﻿ 一类三重非线性积分不等式解的估计

# 一类三重非线性积分不等式解的估计Estimation of Unknown Function of a Class of Triple Integral Inequalities

Abstract: In this paper, the author establishes a class of triple nonlinear integral inequalities with unknown derivative functions, and gives the estimation of unknown functions in inequalities by using the inequality techniques such as variable substitution, amplification, differential and integral.

1. 引言

PACHPATTE [8] 于1998年研究了以下积分不等式：

${u}^{\prime }\left(t\right)\le a\left(t\right)+b\left(t\right){\int }_{0}^{t}c\left(s\right)\left(u\left(s\right)+{u}^{\prime }\left(s\right)\right)\text{d}s\text{ }\left(t\in R\right)$

${u}^{\prime }\left(t\right)\le u\left(0\right)+{\int }_{0}^{t}a\left(s\right)\left(u\left(s\right)+{u}^{\prime }\left(s\right)\right)\text{d}s+{\int }_{0}^{t}a\left(s\right)\left({\int }_{0}^{s}b\left(\sigma \right){u}^{\prime }\left(\sigma \right)\text{d}\sigma \right)\text{d}s\text{ }\left(t\in {R}_{+}\right)$

ZAREEN [9] 于2014年，发表文章介绍了非线性积分不等式：

${u}^{\prime }\left(t\right)\le c+{\int }_{0}^{t}k\left(s\right){u}^{\prime }\left(s\right)\left({{u}^{\prime }}^{p}\left(s\right)+{u}^{2}\left(s\right)\right)\text{d}s\text{ }\left(t\in {R}_{+}\right)$

$\begin{array}{l}{u}^{\prime }\left(t\right)\le w\left(t\right)+p\left(t\right)\left\{u\left(t\right)+{\int }_{{t}_{0}}^{t}\left[a\left(\tau \right)\left[u\left(\tau \right)+{u}^{\prime }\left(\tau \right)\right]\underset{}{}\\ \text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+a\left(\tau \right){\int }_{{t}_{0}}^{\tau }\left[c\left(s\right){u}^{\prime }\left(s\right)\left({u}^{\prime }\left(s\right)+u\left(s\right)\right)+d\left(s\right)\right]\text{d}s\right]\text{d}\tau \right\}\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$

$\begin{array}{l}{u}^{\prime }\left(t\right)\le w\left(t\right)+p\left(t\right)\left\{u\left(t\right)+{\int }_{{t}_{0}}^{t}\left[a\left(\tau \right)\left(u\left(\tau \right)+{u}^{\prime }\left(\tau \right)\right)+a\left(\tau \right){\int }_{{t}_{0}}^{\tau }\left[c\left(s\right)\left(u\left(s\right)+{u}^{\prime }\left(s\right)\right)\\ \text{ }\text{ }\text{ }+c\left(s\right){\int }_{{t}_{0}}^{s}\left[d\left(v\right){u}^{\prime }\left(v\right)\left(u\left(v\right)+{u}^{\prime }\left(v\right)\right)+f\left(v\right)\right]\text{d}v\right]\text{d}s\right]\text{d}\tau \right\}\text{ }\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (1)

2. 预备知识

$u\left(t\right)\le a\left(t\right)+{\int }_{{t}_{0}}^{t}b\left(s\right)u\left(s\right)\text{d}s+{\int }_{{t}_{0}}^{t}c\left(s\right){u}^{2}\left(s\right)\text{d}s\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (2)

$u\left(t\right)\le {\left(\mathrm{exp}\left(-\mathrm{ln}a\left(t\right)-{\int }_{{t}_{0}}^{t}b\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}c\left(s\right)\text{d}s\right)}^{-1}\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (3)

3. 主要结果

$\mathrm{exp}\left(-\mathrm{ln}\left(u\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}A\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}B\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}C\left(s\right)\text{d}s>0\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (4)

$\begin{array}{l}u\left(t\right)\le u\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}\left\{w\left(s\right)\left[1+a\left(s\right)+c\left(s\right)\right]+\left[p\left(s\right)+a\left(s\right)+a\left(s\right)p\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+2c\left(s\right)+c\left(s\right)p\left(s\right)\right]Z\left(s\right)\right\}\text{d}s\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (5)

$Z\left(t\right):={\left(\mathrm{exp}\left(-\mathrm{ln}\left(u\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}A\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}B\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}C\left(s\right)\text{d}s\right)}^{-1}$ (6)

$A\left(t\right):=w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+d\left(t\right){w}^{2}\left(t\right)+f\left(t\right)$ (7)

$B\left(t\right):=p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+2c\left(t\right)+c\left(t\right)p\left(t\right)+d\left(t\right)w\left(t\right)+2d\left(t\right)w\left(t\right)p\left(t\right)$ (8)

$C\left(t\right):=d\left(t\right)p\left(t\right)+d\left(t\right){p}^{2}\left(t\right)$ (9)

$\begin{array}{l}{z}_{1}\left(t\right)=u\left(t\right)+{\int }_{{t}_{0}}^{t}\left[a\left(\tau \right)\left(u\left(\tau \right)+{u}^{\prime }\left(\tau \right)\right)+a\left(\tau \right){\int }_{{t}_{0}}^{\tau }\left[c\left(s\right)\left(u\left(s\right)+{u}^{\prime }\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(s\right){\int }_{{t}_{0}}^{s}\left[d\left(v\right){u}^{\prime }\left(v\right)\left(u\left(v\right)+{u}^{\prime }\left(v\right)\right)+f\left(v\right)\right]\text{d}v\right]\text{d}s\right]\text{d}\tau \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (10)

${z}_{1}\left({t}_{0}\right)=u\left({t}_{0}\right)$$u\left(t\right)\le {z}_{1}\left(t\right)$${u}^{\prime }\left(t\right)\le w\left(t\right)+p\left(t\right){z}_{1}\left(t\right)$ (11)

$\begin{array}{c}{{z}^{\prime }}_{1}\left(t\right)={u}^{\prime }\left(t\right)+a\left(t\right)\left[u\left(t\right)+{u}^{\prime }\left(t\right)\right]+a\left(t\right){\int }_{{t}_{0}}^{t}\left[c\left(s\right)\left(u\left(s\right)+{u}^{\prime }\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(s\right){\int }_{{t}_{0}}^{s}\left[d\left(v\right){u}^{\prime }\left(v\right)\left(u\left(v\right)+{u}^{\prime }\left(v\right)\right)+f\left(v\right)\right]\text{d}v\right]\text{d}s\\ \le w\left(t\right)+p\left(t\right){z}_{1}\left(t\right)+a\left(t\right)\left[{z}_{1}\left(t\right)+w\left(t\right)+p\left(t\right){z}_{1}\left(t\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+a\left(t\right){\int }_{{t}_{0}}^{t}\left[c\left(s\right)\left[{z}_{1}\left(s\right)+w\left(s\right)+p\left(s\right){z}_{1}\left(s\right)\right]+c\left(s\right){\int }_{{t}_{0}}^{s}\left[d\left(v\right)\left(w\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+p\left(v\right){z}_{1}\left(v\right)\right)\left({z}_{1}\left(v\right)+w\left(v\right)+p\left(v\right){z}_{1}\left(v\right)\right)+f\left(v\right)\right]\text{d}v\right]\text{d}s\end{array}$

$\begin{array}{l}=w\left(t\right)\left[1+a\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)\right]{z}_{1}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+a\left(t\right){\int }_{{t}_{0}}^{t}\left[c\left(s\right)w\left(s\right)+\left[c\left(s\right)\left(1+p\left(s\right)\right){z}_{1}\left(s\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(s\right){\int }_{{t}_{0}}^{s}\left[d\left(v\right){w}^{2}\left(v\right)+\left[d\left(v\right)w\left(v\right)+2d\left(v\right)w\left(v\right)p\left(v\right)\right]{z}_{1}\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(v\right)p\left(v\right)+d\left(v\right){p}^{2}\left(v\right)\right]{z}_{1}^{2}\left(v\right)+f\left(v\right)\right]\text{d}v\right]\text{d}s\text{ }\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (12)

$\begin{array}{c}{z}_{2}\left(t\right)={z}_{1}\left(t\right)+{\int }_{{t}_{0}}^{t}\left[c\left(s\right)w\left(s\right)+\left[c\left(s\right)\left(1+p\left(s\right)\right){z}_{1}\left(s\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(s\right){\int }_{{t}_{0}}^{s}\left[d\left(v\right){w}^{2}\left(v\right)+\left[d\left(v\right)w\left(v\right)+2d\left(v\right)w\left(v\right)p\left(v\right)\right]{z}_{1}\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(v\right)p\left(v\right)+d\left(v\right){p}^{2}\left(v\right)\right]{z}_{1}^{2}\left(v\right)+f\left(v\right)\right]\text{d}v\right]\text{d}s\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (13)

${z}_{2}\left({t}_{0}\right)={z}_{1}\left({t}_{0}\right)$${z}_{1}\left(t\right)\le {z}_{2}\left(t\right)$$\left(t\in \left[{t}_{0},\infty \right)\right)$ (14)

$\begin{array}{c}{{z}^{\prime }}_{1}\left(t\right)\le w\left(t\right)\left[1+a\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)\right]{z}_{1}\left(t\right)+a\left(t\right){z}_{2}\left(t\right)\\ \le w\left(t\right)\left[1+a\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+a\left(t\right)\right]{z}_{2}\left(t\right)\text{ }\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (15)

$\begin{array}{c}{{z}^{\prime }}_{2}\left(t\right)={{z}^{\prime }}_{1}\left(t\right)+c\left(t\right)w\left(t\right)+\left[c\left(t\right)\left(1+p\left(t\right)\right){z}_{1}\left(t\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(t\right){\int }_{{t}_{0}}^{t}\left[d\left(v\right){w}^{2}\left(v\right)+\left[d\left(v\right)w\left(v\right)+2d\left(v\right)w\left(v\right)p\left(v\right)\right]{z}_{1}\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(v\right)p\left(v\right)+d\left(v\right){p}^{2}\left(v\right)\right]{z}_{1}^{2}\left(v\right)+f\left(v\right)\right]\text{d}v\\ \le w\left(t\right)\left[1+a\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+a\left(t\right)\right]{z}_{2}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(t\right)w\left(t\right)+\left[c\left(t\right)\left(1+p\left(t\right)\right){z}_{2}\left(t\right)\right]\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(t\right){\int }_{{t}_{0}}^{t}\left[d\left(v\right){w}^{2}\left(v\right)+\left[d\left(v\right)w\left(v\right)+2d\left(v\right)w\left(v\right)p\left(v\right)\right]{z}_{2}\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(v\right)p\left(v\right)+d\left(v\right){p}^{2}\left(v\right)\right]{z}_{2}^{2}\left(v\right)+f\left(v\right)\right]\text{d}v\\ =w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+c\left(t\right)+c\left(t\right)p\left(t\right)\right]{z}_{2}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(t\right){\int }_{{t}_{0}}^{t}\left[d\left(v\right){w}^{2}\left(v\right)+\left[d\left(v\right)w\left(v\right)+2d\left(v\right)w\left(v\right)p\left(v\right)\right]{z}_{2}\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(v\right)p\left(v\right)+d\left(v\right){p}^{2}\left(v\right)\right]{z}_{2}^{2}\left(v\right)+f\left(v\right)\right]\text{d}v\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (16)

$\begin{array}{c}{z}_{3}\left(t\right)={z}_{2}\left(t\right)+{\int }_{{t}_{0}}^{t}\left[d\left(v\right){w}^{2}\left(v\right)+\left[d\left(v\right)w\left(v\right)+2d\left(v\right)w\left(v\right)p\left(v\right)\right]{z}_{2}\left(v\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(v\right)p\left(v\right)+d\left(v\right){p}^{2}\left(v\right)\right]{z}_{2}^{2}\left(v\right)+f\left(v\right)\right]\text{d}v\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (17)

${z}_{3}\left({t}_{0}\right)={z}_{2}\left({t}_{0}\right)$${z}_{2}\left(t\right)\le {z}_{3}\left(t\right)$$\left(t\in \left[{t}_{0},\infty \right)\right)$ (18)

$\begin{array}{c}{{z}^{\prime }}_{2}\left(t\right)\le w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+c\left(t\right)+c\left(t\right)p\left(t\right)\right]{z}_{2}\left(t\right)+c\left(t\right){z}_{3}\left(t\right)\\ \le w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+2c\left(t\right)+c\left(t\right)p\left(t\right)\right]{z}_{3}\left(t\right)\text{ }\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (19)

$\begin{array}{c}{{z}^{\prime }}_{3}\left(t\right)={{z}^{\prime }}_{2}\left(t\right)+d\left(t\right){w}^{2}\left(t\right)+\left[d\left(t\right)w\left(t\right)+2d\left(t\right)w\left(t\right)p\left(t\right)\right]{z}_{2}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(t\right)p\left(t\right)+d\left(t\right){p}^{2}\left(t\right)\right]{z}_{2}^{2}\left(t\right)+f\left(t\right)\\ \le w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+2c\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+c\left(t\right)p\left(t\right)\right]{z}_{3}\left(t\right)+d\left(t\right){w}^{2}\left(t\right)+\left[d\left(t\right)w\left(t\right)+2d\left(t\right)w\left(t\right)p\left(t\right)\right]{z}_{3}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[d\left(t\right)p\left(t\right)+d\left(t\right){p}^{2}\left(t\right)\right]{z}_{3}^{2}\left(t\right)+f\left( t \right)\end{array}$

$\begin{array}{l}=\left[w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+d\left(t\right){w}^{2}\left(t\right)+f\left(t\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+2c\left(t\right)+c\left(t\right)p\left(t\right)+d\left(t\right)w\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2d\left(t\right)w\left(t\right)p\left(t\right)\right]{z}_{3}\left(t\right)+\left[d\left(t\right)p\left(t\right)+d\left(t\right){p}^{2}\left(t\right)\right]{z}_{3}^{2}\left(t\right)\\ =A\left(t\right)+B\left(t\right){z}_{3}\left(t\right)+C\left(t\right){z}_{3}^{2}\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (20)

${z}_{3}\left(t\right)\le {z}_{3}\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}A\left(s\right)\text{d}s+{\int }_{{t}_{0}}^{t}B\left(s\right){z}_{3}\left(s\right)\text{d}s+{\int }_{{t}_{0}}^{t}C\left(s\right){z}_{3}^{2}\left(s\right)\text{d}s\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (21)

${z}_{3}\left(t\right)\le {\left(\mathrm{exp}\left(-\mathrm{ln}\left({z}_{3}\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}A\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}B\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}C\left(s\right)\text{d}s\right)}^{-1}\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (22)

$u\left({t}_{0}\right)={z}_{1}\left({t}_{0}\right)={z}_{2}\left({t}_{0}\right)={z}_{3}\left({t}_{0}\right)$ (23)

${z}_{3}\left(t\right)\le {\left(\mathrm{exp}\left(-\mathrm{ln}\left(u\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}A\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}B\left(s\right)\text{d}s\right)-{\int }_{{t}_{0}}^{t}C\left(s\right)\text{d}s\right)}^{-1}=Z\left(t\right)\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (24)

${{z}^{\prime }}_{2}\left(t\right)\le w\left(t\right)\left[1+a\left(t\right)+c\left(t\right)\right]+\left[p\left(t\right)+a\left(t\right)+a\left(t\right)p\left(t\right)+2c\left(t\right)+c\left(t\right)p\left(t\right)\right]Z\left(t\right)\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)$ (25)

$\begin{array}{l}{z}_{2}\left(t\right)\le {z}_{2}\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}\left\{w\left(s\right)\left[1+a\left(s\right)+c\left(s\right)\right]+\left[p\left(s\right)+a\left(s\right)+a\left(s\right)p\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2c\left(s\right)+c\left(s\right)p\left(s\right)\right]Z\left(s\right)\right\}\text{d}s\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$ (26)

$\begin{array}{l}{z}_{2}\left(t\right)\le u\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}\left\{w\left(s\right)\left[1+a\left(s\right)+c\left(s\right)\right]+\left[p\left(s\right)+a\left(s\right)+a\left(s\right)p\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2c\left(s\right)+c\left(s\right)p\left(s\right)\right]Z\left(s\right)\right\}\text{d}s\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$

$u\left(t\right)\le {z}_{1}\left(t\right)\le {z}_{2}\left(t\right)\le {z}_{3}\left(t\right)$ (27)

$\begin{array}{l}u\left(t\right)\le u\left({t}_{0}\right)+{\int }_{{t}_{0}}^{t}\left\{w\left(s\right)\left[1+a\left(s\right)+c\left(s\right)\right]+\left[p\left(s\right)+a\left(s\right)+a\left(s\right)p\left(s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+2c\left(s\right)+c\left(s\right)p\left(s\right)\right]Z\left(s\right)\right\}\text{d}s\text{ }\left(t\in \left[{t}_{0},\infty \right)\right)\end{array}$

NOTES

*通讯作者。

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