# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 599-612. doi: 10.1016/j.amc.2004.04.067. [2] K. Amano, A stochastic Gronwall inequality and its applications, J. Ineq. Pure Appl. Math., 6 (2005), Art. 17, 5pp. [3] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10 (1943), 643-647. doi: 10.1215/S0012-7094-43-01059-2. [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), 81-94. doi: 10.1007/BF02022967. [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), Art. ID 909156, 12pp. doi: 10.1155/2008/909156. [6] S. Deng and C. Prather, Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay, J. Ineq. Pure Appl. Math., 9 (2008), Art. 34, 11pp. [7] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 2006. [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), 292-296. doi: 10.2307/1967124. [9] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988. doi: 10.1007/BF01218837. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin-New York, 1981. [11] K. Itô, On a stochastic integral equation, Proc. Japan Acad., 22 (1946), 32-35. doi: 10.3792/pja/1195572371. [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Math. Stud., 204, North-Holland, Amsterdam, 2006. [13] O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl., 252 (2000), 389-401. doi: 10.1006/jmaa.2000.7085. [14] Q. Ma and E. Yang, Estimates on solutions of some weakly singular Volterra integral inequalities, Acta Math. Appl. Sinca, 25 (2002), 505-515. [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), 349-366. doi: 10.1006/jmaa.1997.5532. [16] G. Mittag-Leffler, Sur la nouvelle fonction $E_\alpha(x)$, C. R. Acad. Sci. Paris, 137 (1903), 554-558. [17] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, $6^{th}$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [18] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Math. in Sci. and Eng., 197, Academic Press, San Diego, 1998. [19] M. Pinto, Integral inequalities of Bihari-type and applications, Funkcial. Ekvac., 33 (1990), 387-403. [20] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., 43, Princeton University Press, New Jersey, 1993. [21] N.-E. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, J. Inequal. Appl., 2006 (2006), Art. ID 84561, 12pp. doi: 10.1155/JIA/2006/84561. [22] W. Wang, A generalized retarded Gronwall-like inequality in two variables and applications to BVP, Appl. Math. Comput., 191 (2007), 144-154. doi: 10.1016/j.amc.2007.02.099. [23] M. Wu and N. Huang, Stochastic integral inequalities with applications, Math. Inequal. Appl., 13 (2010), 667-677. doi: 10.7153/mia-13-48. [24] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167. [25] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II, J. Math. Kyoto Univ., 11 (1971), 533-563. [26] Y. Yan, Nonlinear Gronwall-Bellman type integral inequalities with Maxima, Math. Inequal. Appl., 16 (2013), 911-928. doi: 10.7153/mia-16-71. [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), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061. [28] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65 (1997), 119-128.

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##### 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), 599-612. doi: 10.1016/j.amc.2004.04.067. [2] K. Amano, A stochastic Gronwall inequality and its applications, J. Ineq. Pure Appl. Math., 6 (2005), Art. 17, 5pp. [3] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10 (1943), 643-647. doi: 10.1215/S0012-7094-43-01059-2. [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), 81-94. doi: 10.1007/BF02022967. [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), Art. ID 909156, 12pp. doi: 10.1155/2008/909156. [6] S. Deng and C. Prather, Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay, J. Ineq. Pure Appl. Math., 9 (2008), Art. 34, 11pp. [7] A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, 2006. [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), 292-296. doi: 10.2307/1967124. [9] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1988. doi: 10.1007/BF01218837. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin-New York, 1981. [11] K. Itô, On a stochastic integral equation, Proc. Japan Acad., 22 (1946), 32-35. doi: 10.3792/pja/1195572371. [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Math. Stud., 204, North-Holland, Amsterdam, 2006. [13] O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl., 252 (2000), 389-401. doi: 10.1006/jmaa.2000.7085. [14] Q. Ma and E. Yang, Estimates on solutions of some weakly singular Volterra integral inequalities, Acta Math. Appl. Sinca, 25 (2002), 505-515. [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), 349-366. doi: 10.1006/jmaa.1997.5532. [16] G. Mittag-Leffler, Sur la nouvelle fonction $E_\alpha(x)$, C. R. Acad. Sci. Paris, 137 (1903), 554-558. [17] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, $6^{th}$ edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6. [18] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Math. in Sci. and Eng., 197, Academic Press, San Diego, 1998. [19] M. Pinto, Integral inequalities of Bihari-type and applications, Funkcial. Ekvac., 33 (1990), 387-403. [20] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., 43, Princeton University Press, New Jersey, 1993. [21] N.-E. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, J. Inequal. Appl., 2006 (2006), Art. ID 84561, 12pp. doi: 10.1155/JIA/2006/84561. [22] W. Wang, A generalized retarded Gronwall-like inequality in two variables and applications to BVP, Appl. Math. Comput., 191 (2007), 144-154. doi: 10.1016/j.amc.2007.02.099. [23] M. Wu and N. Huang, Stochastic integral inequalities with applications, Math. Inequal. Appl., 13 (2010), 667-677. doi: 10.7153/mia-13-48. [24] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11 (1971), 155-167. [25] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations II, J. Math. Kyoto Univ., 11 (1971), 533-563. [26] Y. Yan, Nonlinear Gronwall-Bellman type integral inequalities with Maxima, Math. Inequal. Appl., 16 (2013), 911-928. doi: 10.7153/mia-16-71. [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), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061. [28] W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math., 65 (1997), 119-128.
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