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

作者: 卢钰松 , 黄星寿 :河池学院数学与统计学院,广西 宜州;

关键词: 非线性积分不等式含有未知导函数的三重积分估计Nonlinear Integral Inequality Triple Integral with Unknown Derivative Function Estimation

摘要:
本文研究了一类含有未知导函数的三重非线性积分不等式,利用变量替换、放大、微分、积分等不等式技巧给出了不等式中未知函数的估计,推广了相应的结果。

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

在微分方程和积分方程解存在性、有界性和唯一性等定性性质的研究中,Gronwall [1] 型积分不等式是一种重要的工具,因此人们不断的对它进行研究,将它的形式进行推广,使其在微分方程和积分方程中的应用范围不断增大 [2] - [7]。

PACHPATTE [8] 于1998年研究了以下积分不等式:

u ( t ) a ( t ) + b ( t ) 0 t c ( s ) ( u ( s ) + u ( s ) ) d s ( t R )

u ( t ) u ( 0 ) + 0 t a ( s ) ( u ( s ) + u ( s ) ) d s + 0 t a ( s ) ( 0 s b ( σ ) u ( σ ) d σ ) d s ( t R + )

该积分不等式的积分号内包含未知函数及其导函数。

ZAREEN [9] 于2014年,发表文章介绍了非线性积分不等式:

u ( t ) c + 0 t k ( s ) u ( s ) ( u p ( s ) + u 2 ( s ) ) d s ( t R + )

的解的估计。

黄星寿,王五生等 [10] 于2019年,研究了一类积分号内包含未知函数及其导函数的非线性二重积分不等式:

u ( t ) w ( t ) + p ( t ) { u ( t ) + t 0 t [ a ( τ ) [ u ( τ ) + u ( τ ) ] + a ( τ ) t 0 τ [ c ( s ) u ( s ) ( u ( s ) + u ( s ) ) + d ( s ) ] d s ] d τ } ( t [ t 0 , ) )

受以上研究成果的启发,本文构造了以下积分号内包含未知函数及其导函数的非线性积分不等式:

u ( t ) w ( t ) + p ( t ) { u ( t ) + t 0 t [ a ( τ ) ( u ( τ ) + u ( τ ) ) + a ( τ ) t 0 τ [ c ( s ) ( u ( s ) + u ( s ) ) + c ( s ) t 0 s [ d ( v ) u ( v ) ( u ( v ) + u ( v ) ) + f ( v ) ] d v ] d s ] d τ } ( t [ t 0 , ) ) (1)

此类积分不等式将文献 [10] 推广到了三重积分,文中利用放大、变量替换、微分–积分等不等式技巧给出了不等式(1)中未知函数的估计。

2. 预备知识

引理 [10] 假设函数 a ( t ) , b ( t ) c ( t ) 都是定义在 [ t 0 , ) 上的非负连续已知函数,并且函数 a ( t ) [ t 0 , ) 上的增函数,未知函数 u ( t ) 满足不等式

u ( t ) a ( t ) + t 0 t b ( s ) u ( s ) d s + t 0 t c ( s ) u 2 ( s ) d s ( t [ t 0 , ) ) (2)

如果 exp ( ln a ( t ) t 0 t b ( s ) d s ) t 0 t c ( s ) d s > 0 ,则有未知函数 u ( t ) 的估计式:

u ( t ) ( exp ( ln a ( t ) t 0 t b ( s ) d s ) t 0 t c ( s ) d s ) 1 ( t [ t 0 , ) ) (3)

3. 主要结果

定理 假设已知函数 w ( t ) , p ( t ) , a ( t ) , c ( t ) , d ( t ) , f ( t ) 都是定义在 [ t 0 , ) 上的非负连续函数,未知函数 u ( t ) u ( t ) 定义在 [ t 0 , ) 上,且满足不等式(1)。

如果

exp ( ln ( u ( t 0 ) + t 0 t A ( s ) d s ) t 0 t B ( s ) d s ) t 0 t C ( s ) d s > 0 ( t [ t 0 , ) ) (4)

那么未知函数 u ( t ) 的估计式为:

u ( t ) u ( t 0 ) + t 0 t { w ( s ) [ 1 + a ( s ) + c ( s ) ] + [ p ( s ) + a ( s ) + a ( s ) p ( s ) + 2 c ( s ) + c ( s ) p ( s ) ] Z ( s ) } d s ( t [ t 0 , ) ) (5)

其中

Z ( t ) : = ( exp ( ln ( u ( t 0 ) + t 0 t A ( s ) d s ) t 0 t B ( s ) d s ) t 0 t C ( s ) d s ) 1 (6)

A ( t ) : = w ( t ) [ 1 + a ( t ) + c ( t ) ] + d ( t ) w 2 ( t ) + f ( t ) (7)

B ( t ) : = p ( t ) + a ( t ) + a ( t ) p ( t ) + 2 c ( t ) + c ( t ) p ( t ) + d ( t ) w ( t ) + 2 d ( t ) w ( t ) p ( t ) (8)

C ( t ) : = d ( t ) p ( t ) + d ( t ) p 2 ( t ) (9)

证明:令函数

z 1 ( t ) = u ( t ) + t 0 t [ a ( τ ) ( u ( τ ) + u ( τ ) ) + a ( τ ) t 0 τ [ c ( s ) ( u ( s ) + u ( s ) ) + c ( s ) t 0 s [ d ( v ) u ( v ) ( u ( v ) + u ( v ) ) + f ( v ) ] d v ] d s ] d τ ( t [ t 0 , ) ) (10)

则有

z 1 ( t 0 ) = u ( t 0 ) u ( t ) z 1 ( t ) u ( t ) w ( t ) + p ( t ) z 1 ( t ) (11)

对(10)式两边求导,并将(1)式代入,则可得:

z 1 ( t ) = u ( t ) + a ( t ) [ u ( t ) + u ( t ) ] + a ( t ) t 0 t [ c ( s ) ( u ( s ) + u ( s ) ) + c ( s ) t 0 s [ d ( v ) u ( v ) ( u ( v ) + u ( v ) ) + f ( v ) ] d v ] d s w ( t ) + p ( t ) z 1 ( t ) + a ( t ) [ z 1 ( t ) + w ( t ) + p ( t ) z 1 ( t ) ] + a ( t ) t 0 t [ c ( s ) [ z 1 ( s ) + w ( s ) + p ( s ) z 1 ( s ) ] + c ( s ) t 0 s [ d ( v ) ( w ( v ) + p ( v ) z 1 ( v ) ) ( z 1 ( v ) + w ( v ) + p ( v ) z 1 ( v ) ) + f ( v ) ] d v ] d s

= w ( t ) [ 1 + a ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) ] z 1 ( t ) + a ( t ) t 0 t [ c ( s ) w ( s ) + [ c ( s ) ( 1 + p ( s ) ) z 1 ( s ) ] + c ( s ) t 0 s [ d ( v ) w 2 ( v ) + [ d ( v ) w ( v ) + 2 d ( v ) w ( v ) p ( v ) ] z 1 ( v ) + [ d ( v ) p ( v ) + d ( v ) p 2 ( v ) ] z 1 2 ( v ) + f ( v ) ] d v ] d s ( t [ t 0 , ) ) (12)

又令

z 2 ( t ) = z 1 ( t ) + t 0 t [ c ( s ) w ( s ) + [ c ( s ) ( 1 + p ( s ) ) z 1 ( s ) ] + c ( s ) t 0 s [ d ( v ) w 2 ( v ) + [ d ( v ) w ( v ) + 2 d ( v ) w ( v ) p ( v ) ] z 1 ( v ) + [ d ( v ) p ( v ) + d ( v ) p 2 ( v ) ] z 1 2 ( v ) + f ( v ) ] d v ] d s ( t [ t 0 , ) ) (13)

则有

z 2 ( t 0 ) = z 1 ( t 0 ) z 1 ( t ) z 2 ( t ) ( t [ t 0 , ) ) (14)

根据(12)~(14)式,可得:

z 1 ( t ) w ( t ) [ 1 + a ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) ] z 1 ( t ) + a ( t ) z 2 ( t ) w ( t ) [ 1 + a ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + a ( t ) ] z 2 ( t ) ( t [ t 0 , ) ) (15)

又对(13)式两边求导,并将(15)式代入,可得:

z 2 ( t ) = z 1 ( t ) + c ( t ) w ( t ) + [ c ( t ) ( 1 + p ( t ) ) z 1 ( t ) ] + c ( t ) t 0 t [ d ( v ) w 2 ( v ) + [ d ( v ) w ( v ) + 2 d ( v ) w ( v ) p ( v ) ] z 1 ( v ) + [ d ( v ) p ( v ) + d ( v ) p 2 ( v ) ] z 1 2 ( v ) + f ( v ) ] d v w ( t ) [ 1 + a ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + a ( t ) ] z 2 ( t ) + c ( t ) w ( t ) + [ c ( t ) ( 1 + p ( t ) ) z 2 ( t ) ]

+ c ( t ) t 0 t [ d ( v ) w 2 ( v ) + [ d ( v ) w ( v ) + 2 d ( v ) w ( v ) p ( v ) ] z 2 ( v ) + [ d ( v ) p ( v ) + d ( v ) p 2 ( v ) ] z 2 2 ( v ) + f ( v ) ] d v = w ( t ) [ 1 + a ( t ) + c ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + c ( t ) + c ( t ) p ( t ) ] z 2 ( t ) + c ( t ) t 0 t [ d ( v ) w 2 ( v ) + [ d ( v ) w ( v ) + 2 d ( v ) w ( v ) p ( v ) ] z 2 ( v ) + [ d ( v ) p ( v ) + d ( v ) p 2 ( v ) ] z 2 2 ( v ) + f ( v ) ] d v ( t [ t 0 , ) ) (16)

再令

z 3 ( t ) = z 2 ( t ) + t 0 t [ d ( v ) w 2 ( v ) + [ d ( v ) w ( v ) + 2 d ( v ) w ( v ) p ( v ) ] z 2 ( v ) + [ d ( v ) p ( v ) + d ( v ) p 2 ( v ) ] z 2 2 ( v ) + f ( v ) ] d v ( t [ t 0 , ) ) (17)

则有

z 3 ( t 0 ) = z 2 ( t 0 ) z 2 ( t ) z 3 ( t ) ( t [ t 0 , ) ) (18)

根据(16)~(18)式,可得:

z 2 ( t ) w ( t ) [ 1 + a ( t ) + c ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + c ( t ) + c ( t ) p ( t ) ] z 2 ( t ) + c ( t ) z 3 ( t ) w ( t ) [ 1 + a ( t ) + c ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + 2 c ( t ) + c ( t ) p ( t ) ] z 3 ( t ) ( t [ t 0 , ) ) (19)

再对(17)式两边求导,并将(19)式代入,可得:

z 3 ( t ) = z 2 ( t ) + d ( t ) w 2 ( t ) + [ d ( t ) w ( t ) + 2 d ( t ) w ( t ) p ( t ) ] z 2 ( t ) + [ d ( t ) p ( t ) + d ( t ) p 2 ( t ) ] z 2 2 ( t ) + f ( t ) w ( t ) [ 1 + a ( t ) + c ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + 2 c ( t ) + c ( t ) p ( t ) ] z 3 ( t ) + d ( t ) w 2 ( t ) + [ d ( t ) w ( t ) + 2 d ( t ) w ( t ) p ( t ) ] z 3 ( t ) + [ d ( t ) p ( t ) + d ( t ) p 2 ( t ) ] z 3 2 ( t ) + f ( t )

= [ w ( t ) [ 1 + a ( t ) + c ( t ) ] + d ( t ) w 2 ( t ) + f ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + 2 c ( t ) + c ( t ) p ( t ) + d ( t ) w ( t ) + 2 d ( t ) w ( t ) p ( t ) ] z 3 ( t ) + [ d ( t ) p ( t ) + d ( t ) p 2 ( t ) ] z 3 2 ( t ) = A ( t ) + B ( t ) z 3 ( t ) + C ( t ) z 3 2 ( t ) ( t [ t 0 , ) ) (20)

其中, A ( t ) , B ( t ) , C ( t ) 的定义为(7)~(9)式。

将(20)式中的t改写为s,然后两边关于s从 t 0 到t积分,得:

z 3 ( t ) z 3 ( t 0 ) + t 0 t A ( s ) d s + t 0 t B ( s ) z 3 ( s ) d s + t 0 t C ( s ) z 3 2 ( s ) d s ( t [ t 0 , ) ) (21)

由于(21)式满足了引理要求的条件,根据引理可以得到(21)式中 z 3 ( t ) 的估计:

z 3 ( t ) ( exp ( ln ( z 3 ( t 0 ) + t 0 t A ( s ) d s ) t 0 t B ( s ) d s ) t 0 t C ( s ) d s ) 1 ( t [ t 0 , ) ) (22)

根据(11)、(14)、(18)式,可以知道

u ( t 0 ) = z 1 ( t 0 ) = z 2 ( t 0 ) = z 3 ( t 0 ) (23)

所以

z 3 ( t ) ( exp ( ln ( u ( t 0 ) + t 0 t A ( s ) d s ) t 0 t B ( s ) d s ) t 0 t C ( s ) d s ) 1 = Z ( t ) ( t [ t 0 , ) ) (24)

其中 Z ( t ) 定义为(6)式。

把(24)式代入(19)式,可得:

z 2 ( t ) w ( t ) [ 1 + a ( t ) + c ( t ) ] + [ p ( t ) + a ( t ) + a ( t ) p ( t ) + 2 c ( t ) + c ( t ) p ( t ) ] Z ( t ) ( t [ t 0 , ) ) (25)

对(25)式两边求积分,可得:

z 2 ( t ) z 2 ( t 0 ) + t 0 t { w ( s ) [ 1 + a ( s ) + c ( s ) ] + [ p ( s ) + a ( s ) + a ( s ) p ( s ) + 2 c ( s ) + c ( s ) p ( s ) ] Z ( s ) } d s ( t [ t 0 , ) ) (26)

根据(23)式知 u ( t 0 ) = z 2 ( t 0 ) ,因此得到:

z 2 ( t ) u ( t 0 ) + t 0 t { w ( s ) [ 1 + a ( s ) + c ( s ) ] + [ p ( s ) + a ( s ) + a ( s ) p ( s ) + 2 c ( s ) + c ( s ) p ( s ) ] Z ( s ) } d s ( t [ t 0 , ) )

又根据(11)、(14)、(18)式,可以知道

u ( t ) z 1 ( t ) z 2 ( t ) z 3 ( t ) (27)

所以得估计式:

u ( t ) u ( t 0 ) + t 0 t { w ( s ) [ 1 + a ( s ) + c ( s ) ] + [ p ( s ) + a ( s ) + a ( s ) p ( s ) + 2 c ( s ) + c ( s ) p ( s ) ] Z ( s ) } d s ( t [ t 0 , ) )

证毕。

基金项目

国家自然科学基金资助项目(11961021,11561019);广西高校中青年教师科研基础能力提升项目(2019KY0625)。

NOTES

*通讯作者。

文章引用: 卢钰松 , 黄星寿 (2020) 一类三重非线性积分不等式解的估计。 应用数学进展, 9, 1200-1205. doi: 10.12677/AAM.2020.98140

参考文献

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https://doi.org/10.2307/1967124

[2] Bellman, R. (1943) The Stability of Solutions of Linear Differential Equations. Duke Mathematical Journal, 10, 643-647.
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[3] Ma, Q.H. and Pecaric, J. (2008) Estimates on Solutions of Some New Nonlinear Retarded Volterra-Fredholm Type Integral Inequalities. Nonlinear Analysis: Theory, Methods & Applications, 69, 393-407.
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[4] 王五生, 李自尊. 一类新的非线性和差分不等式及其应用[J]. 系统科学与数学, 2012, 32(2): 181-189.

[5] Zheng, B. (2012) (G’/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics. Communications in Theoretical Physics, 58, 623-630.
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[7] 卢钰松, 王五生. 一类含有p次幂的Volterra-Fredholm型非线性迭代积分不等式[J]. 西南大学学报(自然科学版), 2015, 27(8): 76-80.

[8] Pachpatte, B.G. (1998) Inequalities for Differential and Integral Equations. Academic Press, New York, p. 45.

[9] Zareen, A.K. (2014) On Some Fundamental Integrodifferential Inequalities. Applied Mathematics, 5, 2968-2973.
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[10] 黄星寿, 王五生, 罗日才. 一类二重积分不等式中未知函数的估计[J]. 华南师范大学学报(自然科学版), 2019, 51(3): 108-111.

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