一类含有未知导函数的三重积分不等式中未知函数的估计
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摘要
研究了一类非线性三重积分不等式,其中被积函数中含有未知函数及其导函数,积分项外包含了非常数项.利用变量替换技巧、放大技巧和反函数技巧等分析手段,给出了三重积分-微分不等式中未知函数的显上界估计,推广了已有结果.最后举例说明所得结果可以用来研究微分-积分方程解的估计.
Gronwall type integral inequality is the important tool in the study of existence, uniqueness, boundedness and other qualitative properties of solutions of differential equations, integral equation and integro-differential equations. In this paper, a class of nonlinear triple integral inequality is studied, which includes an unknown function and its derivative function in integrand function, and a nonconstant factor outside integral sign. The upper bounds of the unknown function in the integro-differential inequality is estimated explicitly using the techniques of change of variable, the method of amplification, and inverse function technique, which generalized some known results. The derived results can be applied in the study of the explicit upper bounds of solutions of a class of integro-differential equations.
Gronwall type integral inequality is the important tool in the study of existence, uniqueness, boundedness and other qualitative properties of solutions of differential equations, integral equation and integro-differential equations. In this paper, a class of nonlinear triple integral inequality is studied, which includes an unknown function and its derivative function in integrand function, and a nonconstant factor outside integral sign. The upper bounds of the unknown function in the integro-differential inequality is estimated explicitly using the techniques of change of variable, the method of amplification, and inverse function technique, which generalized some known results. The derived results can be applied in the study of the explicit upper bounds of solutions of a class of integro-differential equations.
引文
[1] GRONWALL T H.Note on the Derivatives with Respect to a Parameter of the Solutions of a System of Differential Equations [J].The Annals of Mathematics,1919,20(4):292.
[2] BELLMAN R.The Stability of Solutions of Linear Differential Equations [J].Duke Mathematical Journal,1943,10(4):643-647.
[3] AGARWAL R P,DENG S F,ZHANG W N.Generalization of a Retarded Gronwall-Like Inequality and Its Applications [J].Applied Mathematics and Computation,2005,165(3):599-612.
[4] MA Q H,PE■.Some New Explicit Bounds for Weakly Singular Integral Inequalities with Applications to Fractional Differential and Integral Equations [J].Journal of Mathematical Analysis and Applications,2008,341(2):894-905.
[5] 欧阳云,王五生.一类非线性弱奇异三重积分不等式中未知函数的估计及其应用 [J].西南大学学报(自然科学版),2017,39(3):69-74.
[6] 侯宗毅,王五生.一类变下限非线性Volterra-Fredholm型积分不等式及其应用 [J].西南师范大学学报(自然科学版),2016,41(2):21-25.
[7] 梁英.一类时滞弱奇异Wendroff型积分不等式 [J].四川师范大学学报(自然科学版),2014,37(4):493-496.
[8] ABDELDAIM A.Nonlinear Retarded Integral Inequalities of Gronwall-Bellman Type and Applications [J].Journal of Mathematical Inequalities,2016(1):285-299.
[9] XU R,MENG F W.Some New Weakly Singular Integral Inequalities and Their Applications to Fractional Differential Equations [J].Journal of Inequalities and Applications,2016,2016:78.
[10] 黄春妙,王五生.弱奇异迭代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),2017,49(4):111-114.
[11] EN MI Y.A Generalized Gronwall-Bellman Type Delay Integral Inequality with Two Independent Variables on Time Scales [J].Journal of Mathematical Inequalities,2017(4):1151-1160.
[12] ZHOU J,SHEN J,ZHANG W N.A Powered Gronwall-Type Inequality and Applications to Stochastic Differential Equations [J].Discrete and Continuous Dynamical Systems,2016,36(12):7207-7234.
[13] PACHPATTE B G.Inequalities for Differential and Integral Equations [M].New York:Academic Press,1998.
[14] AKIN-BOHNER E,BOHNER M,AKIN F.Pachpatte Inequalities on Time Scales [J].Journal of Inequalities in Pure and Applied Mathematics,2005,6(1):1-23.
[15] KHAN Z A.On some Fundamental Integrodifferential Inequalities [J].Applied Mathematics,2014,5(19):2968-2973.
[16] JIANG F C,MENG F W.Explicit Bounds on Some New Nonlinear Integral Inequalities with Delay [J].Journal of Computational and Applied Mathematics,2007,205(1):479-486.
[17] MA Q H,PE■.Some New Explicit Bounds for Weakly Singular Integral Inequalities with Applications to Fractional Differential and Integral Equations [J].Journal of Mathematical Analysis and Applications,2008,341(2):894-905.
[2] BELLMAN R.The Stability of Solutions of Linear Differential Equations [J].Duke Mathematical Journal,1943,10(4):643-647.
[3] AGARWAL R P,DENG S F,ZHANG W N.Generalization of a Retarded Gronwall-Like Inequality and Its Applications [J].Applied Mathematics and Computation,2005,165(3):599-612.
[4] MA Q H,PE■.Some New Explicit Bounds for Weakly Singular Integral Inequalities with Applications to Fractional Differential and Integral Equations [J].Journal of Mathematical Analysis and Applications,2008,341(2):894-905.
[5] 欧阳云,王五生.一类非线性弱奇异三重积分不等式中未知函数的估计及其应用 [J].西南大学学报(自然科学版),2017,39(3):69-74.
[6] 侯宗毅,王五生.一类变下限非线性Volterra-Fredholm型积分不等式及其应用 [J].西南师范大学学报(自然科学版),2016,41(2):21-25.
[7] 梁英.一类时滞弱奇异Wendroff型积分不等式 [J].四川师范大学学报(自然科学版),2014,37(4):493-496.
[8] ABDELDAIM A.Nonlinear Retarded Integral Inequalities of Gronwall-Bellman Type and Applications [J].Journal of Mathematical Inequalities,2016(1):285-299.
[9] XU R,MENG F W.Some New Weakly Singular Integral Inequalities and Their Applications to Fractional Differential Equations [J].Journal of Inequalities and Applications,2016,2016:78.
[10] 黄春妙,王五生.弱奇异迭代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),2017,49(4):111-114.
[11] EN MI Y.A Generalized Gronwall-Bellman Type Delay Integral Inequality with Two Independent Variables on Time Scales [J].Journal of Mathematical Inequalities,2017(4):1151-1160.
[12] ZHOU J,SHEN J,ZHANG W N.A Powered Gronwall-Type Inequality and Applications to Stochastic Differential Equations [J].Discrete and Continuous Dynamical Systems,2016,36(12):7207-7234.
[13] PACHPATTE B G.Inequalities for Differential and Integral Equations [M].New York:Academic Press,1998.
[14] AKIN-BOHNER E,BOHNER M,AKIN F.Pachpatte Inequalities on Time Scales [J].Journal of Inequalities in Pure and Applied Mathematics,2005,6(1):1-23.
[15] KHAN Z A.On some Fundamental Integrodifferential Inequalities [J].Applied Mathematics,2014,5(19):2968-2973.
[16] JIANG F C,MENG F W.Explicit Bounds on Some New Nonlinear Integral Inequalities with Delay [J].Journal of Computational and Applied Mathematics,2007,205(1):479-486.
[17] MA Q H,PE■.Some New Explicit Bounds for Weakly Singular Integral Inequalities with Applications to Fractional Differential and Integral Equations [J].Journal of Mathematical Analysis and Applications,2008,341(2):894-905.