December  2016, 36(12): 7207-7234. doi: 10.3934/dcds.2016114

A powered Gronwall-type inequality and applications to stochastic differential equations

1. 

Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan 610066, China

2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received  February 2016 Revised  May 2016 Published  October 2016

In this paper we study a powered integral inequality involving a finite sum, which can be used to solve the inequalities with singular kernels. We present that the solution of the inequality is decided by a finite recursion, whose result is proved to be a continuous, bounded or asymptotic function. Meanwhile, in order to overcome an obstacle from powers of integrals, we modify the method of monotonization into the powered monotonization. Furthermore, relying on the result and our technique of concavification, we discuss a generalized stochastic integral inequality, and give an estimate of the mean square. In the end, as applications, we study uniform boundedness and continuous dependence of solutions for a class of stochastic differential equation in mean square.
Citation: Jun Zhou, Jun Shen, Weinian Zhang. A powered Gronwall-type inequality and applications to stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7207-7234. doi: 10.3934/dcds.2016114
References:
[1]

R. P. Agarwal, S. Deng and W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications,, Appl. Math. Comput., 165 (2005), 599.  doi: 10.1016/j.amc.2004.04.067.  Google Scholar

[2]

K. Amano, A stochastic Gronwall inequality and its applications,, J. Ineq. Pure Appl. Math., 6 (2005).   Google Scholar

[3]

R. Bellman, The stability of solutions of linear differential equations,, Duke Math. J., 10 (1943), 643.  doi: 10.1215/S0012-7094-43-01059-2.  Google Scholar

[4]

I. A. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation,, Acta Math. Acad. Sci. Hung., 7 (1956), 81.  doi: 10.1007/BF02022967.  Google Scholar

[5]

W. Cheung, Q. Ma and S. Tseng, Some new nonlinear weakly singular integral inequalities of Wendroff type with applications,, J. Inequal. Appl., 2008 (2008).  doi: 10.1155/2008/909156.  Google Scholar

[6]

S. Deng and C. Prather, Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay,, J. Ineq. Pure Appl. Math., 9 (2008).   Google Scholar

[7]

A. Friedman, Stochastic Differential Equations and Applications,, Academic Press, (2006).   Google Scholar

[8]

T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,, Ann. Math., 20 (1919), 292.  doi: 10.2307/1967124.  Google Scholar

[9]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge University Press, (1988).  doi: 10.1007/BF01218837.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Math., 840 (1981).   Google Scholar

[11]

K. Itô, On a stochastic integral equation,, Proc. Japan Acad., 22 (1946), 32.  doi: 10.3792/pja/1195572371.  Google Scholar

[12]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Math. Stud., 204 (2006).   Google Scholar

[13]

O. Lipovan, A retarded Gronwall-like inequality and its applications,, J. Math. Anal. Appl., 252 (2000), 389.  doi: 10.1006/jmaa.2000.7085.  Google Scholar

[14]

Q. Ma and E. Yang, Estimates on solutions of some weakly singular Volterra integral inequalities,, Acta Math. Appl. Sinca, 25 (2002), 505.   Google Scholar

[15]

M. Medved, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,, J. Math. Anal. Appl., 214 (1997), 349.  doi: 10.1006/jmaa.1997.5532.  Google Scholar

[16]

G. Mittag-Leffler, Sur la nouvelle fonction $E_\alpha(x)$,, C. R. Acad. Sci. Paris, 137 (1903), 554.   Google Scholar

[17]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, $6^{th}$ edition, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[18]

B. G. Pachpatte, Inequalities for Differential and Integral Equations,, Math. in Sci. and Eng., 197 (1998).   Google Scholar

[19]

M. Pinto, Integral inequalities of Bihari-type and applications,, Funkcial. Ekvac., 33 (1990), 387.   Google Scholar

[20]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Math. Ser., 43 (1993).   Google Scholar

[21]

N.-E. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality,, J. Inequal. Appl., 2006 (2006).  doi: 10.1155/JIA/2006/84561.  Google Scholar

[22]

W. Wang, A generalized retarded Gronwall-like inequality in two variables and applications to BVP,, Appl. Math. Comput., 191 (2007), 144.  doi: 10.1016/j.amc.2007.02.099.  Google Scholar

[23]

M. Wu and N. Huang, Stochastic integral inequalities with applications,, Math. Inequal. Appl., 13 (2010), 667.  doi: 10.7153/mia-13-48.  Google Scholar

[24]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations,, J. Math. Kyoto Univ., 11 (1971), 155.   Google Scholar

[25]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II,, J. Math. Kyoto Univ., 11 (1971), 533.   Google Scholar

[26]

Y. Yan, Nonlinear Gronwall-Bellman type integral inequalities with Maxima,, Math. Inequal. Appl., 16 (2013), 911.  doi: 10.7153/mia-16-71.  Google Scholar

[27]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation,, J. Math. Anal. Appl., 328 (2007), 1075.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[28]

W. Zhang, PM functions, their characteristic intervals and iterative roots,, Ann. Polon. Math., 65 (1997), 119.   Google Scholar

show all references

References:
[1]

R. P. Agarwal, S. Deng and W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications,, Appl. Math. Comput., 165 (2005), 599.  doi: 10.1016/j.amc.2004.04.067.  Google Scholar

[2]

K. Amano, A stochastic Gronwall inequality and its applications,, J. Ineq. Pure Appl. Math., 6 (2005).   Google Scholar

[3]

R. Bellman, The stability of solutions of linear differential equations,, Duke Math. J., 10 (1943), 643.  doi: 10.1215/S0012-7094-43-01059-2.  Google Scholar

[4]

I. A. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation,, Acta Math. Acad. Sci. Hung., 7 (1956), 81.  doi: 10.1007/BF02022967.  Google Scholar

[5]

W. Cheung, Q. Ma and S. Tseng, Some new nonlinear weakly singular integral inequalities of Wendroff type with applications,, J. Inequal. Appl., 2008 (2008).  doi: 10.1155/2008/909156.  Google Scholar

[6]

S. Deng and C. Prather, Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay,, J. Ineq. Pure Appl. Math., 9 (2008).   Google Scholar

[7]

A. Friedman, Stochastic Differential Equations and Applications,, Academic Press, (2006).   Google Scholar

[8]

T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,, Ann. Math., 20 (1919), 292.  doi: 10.2307/1967124.  Google Scholar

[9]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge University Press, (1988).  doi: 10.1007/BF01218837.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Math., 840 (1981).   Google Scholar

[11]

K. Itô, On a stochastic integral equation,, Proc. Japan Acad., 22 (1946), 32.  doi: 10.3792/pja/1195572371.  Google Scholar

[12]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Math. Stud., 204 (2006).   Google Scholar

[13]

O. Lipovan, A retarded Gronwall-like inequality and its applications,, J. Math. Anal. Appl., 252 (2000), 389.  doi: 10.1006/jmaa.2000.7085.  Google Scholar

[14]

Q. Ma and E. Yang, Estimates on solutions of some weakly singular Volterra integral inequalities,, Acta Math. Appl. Sinca, 25 (2002), 505.   Google Scholar

[15]

M. Medved, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,, J. Math. Anal. Appl., 214 (1997), 349.  doi: 10.1006/jmaa.1997.5532.  Google Scholar

[16]

G. Mittag-Leffler, Sur la nouvelle fonction $E_\alpha(x)$,, C. R. Acad. Sci. Paris, 137 (1903), 554.   Google Scholar

[17]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, $6^{th}$ edition, (2003).  doi: 10.1007/978-3-642-14394-6.  Google Scholar

[18]

B. G. Pachpatte, Inequalities for Differential and Integral Equations,, Math. in Sci. and Eng., 197 (1998).   Google Scholar

[19]

M. Pinto, Integral inequalities of Bihari-type and applications,, Funkcial. Ekvac., 33 (1990), 387.   Google Scholar

[20]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,, Princeton Math. Ser., 43 (1993).   Google Scholar

[21]

N.-E. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality,, J. Inequal. Appl., 2006 (2006).  doi: 10.1155/JIA/2006/84561.  Google Scholar

[22]

W. Wang, A generalized retarded Gronwall-like inequality in two variables and applications to BVP,, Appl. Math. Comput., 191 (2007), 144.  doi: 10.1016/j.amc.2007.02.099.  Google Scholar

[23]

M. Wu and N. Huang, Stochastic integral inequalities with applications,, Math. Inequal. Appl., 13 (2010), 667.  doi: 10.7153/mia-13-48.  Google Scholar

[24]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations,, J. Math. Kyoto Univ., 11 (1971), 155.   Google Scholar

[25]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II,, J. Math. Kyoto Univ., 11 (1971), 533.   Google Scholar

[26]

Y. Yan, Nonlinear Gronwall-Bellman type integral inequalities with Maxima,, Math. Inequal. Appl., 16 (2013), 911.  doi: 10.7153/mia-16-71.  Google Scholar

[27]

H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation,, J. Math. Anal. Appl., 328 (2007), 1075.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar

[28]

W. Zhang, PM functions, their characteristic intervals and iterative roots,, Ann. Polon. Math., 65 (1997), 119.   Google Scholar

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