\int }_{{t}_{0}}^{s}\underset{l=1}{\overset{L}{\sum }}{x}_{l}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{y}_{l}\left(\tau \right){u}_{2}\left(\tau \right){\text{e}}^{\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}\theta \text{d}s\\ +{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(s\right){u}_{1}^{{\alpha }_{i}}\left(s\right){\text{e}}^{-{\alpha }_{i}\mu \left(s-{t}_{0}\right)}{\int }_{{t}_{0}}^{s}\underset{m=1}{\overset{M}{\sum }}{A}_{m}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{B}_{m}\left(\tau \right){u}_{1}^{{\gamma }_{m}+1}\left(\tau \right){\text{e}}^{-{\gamma }_{m}\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}\theta \text{d}s\\ +{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(s\right){u}_{2}^{{\alpha }_{i}}\left(s\right){\int }_{{t}_{0}}^{s}\underset{m=1}{\overset{M}{\sum }}{D}_{m}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{E}_{m}\left(\tau \right){u}_{2}^{{\gamma }_{m}+1}\left(\tau \right){\text{e}}^{\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}\theta \text{d}s\end{array}$

2)

$\begin{array}{c}{u}_{2}\left(t\right)\le {k}_{2}+{\int }_{{t}_{0}}^{t}\stackrel{¯}{a}\left(s\right){u}_{1}\left(s\right){\text{e}}^{-\mu \left(s-{t}_{0}\right)}\text{d}s+{\int }_{{t}_{0}}^{t}\stackrel{¯}{b}\left(s\right){u}_{2}\left(s\right)\text{d}s\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{c}}_{i}\left(s\right){u}_{1}^{{\alpha }_{i}+1}\left(s\right){\text{e}}^{-\left({\alpha }_{i}+1\right)\mu \left(s-{t}_{0}\right)}\text{d}s+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{d}}_{i}\left(s\right){u}_{2}^{{\alpha }_{i}+1}\left(s\right)\text{d}s\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{{t}_{0}}^{t}\underset{j=1}{\overset{J}{\sum }}{\stackrel{¯}{e}}_{j}\left(s\right){\int }_{{t}_{0}}^{s}{\stackrel{¯}{f}}_{j}\left(\tau \right){u}_{1}\left(\tau \right){\text{e}}^{-\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{{t}_{0}}^{t}\underset{j=1}{\overset{J}{\sum }}{\stackrel{¯}{g}}_{j}\left(s\right)\left[{\int }_{{t}_{0}}^{s}{\stackrel{¯}{h}}_{j}\left(\tau \right){u}_{2}\left(\tau \right)\text{d}\tau \right]\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{{t}_{0}}^{t}\underset{k=1}{\overset{K}{\sum }}{\stackrel{¯}{o}}_{k}\left(s\right){\int }_{{t}_{0}}^{s}{\stackrel{¯}{p}}_{k}\left(\tau \right){u}_{1}^{{\beta }_{k}+1}\left(\tau \right)\text{ }\text{ }{\text{e}}^{-\left({\beta }_{k}+1\right)\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}s\end{array}$

$\begin{array}{l}\text{ }+{\int }_{{t}_{0}}^{t}\underset{k=1}{\overset{K}{\sum }}{\stackrel{¯}{q}}_{k}\left(s\right){\int }_{{t}_{0}}^{s}{\stackrel{¯}{r}}_{k}\left(\tau \right){u}_{2}^{{\beta }_{k}+1}\left(\tau \right)\text{ }\text{d}\tau \text{d}s\\ \text{ }+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{c}}_{i}\left(s\right){u}_{1}^{{\alpha }_{i}}\left(s\right){\text{e}}^{-{\alpha }_{i}\mu \left(s-{t}_{0}\right)}{\int }_{{t}_{0}}^{s}\underset{l=1}{\overset{L}{\sum }}{\stackrel{¯}{v}}_{l}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{\stackrel{¯}{w}}_{l}\left(\tau \right){u}_{1}\left(\tau \right){\text{e}}^{-\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}\theta \text{d}s\\ \text{ }+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{d}}_{i}\left(s\right){u}_{2}^{{\alpha }_{i}}\left(s\right){\int }_{{t}_{0}}^{s}\underset{l=1}{\overset{L}{\sum }}{\stackrel{¯}{x}}_{l}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{\stackrel{¯}{y}}_{l}\left(\tau \right){u}_{2}\left(\tau \right)\text{d}\tau \text{d}\theta \text{d}s\\ \text{ }+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{c}}_{i}\left(s\right){u}_{1}^{{\alpha }_{i}}\left(s\right){\text{e}}^{-{\alpha }_{i}\mu \left(s-{t}_{0}\right)}{\int }_{{t}_{0}}^{s}\underset{m=1}{\overset{M}{\sum }}{\stackrel{¯}{A}}_{m}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{\stackrel{¯}{B}}_{m}\left(\tau \right){u}_{1}^{{\gamma }_{m}+1}\left(\tau \right){\text{e}}^{-\left({\gamma }_{m}+1\right)\mu \left(\tau -{t}_{0}\right)}\text{d}\tau \text{d}\theta \text{d}s\\ \text{ }+{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{d}}_{i}\left(s\right){u}_{2}^{{\alpha }_{i}}\left(s\right){\int }_{{t}_{0}}^{s}\underset{m=1}{\overset{M}{\sum }}{\stackrel{¯}{D}}_{m}\left(\theta \right){\int }_{{t}_{0}}^{\theta }{\stackrel{¯}{E}}_{m}\left(\tau \right){u}_{2}^{{\gamma }_{m}+1}\left(\tau \right)\text{d}\tau \text{d}\theta \text{d}s\end{array}$

where: ${u}_{1}\left(t\right),{u}_{2}\left(t\right),a\left(t\right),\stackrel{¯}{a}\left(t\right),b\left(t\right),\stackrel{¯}{b}\left(t\right),{c}_{i}\left(t\right),{\stackrel{¯}{c}}_{i}\left(t\right),{d}_{i}\left(t\right),{\stackrel{¯}{d}}_{i}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(i=1,2\text{\hspace{0.17em}},\cdots ,I\right)$, ${e}_{j}\left(t\right),{\stackrel{¯}{e}}_{j}\left(t\right),{f}_{j}\left(t\right),{\stackrel{¯}{f}}_{j}\left(t\right),{g}_{j}\left(t\right),{\stackrel{¯}{g}}_{j}\left(t\right),{h}_{j}\left(t\right),{\stackrel{¯}{h}}_{j}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(j=1,2,\cdots ,J\right)$, ${o}_{k}\left(t\right),{\stackrel{¯}{o}}_{k}\left(t\right),{p}_{k}\left(t\right),{\stackrel{¯}{p}}_{k}\left(t\right),{q}_{k}\left(t\right),{\stackrel{¯}{q}}_{k}\left(t\right),{r}_{k}\left(t\right),{\stackrel{¯}{r}}_{k}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(k=1,2,\cdots ,K\right)$, ${v}_{l}\left(t\right),{\stackrel{¯}{v}}_{l}\left(t\right),{w}_{l}\left(t\right),{\stackrel{¯}{w}}_{l}\left(t\right),{x}_{l}\left(t\right),{\stackrel{¯}{x}}_{l}\left(t\right),{y}_{l}\left(t\right),{\stackrel{¯}{y}}_{l}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(l=1,2,\cdots ,L\right)$, ${A}_{m}\left(t\right),{\stackrel{¯}{A}}_{m}\left(t\right),{B}_{m}\left(t\right),{\stackrel{¯}{B}}_{m}\left(t\right),{D}_{m}\left(t\right),{\stackrel{¯}{D}}_{m}\left(t\right),{E}_{m}\left(t\right),{\stackrel{¯}{E}}_{m}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(m=1,2,\cdots ,M\right)$ are non-negative continuous function on ${R}_{+}$, and: ${\alpha }_{i}\text{\hspace{0.17em}}\left(i=1,2,\cdots ,I\right)$, ${\beta }_{k}\text{\hspace{0.17em}}\left(k=1,2,\cdots ,K\right)$, ${\gamma }_{m}\text{\hspace{0.17em}}\left(m=1,2,\cdots ,M\right)$ are all constants greater than 1, and $1\le {\alpha }_{1}\le {\alpha }_{2}\le \cdots \le {\alpha }_{I}$, $1\le {\beta }_{1}\le {\beta }_{2}\le \cdots \le {\beta }_{K}$, $1\le {\gamma }_{1}\le {\gamma }_{2}\le \cdots \le {\gamma }_{M}$ set: $\stackrel{¯}{\alpha }=\mathrm{max}\left({\alpha }_{I},{\beta }_{K},{\gamma }_{M}\right)$, $\underset{_}{\alpha }=\mathrm{min}\left({\alpha }_{1},{\beta }_{1},{\gamma }_{1}\right)$

iii) let: $k={k}_{1}+{k}_{2}$

$F\left(t\right)=\mathrm{max}\left\{a\left(t\right)+\stackrel{¯}{a}\left(t\right),b\left(t\right)+\stackrel{¯}{b}\left(t\right)\right\}$, ${G}_{i}\left(t\right)=\mathrm{max}\left\{{c}_{i}\left(t\right)+{\stackrel{¯}{c}}_{i}\left(t\right),{d}_{i}\left(t\right)+{\stackrel{¯}{d}}_{i}\left(t\right)\right\}$,

${\stackrel{¯}{\stackrel{¯}{e}}}_{j}\left(t\right)=\mathrm{max}\left\{{e}_{j}\left(t\right),{\stackrel{¯}{e}}_{j}\left(t\right)\right\}$, ${\stackrel{¯}{\stackrel{¯}{g}}}_{j}\left(t\right)=\mathrm{max}\left\{{g}_{j}\left(t\right),{\stackrel{¯}{g}}_{j}\left(t\right)\right\}$,

${H}_{j}\left(t\right)=\mathrm{max}\left\{{\stackrel{¯}{\stackrel{¯}{e}}}_{j}\left(t\right),{\stackrel{¯}{\stackrel{¯}{g}}}_{j}\left(t\right)\right\}$, ${N}_{j}\left(t\right)=\mathrm{max}\left\{{f}_{j}\left(t\right)+{\stackrel{¯}{f}}_{j}\left(t\right),{h}_{j}\left(t\right)+{\stackrel{¯}{h}}_{j}\left(t\right)\right\}$,

${\stackrel{¯}{\stackrel{¯}{o}}}_{k}\left(t\right)=\mathrm{max}\left\{{o}_{k}\left(t\right),{\stackrel{¯}{o}}_{k}\left(t\right)\right\}$, ${\stackrel{¯}{\stackrel{¯}{q}}}_{k}\left(t\right)=\mathrm{max}\left\{{q}_{k}\left(t\right),{\stackrel{¯}{q}}_{k}\left(t\right)\right\}$,

${Q}_{k}\left(t\right)=\mathrm{max}\left\{{\stackrel{¯}{\stackrel{¯}{o}}}_{k}\left(t\right),{\stackrel{¯}{\stackrel{¯}{q}}}_{k}\left(t\right)\right\}$, ${R}_{k}\left(t\right)=\mathrm{max}\left\{{p}_{k}\left(t\right)+{\stackrel{¯}{p}}_{k}\left(t\right),{r}_{k}\left(t\right)+{\stackrel{¯}{r}}_{k}\left(t\right)\right\}$,

${\stackrel{¯}{\stackrel{¯}{v}}}_{l}\left(t\right)=\mathrm{max}\left\{{v}_{l}\left(t\right),{\stackrel{¯}{v}}_{l}\left(t\right)\right\}$, ${\stackrel{¯}{\stackrel{¯}{x}}}_{l}\left(t\right)=\mathrm{max}\left\{{x}_{l}\left(t\right),{\stackrel{¯}{x}}_{l}\left(t\right)\right\}$,

${T}_{l}\left(t\right)=\mathrm{max}\left\{{\stackrel{¯}{\stackrel{¯}{v}}}_{l}\left(t\right),{\stackrel{¯}{\stackrel{¯}{x}}}_{l}\left(t\right)\right\}$, ${W}_{l}\left(t\right)=\mathrm{max}\left\{{w}_{l}\left(t\right)+{\stackrel{¯}{w}}_{l}\left(t\right),{y}_{l}\left(t\right)+{\stackrel{¯}{y}}_{l}\left(t\right)\right\}$,

${\stackrel{¯}{\stackrel{¯}{A}}}_{m}\left(t\right)=\mathrm{max}\left\{{A}_{m}\left(t\right),{\stackrel{¯}{A}}_{m}\left(t\right)\right\}$, ${\stackrel{¯}{\stackrel{¯}{D}}}_{m}\left(t\right)=\mathrm{max}\left\{{D}_{m}\left(t\right),{\stackrel{¯}{D}}_{m}\left(t\right)\right\}$,

${Y}_{m}\left(t\right)=\mathrm{max}\left\{{\stackrel{¯}{\stackrel{¯}{A}}}_{m}\left(t\right),{\stackrel{¯}{\stackrel{¯}{D}}}_{m}\left(t\right)\right\}$, ${Z}_{m}\left(t\right)=\mathrm{max}\left\{{B}_{m}\left(t\right)+{\stackrel{¯}{B}}_{m}\left(t\right),{E}_{m}\left(t\right)+{\stackrel{¯}{E}}_{m}\left(t\right)\right\}$,

iv) Let:

$\begin{array}{c}\Delta \left(t\right)=F\left(t\right)+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right)+\underset{j=1}{\overset{J}{\sum }}{H}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{N}_{j}\left(s\right)\text{d}s+2\underset{k=1}{\overset{K}{\sum }}{Q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{R}_{k}\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{T}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{W}_{l}\left(\sigma \right)\text{d}\sigma \text{d}s+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{Z}_{m}\left(\sigma \right)\text{d}\sigma \text{d}s\end{array}$

$\begin{array}{c}\Phi \left(t\right)=\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right)+\underset{k=1}{\overset{K}{\sum }}{Q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{R}_{k}\left(s\right)\text{d}s+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{T}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{W}_{l}\left(\sigma \right)\text{d}\sigma \text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{Z}_{m}\left(\sigma \right)\text{d}\sigma \text{d}s\end{array}$

$\Gamma \left(t\right)={\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{Z}_{m}\left(\sigma \right)\text{d}\sigma \text{d}s$

${\Lambda }_{1}\left(t\right)=1-\stackrel{¯}{\alpha }{c}^{\stackrel{¯}{\alpha }}{\int }_{{t}_{0}}^{t}\left[\Phi \left(\tau \right)+\Gamma \left(\tau \right)\right]\mathrm{exp}\left(\stackrel{¯}{\alpha }{\int }_{{t}_{0}}^{\tau }\Delta \left(\sigma \right)\text{d}\sigma \right)\text{d}\tau$

$\Pi \left(t\right)=F\left(t\right)+\underset{j=1}{\overset{J}{\sum }}{H}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{N}_{j}\left(t\right)\text{d}t$

$\begin{array}{c}\Theta \left(t\right)=\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){Щ}^{{\alpha }_{i}-\underset{_}{\alpha }}+\underset{k=1}{\overset{K}{\sum }}{Q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{R}_{k}\left(s\right){Щ}^{{\beta }_{k}-\underset{_}{\alpha }}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){Щ}^{{\alpha }_{i}-\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{T}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{W}_{l}\left(\sigma \right)\text{d}\sigma \end{array}$

$\Sigma \left(t\right)=\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(t\right){\int }_{{t}_{0}}^{t}{Z}_{m}\left(s\right){Щ}^{{\gamma }_{m}-\underset{_}{\alpha }}\text{d}s$

${\Lambda }_{2}\left(t\right)=1-\underset{_}{\alpha }{c}^{\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\left[\Theta \left(\tau \right)+\Sigma \left(\tau \right)\right]\mathrm{exp}\left(\underset{_}{\alpha }{\int }_{{t}_{0}}^{\tau }\Pi \left(\sigma \right)\text{d}\sigma \right)\text{d}\tau$

And ${\int }_{{t}_{0}}^{+\infty }\Delta \left(s\right)\text{d}s<+\infty$,

${\Lambda }_{1}\left(t\right)>0$, $\left[1-\left(\stackrel{¯}{\alpha }-1\right){c}^{\stackrel{¯}{\alpha }}{\int }_{{t}_{0}}^{t}\Phi \left(s\right){\Lambda }_{1}^{-\frac{1}{\stackrel{¯}{\alpha }}}\left(s\right)\mathrm{exp}\left(\stackrel{¯}{\alpha }{\int }_{{t}_{0}}^{s}\Delta \left(\tau \right)\text{d}\tau \right)\text{d}s\right]>0$

${\Lambda }_{2}\left(t\right)>0$, $\left[1-\left(\underset{_}{\alpha }-1\right){c}^{\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\Theta \left(s\right){\Lambda }_{2}^{-\frac{1}{\underset{_}{\alpha }}}\left(s\right)\mathrm{exp}\left(\underset{_}{\alpha }{\int }_{{t}_{0}}^{s}\Sigma \left(\tau \right)\text{d}\tau \right)\text{d}s\right]>0$

v) Assume:

$\Omega \left(t\right)\le k\mathrm{exp}\left({\int }_{{t}_{0}}^{t}\Pi \left(s\right)\text{d}s\right)\cdot {\left[1-\left(\underset{_}{\alpha }-1\right){k}^{\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\Theta \left(s\right){\Lambda }_{2}^{-\frac{1}{\underset{_}{\alpha }}}\left(s\right)\mathrm{exp}\left(\underset{_}{\alpha }{\int }_{{t}_{0}}^{s}\Pi \left(\tau \right)\text{d}\tau \right)\text{d}s\right]}^{-\frac{1}{\underset{_}{\alpha }-1}}$ then: ${u}_{1}\left(t\right)\le \Omega \left(t\right){\text{e}}^{\mu \left(t-{t}_{0}\right)}$, ${u}_{2}\left(t\right)\le \Omega \left(t\right)$.

3. Main Results

Consider the following system

$\left\{\begin{array}{l}\begin{array}{c}\frac{\text{d}y}{\text{d}t}=B\left(t\right)y+C\left(t\right)z\\ +Y\left(s,y\left(s\right),z\left(s\right),{\int }_{0}^{t}{h}_{1}\left(s,y\left(s\right),z\left(s\right),y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)\end{array}\hfill \\ \begin{array}{c}\frac{\text{d}z}{\text{d}t}=D\left(t\right)y+E\left(t\right)z\\ +Z\left(s,y\left(s\right),z\left(s\right),{\int }_{0}^{t}{h}_{2}\left(s,y\left(s\right),z\left(s\right),y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)\end{array}\hfill \end{array}$ (2)

where $\tau \ge 0$ is a constant, initial condition is:

$x\left(t\right)=\varphi \left(t\right),\text{\hspace{0.17em}}\stackrel{˙}{x}\left(t\right)=\stackrel{˙}{\varphi }\left(t\right),\text{\hspace{0.17em}}{t}_{0}-\tau \le t\le {t}_{0},$

$B\left(t\right)$ is an $m×m$ matrix, $Y\left(s,y\left(s\right),z\left(s\right),{\int }_{0}^{t}{h}_{1}\left(s,y\left(s\right),z\left(s\right),y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)$ is an $m×1$ matrix, $Z\left(s,y\left(s\right),z\left(s\right),{\int }_{0}^{t}{h}_{2}\left(s,y\left(s\right),z\left(s\right),y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)$ is an $\left(n-m\right)×1$ matrix, they are all continuous for $t\in I$ and satisfy the condition of existence and uniqueness theorem.

Set $Y\left(t,s\right)$ and $Z\left(t,s\right)$ satisfied:

$\left\{\begin{array}{l}\frac{\partial Y\left(t,s\right)}{\partial t}=B\left(t\right)Y\left(t,s\right)\\ Y\left(s,s\right)=I\end{array}$,

$\left\{\begin{array}{l}\frac{\partial Z\left(t,s\right)}{\partial t}=E\left(t\right)Z\left(t,s\right)\\ Z\left(s,s\right)=I\end{array}$

Theorem If (2) satisfies the following conditions:

i) $‖Y\left(t,s\right)‖\le {m}_{1}{\text{e}}^{-\lambda \left(t-s\right)}$, $‖Z\left(t,s\right)‖\le {m}_{2}$ ;

ii) $\begin{array}{l}‖Y\left(t,y,z,y\left(t-\tau \right),z\left(t-\tau \right),{\int }_{0}^{t}{h}_{1}\left(s,y,z,y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)‖\\ \le a\left(t\right)\left(‖y\left(t-\delta \left(t\right)\right)‖+‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+b\left(t\right)\left(‖z\left(t-\delta \left(t\right)\right)‖+‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖\right){\text{e}}^{-\epsilon \left(t-{t}_{0}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(t\right)\left({‖y\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}+{‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}\right){\text{e}}^{{\alpha }_{i}\epsilon \left(t-{t}_{0}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(t\right)\left({‖z\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}+{‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}\right){\text{e}}^{-\epsilon \left(t-{t}_{0}\right)}\end{array}$

$\begin{array}{l}+\underset{j=1}{\overset{J}{\sum }}{e}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{f}_{j}\left(s\right)\left(‖y\left(s-\delta \left(s\right)\right)‖+‖\stackrel{˙}{y}\left(s-\delta \left(s\right)\right)‖\right)\text{d}s\\ \text{ }+\underset{j=1}{\overset{J}{\sum }}{g}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{h}_{j}\left(s\right)\left(‖z\left(s-\delta \left(s\right)\right)‖+‖\stackrel{˙}{z}\left(s-\delta \left(s\right)\right)‖\right){\text{e}}^{-\epsilon \left(s-{t}_{0}\right)}\text{d}s\\ \text{ }+\underset{k=1}{\overset{K}{\sum }}{o}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{p}_{k}\left(s\right)\left({‖y\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}+{‖\stackrel{˙}{y}\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}\right){\text{e}}^{{\beta }_{k}\epsilon \left(s-{t}_{0}\right)}\text{d}s\\ \text{ }+\underset{k=1}{\overset{K}{\sum }}{q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{r}_{k}\left(s\right)\left({‖z\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}+{‖\stackrel{˙}{z}\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}\right){\text{e}}^{-\epsilon \left(s-{t}_{1}\right)}\text{d}s\end{array}$

$\begin{array}{l}\text{ }+\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(t\right)\left({‖y\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right){\text{e}}^{{\alpha }_{i}\epsilon \left(t-{t}_{0}\right)}\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{v}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{w}_{l}\left(\theta \right)\left(‖y\left(\theta -\delta \left(\theta \right)\right)‖+‖\stackrel{˙}{y}\left(\theta -\delta \left(\theta \right)\right)‖\right)\text{d}\theta \text{d}s\\ \text{ }+\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(t\right)\left({‖z\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right)\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{x}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{y}_{l}\left(\theta \right)\left(‖z\left(\theta -\delta \left(\theta \right)\right)‖+‖\stackrel{˙}{z}\left(\theta -\delta \left(\theta \right)\right)‖\right){\text{e}}^{-\epsilon \left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\end{array}$

$\begin{array}{l}\text{ }+\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(t\right)\left({‖y\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right){\text{e}}^{{\alpha }_{i}\epsilon \left(t-{t}_{0}\right)}\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{A}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{B}_{m}\left(\theta \right)\left({‖y\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}+{‖\stackrel{˙}{y}\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}\right){\text{e}}^{{\gamma }_{m}\epsilon \left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\\ \text{ }+\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(t\right)\left({‖z\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right)\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{D}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{E}_{m}\left(\theta \right)\left({‖z\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}+{‖\stackrel{˙}{z}\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}\right){\text{e}}^{-\epsilon \left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\end{array}$

iv) $\begin{array}{l}‖Z\left(t,y,z,y\left(t-\tau \right),z\left(t-\tau \right),{\int }_{0}^{t}{h}_{2}\left(s,y,z,y\left(s-\tau \right),\stackrel{˙}{z}\left(s-\tau \right)\right)\text{d}s\right)‖\\ \le \stackrel{¯}{a}\left(t\right)\left(‖y\left(t-\delta \left(t\right)\right)‖+‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖\right){\text{e}}^{\epsilon \left(t-{t}_{0}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\stackrel{¯}{b}\left(t\right)\left(‖z\left(t-\delta \left(t\right)\right)‖+‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{c}}_{i}\left(t\right)\left({‖y\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}+{‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}\right){\text{e}}^{\left({\alpha }_{i}+1\right)\epsilon \left(t-{t}_{0}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{d}}_{i}\left(t\right)\left({‖z\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}+{‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}+1}\right)\end{array}$

$\begin{array}{l}\text{ }+\underset{j=1}{\overset{J}{\sum }}{\stackrel{¯}{e}}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{\stackrel{¯}{f}}_{j}\left(s\right)\left(‖y\left(s-\delta \left(s\right)\right)‖+‖\stackrel{˙}{y}\left(s-\delta \left(s\right)\right)‖\right){\text{e}}^{\epsilon \left(s-{t}_{0}\right)}\text{d}s\\ \text{ }+\underset{j=1}{\overset{J}{\sum }}{\stackrel{¯}{g}}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{\stackrel{¯}{h}}_{j}\left(s\right)\left(‖z\left(s-\delta \left(s\right)\right)‖+‖\stackrel{˙}{z}\left(s-\delta \left(s\right)\right)‖\right)\text{d}s\\ \text{ }+\underset{k=1}{\overset{K}{\sum }}{\stackrel{¯}{o}}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{\stackrel{¯}{p}}_{k}\left(s\right)\left({‖y\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}+{‖\stackrel{˙}{y}\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}\right){\text{e}}^{\left({\beta }_{k}+1\right)\epsilon \left(s-{t}_{0}\right)}\text{d}s\\ \text{ }+\underset{k=1}{\overset{K}{\sum }}{\stackrel{¯}{q}}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{\stackrel{¯}{r}}_{k}\left(s\right)\left({‖z\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}+{‖\stackrel{˙}{z}\left(s-\delta \left(s\right)\right)‖}^{{\beta }_{k}+1}\right)\text{d}s\end{array}$

$\begin{array}{l}\text{ }+\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{c}}_{i}\left(t\right)\left({‖y\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right){\text{e}}^{{\alpha }_{i}\epsilon \left(t-{t}_{0}\right)}\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{v}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{w}_{l}\left(\theta \right)\left(‖y\left(\theta -\delta \left(\theta \right)\right)‖+‖\stackrel{˙}{y}\left(\theta -\delta \left(\theta \right)\right)‖\right){\text{e}}^{\epsilon \left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\\ \text{ }+\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{d}}_{i}\left(t\right)\left({‖z\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right)\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{\stackrel{¯}{x}}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{\stackrel{¯}{y}}_{l}\left(\theta \right)\left(‖z\left(\theta -\delta \left(\theta \right)\right)‖+‖\stackrel{˙}{z}\left(\theta -\delta \left(\theta \right)\right)‖\right)\text{d}\theta \text{d}s\end{array}$

$\begin{array}{l}\text{ }+\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{c}}_{i}\left(t\right)\left({‖y\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{y}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right){\text{e}}^{{\alpha }_{i}\epsilon \left(t-{t}_{0}\right)}\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{\stackrel{¯}{A}}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{\stackrel{¯}{B}}_{m}\left(\theta \right)\left({‖y\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}+{‖\stackrel{˙}{y}\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}\right){\text{e}}^{\left({\gamma }_{m}+1\right)\epsilon \left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\\ \text{ }+\underset{i=1}{\overset{I}{\sum }}{\stackrel{¯}{d}}_{i}\left(t\right)\left({‖z\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}+{‖\stackrel{˙}{z}\left(t-\delta \left(t\right)\right)‖}^{{\alpha }_{i}}\right)\\ \text{ }\cdot {\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{\stackrel{¯}{D}}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{\stackrel{¯}{E}}_{m}\left(\theta \right)\left({‖z\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}+{‖\stackrel{˙}{z}\left(\theta -\delta \left(\theta \right)\right)‖}^{{\gamma }_{m}+1}\right)\text{d}\theta \text{d}s\end{array}$

where： $a\left(t\right),\stackrel{¯}{a}\left(t\right),b\left(t\right),\stackrel{¯}{b}\left(t\right),{c}_{i}\left(t\right),{\stackrel{¯}{c}}_{i}\left(t\right),{d}_{i}\left(t\right),{\stackrel{¯}{d}}_{i}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(i=1,2\text{\hspace{0.17em}},\cdots ,I\right)$,

${e}_{j}\left(t\right),{\stackrel{¯}{e}}_{j}\left(t\right),{f}_{j}\left(t\right),{\stackrel{¯}{f}}_{j}\left(t\right),{g}_{j}\left(t\right),{\stackrel{¯}{g}}_{j}\left(t\right),{h}_{j}\left(t\right),{\stackrel{¯}{h}}_{j}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(j=1,2,\cdots ,J\right)$,

${o}_{k}\left(t\right),{\stackrel{¯}{o}}_{k}\left(t\right),{p}_{k}\left(t\right),{\stackrel{¯}{p}}_{k}\left(t\right),{q}_{k}\left(t\right),{\stackrel{¯}{q}}_{k}\left(t\right),{r}_{k}\left(t\right),{\stackrel{¯}{r}}_{k}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(k=1,2,\cdots ,K\right)$,

${v}_{l}\left(t\right),{\stackrel{¯}{v}}_{l}\left(t\right),{w}_{l}\left(t\right),{\stackrel{¯}{w}}_{l}\left(t\right),{x}_{l}\left(t\right),{\stackrel{¯}{x}}_{l}\left(t\right),{y}_{l}\left(t\right),{\stackrel{¯}{y}}_{l}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(l=1,2,\cdots ,L\right)$,

${A}_{m}\left(t\right),{\stackrel{¯}{A}}_{m}\left(t\right),{B}_{m}\left(t\right),{\stackrel{¯}{B}}_{m}\left(t\right),{D}_{m}\left(t\right),{\stackrel{¯}{D}}_{m}\left(t\right),{E}_{m}\left(t\right),{\stackrel{¯}{E}}_{m}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(m=1,2,\cdots ,M\right)$ are non-negative continuous monotonic non-increasing functions on ${R}_{+}$, and: ${\alpha }_{i}\text{\hspace{0.17em}}\left(i=1,2,\cdots ,I\right)$, ${\beta }_{k}\text{\hspace{0.17em}}\left(k=1,2,\cdots ,K\right)$, ${\gamma }_{m}\text{\hspace{0.17em}}\left(m=1,2,\cdots ,M\right)$ are all constants greater than 1, $1\le {\alpha }_{1}\le {\alpha }_{2}\le \cdots \le {\alpha }_{I}$, $1\le {\beta }_{1}\le {\beta }_{2}\le \cdots \le {\beta }_{K}$, $1\le {\gamma }_{1}\le {\gamma }_{2}\le \cdots \le {\gamma }_{M}$ let: $\stackrel{¯}{\alpha }=\mathrm{max}\left({\alpha }_{I},{\beta }_{K},{\gamma }_{M}\right)$, $\underset{_}{\alpha }=\mathrm{min}\left({\alpha }_{1},{\beta }_{1},{\gamma }_{1}\right)$

iv) Set:

$\begin{array}{c}\Delta \left(t\right)=F\left(t\right)+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right)+\underset{j=1}{\overset{J}{\sum }}{H}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{N}_{j}\left(s\right)\text{d}s+2\underset{k=1}{\overset{K}{\sum }}{Q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{R}_{k}\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{T}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{W}_{l}\left(\sigma \right)\text{d}\sigma \text{d}s+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{Z}_{m}\left(\sigma \right)\text{d}\sigma \text{d}s\end{array}$

$\begin{array}{c}\Phi \left(t\right)=\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right)+\underset{k=1}{\overset{K}{\sum }}{Q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{R}_{k}\left(s\right)\text{d}s+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{T}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{W}_{l}\left(\sigma \right)\text{d}\sigma \text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{Z}_{m}\left(\sigma \right)\text{d}\sigma \text{d}s\end{array}$

$\Gamma \left(t\right)={\int }_{{t}_{0}}^{t}\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(s\right){\int }_{{t}_{0}}^{s}{Z}_{m}\left(\sigma \right)\text{d}\sigma \text{d}s$

${\Lambda }_{1}\left(t\right)=1-\stackrel{¯}{\alpha }{c}^{\stackrel{¯}{\alpha }}{\int }_{{t}_{0}}^{t}\left[\Phi \left(\tau \right)+\Gamma \left(\tau \right)\right]\mathrm{exp}\left(\stackrel{¯}{\alpha }{\int }_{{t}_{0}}^{\tau }\Delta \left(\sigma \right)\text{d}\sigma \right)\text{d}\tau$

$\Pi \left(t\right)=F\left(t\right)+\underset{j=1}{\overset{J}{\sum }}{H}_{j}\left(t\right){\int }_{{t}_{0}}^{t}{N}_{j}\left(t\right)\text{d}t$

$\begin{array}{c}\Theta \left(t\right)=\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){Щ}^{{\alpha }_{i}-\underset{_}{\alpha }}+\underset{k=1}{\overset{K}{\sum }}{Q}_{k}\left(t\right){\int }_{{t}_{0}}^{t}{R}_{k}\left(s\right){Щ}^{{\beta }_{k}-\underset{_}{\alpha }}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{i=1}{\overset{I}{\sum }}{G}_{i}\left(t\right){Щ}^{{\alpha }_{i}-\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\underset{l=1}{\overset{L}{\sum }}{T}_{l}\left(s\right){\int }_{{t}_{0}}^{s}{W}_{l}\left(\sigma \right)\text{d}\sigma \end{array}$

$\Sigma \left(t\right)=\underset{m=1}{\overset{M}{\sum }}{Y}_{m}\left(t\right){\int }_{{t}_{0}}^{t}{Z}_{m}\left(s\right){Щ}^{{\gamma }_{m}-\underset{_}{\alpha }}\text{d}s$

${\Lambda }_{2}\left(t\right)=1-\underset{_}{\alpha }{c}^{\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\left[\Theta \left(\tau \right)+\Sigma \left(\tau \right)\right]\mathrm{exp}\left(\underset{_}{\alpha }{\int }_{{t}_{0}}^{\tau }\Pi \left(\sigma \right)\text{d}\sigma \right)\text{d}\tau$

and ${\int }_{{t}_{0}}^{+\infty }\Delta \left(s\right)\text{d}s<+\infty$,

${\Lambda }_{1}\left(t\right)>0$, $\left[1-\left(\stackrel{¯}{\alpha }-1\right){c}^{\stackrel{¯}{\alpha }}{\int }_{{t}_{0}}^{t}\Phi \left(s\right){\Lambda }_{1}^{-\frac{1}{\stackrel{¯}{\alpha }}}\left(s\right)\mathrm{exp}\left(\stackrel{¯}{\alpha }{\int }_{{t}_{0}}^{s}\Delta \left(\tau \right)\text{d}\tau \right)\text{d}s\right]>0$

${\Lambda }_{2}\left(t\right)>0$, $\left[1-\left(\underset{_}{\alpha }-1\right){c}^{\underset{_}{\alpha }}{\int }_{{t}_{0}}^{t}\Theta \left(s\right){\Lambda }_{2}^{-\frac{1}{\underset{_}{\alpha }}}\left(s\right)\mathrm{exp}\left(\underset{_}{\alpha }{\int }_{{t}_{0}}^{s}\Sigma \left(\tau \right)\text{d}\tau \right)\text{d}s\right]>0$

v) Set:

$\Omega \left(t\right)=k\mathrm{exp}\left({\int }_{0}^{t}\Pi \left(s\right)\text{d}s\right)\cdot {\left[1-\left(\underset{_}{\alpha }-1\right){k}^{\underset{_}{\alpha }}{\int }_{0}^{t}\Theta \left(s\right){\Lambda }_{2}^{-\frac{1}{\underset{_}{\alpha }}}\left(s\right)\mathrm{exp}\left(\underset{_}{\alpha }{\int }_{0}^{s}\Pi \left(\tau \right)\text{d}\tau \right)\text{d}s\right]}^{-\frac{1}{\underset{_}{\alpha }-1}}$ then:

1) when $\lambda >\epsilon$, the trivial solution of (2) is $LS$, $G{E}_{q}ELAS$ with respect to y;

2) when $\lambda =\epsilon$, the trivial solution of (2) is, $GUELAS$ with respect to.

Proof Apply constant variation method to system (2), it can be deduced that:

$y\left(t\right)=Y\left(t,{t}_{0}\right){y}_{0}+{\int }_{{t}_{0}}^{t}Y\left(t,s\right){F}_{1}\left(s,x\left(s-\delta \left(s\right)\right),{\int }_{{t}_{0}}^{s}{h}_{1}\left(\tau ,x\left(\tau -\delta \left(\tau \right)\right)\right)\text{d}\tau \right)\text{d}s$ (3)

$z\left(t\right)=Z\left(t,{t}_{0}\right){z}_{0}+{\int }_{{t}_{0}}^{t}Z\left(t,s\right){F}_{2}\left(s,x\left(s-\delta \left(s\right)\right),{\int }_{{t}_{0}}^{s}{h}_{2}\left(\tau ,x\left(\tau -\delta \left(\tau \right)\right)\right)\text{d}\tau \right)\text{d}s$ (4)

By the condition of the theory, available from (3): (5)

however, set

${u}_{1}\left(t\right)=\left(‖y\left(t\right)‖+‖\stackrel{˙}{y}\left(t\right)‖\right){\text{e}}^{\lambda \left(t-{t}_{0}\right)}$, ${u}_{2}\left(t\right)=‖z\left(t\right)‖+‖\stackrel{˙}{z}\left(t\right)‖$

${\phi }_{1}^{\left(1\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}‖{\phi }_{1}\left(t\right)‖$, ${\phi }_{2}^{\left(1\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}{‖{\phi }_{1}\left(t\right)‖}^{{\alpha }_{i}+1}$

${\phi }_{3}^{\left(1\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}{‖{\phi }_{1}\left(t\right)‖}^{{\beta }_{k}+1}$, ${\phi }_{4}^{\left(1\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}{‖{\phi }_{1}\left(t\right)‖}^{{\gamma }_{m}+1}$

${\phi }^{\left(1\right)}=\mathrm{max}\left\{{\phi }_{1}^{\left(1\right)},{\phi }_{2}^{\left(1\right)},{\phi }_{3}^{\left(1\right)},{\phi }_{4}^{\left(1\right)},{\phi }_{2}^{\left(1\right)}{\phi }_{4}^{\left(1\right)}\right\}$

${\phi }_{1}^{\left(2\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}‖{\phi }_{2}\left(t\right)‖$, ${\phi }_{2}^{\left(2\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}{‖{\phi }_{2}\left(t\right)‖}^{{\alpha }_{i}+1}$

${\phi }_{3}^{\left(2\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}{‖{\phi }_{2}\left(t\right)‖}^{{\beta }_{k}+1}$, ${\phi }_{4}^{\left(2\right)}=\underset{-\delta \le t\le 0}{\mathrm{sup}}{‖{\phi }_{2}\left(t\right)‖}^{{\gamma }_{m}+1}$

${\phi }^{\left(2\right)}=\mathrm{max}\left\{{\phi }_{1}^{\left(2\right)},{\phi }_{2}^{\left(2\right)},{\phi }_{3}^{\left(2\right)},{\phi }_{4}^{\left(2\right)},{\phi }_{2}^{\left(2\right)}{\phi }_{4}^{\left(2\right)}\right\}$

Set:

$‖{u}_{1}\left(t\right)‖=\left\{\begin{array}{l}\mathrm{max}\left\{{\phi }_{1}^{\left(1\right)},\mathrm{max}\left(u\left(\xi \right)\right)\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le \xi \le t\\ {\phi }_{1}^{\left(1\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\delta \le t\le 0\end{array}$

$‖{u}_{2}\left(t\right)‖=\left\{\begin{array}{l}\mathrm{max}\left\{{\phi }_{1}^{\left(2\right)},\mathrm{max}\left(u\left(\xi \right)\right)\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le \xi \le t\\ {\phi }_{1}^{\left(2\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\delta \le t\le 0\end{array}$

Obviously, $‖{u}_{1}\left(t\right)‖$, $‖{u}_{2}\left(t\right)‖$ are monotonous, and defined by, ${u}_{1}\left(t\right)$, ${u}_{2}\left(t\right)$, we have

${u}_{1}\left(t-\delta \left(t\right)\right)\le {\phi }_{1}^{\left(1\right)}$,

${u}_{2}\left(t-\delta \left(t\right)\right)\le {\phi }_{1}^{\left( 2 \right)}$

So substituting (5) into (2) gives:

$\begin{array}{c}{u}_{1}\left(t\right)\\ \le {\phi }^{\left(1\right)}+{M}_{2}{\int }_{{t}_{0}}^{t}a\left(s\right)‖{u}_{1}\left(s\right)‖\text{d}s+{M}_{2}{\int }_{{t}_{0}}^{t}b\left(s\right)‖{u}_{2}\left(s\right)‖{\text{e}}^{\left(\lambda -\epsilon \right)\left(s-{t}_{0}\right)}\text{d}s\\ +{M}_{2}{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(s\right){‖{u}_{1}\left(s\right)‖}^{{\alpha }_{i}+1}{\text{e}}^{-{\alpha }_{i}\left(\lambda -\epsilon \right)\left(s-{t}_{0}\right)}\text{d}s+{M}_{2}{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(s\right){‖{u}_{2}\left(s\right)‖}^{{\alpha }_{i}+1}{\text{e}}^{\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}s\\ +{M}_{2}{\int }_{{t}_{0}}^{t}\underset{j=1}{\overset{J}{\sum }}{e}_{j}\left(s\right){\int }_{{t}_{0}}^{s}{f}_{j}\left(\theta \right)‖{u}_{1}\left(\theta \right)‖\text{d}\theta \text{d}s\\ +{M}_{2}{\int }_{{t}_{0}}^{t}\underset{j=1}{\overset{J}{\sum }}{g}_{j}\left(s\right){\int }_{{t}_{0}}^{s}{h}_{j}\left(\theta \right)‖{u}_{2}\left(\theta \right)‖{\text{e}}^{\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\\ +{M}_{2}{\int }_{{t}_{0}}^{t}\underset{k=1}{\overset{K}{\sum }}{o}_{k}\left(s\right){\int }_{{t}_{0}}^{s}{p}_{k}\left(\theta \right){‖{u}_{1}\left(\theta \right)‖}^{{\beta }_{k}+1}{\text{e}}^{-{\beta }_{k}\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\end{array}$

$\begin{array}{l}\text{ }+{M}_{2}{\int }_{{t}_{0}}^{t}\underset{k=1}{\overset{K}{\sum }}{q}_{k}\left(s\right){\int }_{{t}_{0}}^{s}{r}_{k}\left(\theta \right){‖{u}_{2}\left(\theta \right)‖}^{{\beta }_{k}+1}{\text{e}}^{\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}s\\ \text{ }+{M}_{2}{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(s\right){‖{u}_{1}\left(s\right)‖}^{{\alpha }_{i}}{\text{e}}^{-{\alpha }_{i}\left(\lambda -\epsilon \right)\left(s-{t}_{0}\right)}{\int }_{{t}_{0}}^{s}\underset{l=1}{\overset{L}{\sum }}{v}_{l}\left(\sigma \right){\int }_{{t}_{0}}^{\sigma }{w}_{l}\left(\theta \right)‖{u}_{1}\left(\theta \right)‖\text{d}\theta \text{d}\sigma \text{d}s\\ \text{ }+{M}_{2}{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(s\right){‖{u}_{2}\left(s\right)‖}^{{\alpha }_{i}}{\int }_{{t}_{0}}^{s}\underset{l=1}{\overset{L}{\sum }}{x}_{l}\left(\sigma \right){\int }_{{t}_{0}}^{\sigma }{y}_{l}\left(\theta \right)‖{u}_{2}\left(\theta \right)‖{\text{e}}^{\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}\sigma \text{d}s\\ \text{ }+{M}_{2}{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{c}_{i}\left(s\right){‖{u}_{2}\left(s\right)‖}^{{\alpha }_{i}}{\text{e}}^{-{\alpha }_{i}\left(\lambda -\epsilon \right)\left(s-{t}_{0}\right)}{\int }_{{t}_{0}}^{s}\underset{m=1}{\overset{M}{\sum }}{A}_{m}\left(\sigma \right){\int }_{{t}_{0}}^{\sigma }{B}_{m}\left(\theta \right){‖{u}_{1}\left(\theta \right)‖}^{{\gamma }_{m}+1}{\text{e}}^{-{\gamma }_{m}\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}\sigma \text{d}s\\ \text{ }+{M}_{2}{\int }_{{t}_{0}}^{t}\underset{i=1}{\overset{I}{\sum }}{d}_{i}\left(s\right){‖{u}_{2}\left(s\right)‖}^{{\alpha }_{i}}{\int }_{\text{\hspace{0.17em}}{t}_{0}}^{\text{\hspace{0.17em}}s}\underset{m=1}{\overset{M}{\sum }}{D}_{m}\left(\sigma \right){\int }_{{t}_{0}}^{\sigma }{E}_{m}\left(\theta \right){‖{u}_{2}\left(\theta \right)‖}^{{\gamma }_{m}+1}{\text{e}}^{\left(\lambda -\epsilon \right)\left(\theta -{t}_{0}\right)}\text{d}\theta \text{d}\sigma \text{d}s\end{array}$

Similarly available: where ${M}_{2}=\mathrm{max}\left\{{M}_{1},{M}_{1}^{{\alpha }_{i}+1},{M}_{1}^{{\beta }_{k}+1},{M}_{1}^{{\gamma }_{m}+1},{M}_{1}^{{\alpha }_{i}+{\gamma }_{m}+1}\right\}$.

So it can be obtained from Lemma: ${u}_{1}\left(t\right)\le k{\text{e}}^{\left(\lambda -\epsilon \right)\left(t-{t}_{0}\right)}\Omega \left(t\right)$, ${u}_{2}\left(t\right)\le k\Omega \left(t\right)$ here $k={\text{e}}^{\lambda {t}_{0}}\varphi$, $\varphi =\mathrm{max}\left({\phi }^{\left(1\right)},{\phi }^{\left(2\right)}\right)$, $\Omega \left(t\right)$ As the lemma states, then:

$‖y\left(t\right)‖+‖\stackrel{˙}{y}\left(t\right)‖\le {M}_{3}\phi {\text{e}}^{-\epsilon \left(t-{t}_{0}\right)}\text{\hspace{0.17em}}\left(\lambda >\epsilon \right)$, $‖y\left(t\right)‖+‖\stackrel{˙}{y}\left(t\right)‖\le {M}_{4}\phi {\text{e}}^{-\epsilon \left(t-{t}_{0}\right)}\text{\hspace{0.17em}}\left(\lambda =\epsilon \right)$

$‖z\left(t\right)‖+‖\stackrel{˙}{z}\left(t\right)‖\le {M}_{5}\phi$ (6)

here: ${M}_{3}={\text{e}}^{\left(\lambda -\epsilon \right){t}_{0}}\Omega \left(t\right)$ ; ${M}_{4}=\Omega \left(t\right)$ ; ${M}_{5}={\text{e}}^{\lambda {t}_{0}}\Omega \left(t\right)$.

Notice the theorem conditions, we have ${M}_{1}$ is a constant that has nothing to do with ${t}_{0}$, ${M}_{2}$ and ${M}_{3}$ are constants that has nothing to do with ${t}_{0}$.

Therefore, when $\lambda >\epsilon$, (6) means the trivial solution of (2) is $LS$, $G{E}_{q}ELAS$ with respect to y; when $\lambda =\epsilon$, (6) means the trivial solution of (2) is $LS$, $GUELAS$ with respect to y.

Note: The differential system discussed in this paper is the time-differential form of the ordinary differential system in  . The time-differential system in  is generalized to a neutral system, and the Lipschitz stability in  is further extended to equi-exponential Lipschitz asymptotic stability and uniform exponential Lipschitz asymptotic stability and added global results

4. Conclusion

In this paper, we use the method of integral inequalities to establish double stability criteria. As a result, studying the partial stability of differential equations becomes more important. In addition, the partial stability of differential equations is widely used in science and technology.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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