February  2022, 5(1): 45-55. doi: 10.3934/mfc.2021022

Some generalizations of delay integral inequalities of Gronwall-Bellman type with power and their applications

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

* Corresponding author: Fanwei Meng

Received  March 2021 Revised  August 2021 Published  February 2022 Early access  October 2021

Noting the diverse generalizations of the Gronwall-Bellman inequality, this paper investigates some new delay integral inequalities with power, deriving explicit bound on the solution and providing an example. The inequalities given here can act as powerful tools for studying qualitative properties such as existence, uniqueness, boundedness, stability and asymptotics of solutions of differential and integral equations.

Citation: Haoyue Song, Fanwei Meng. Some generalizations of delay integral inequalities of Gronwall-Bellman type with power and their applications. Mathematical Foundations of Computing, 2022, 5 (1) : 45-55. doi: 10.3934/mfc.2021022
References:
[1]

A. Abdeldaim, Nonlinear retarded integral inequalities of Gronwall-Bellman type and applications, J. Math. Inequal., 10 (2016), 285-299.  doi: 10.7153/jmi-10-24.

[2]

A. Abdeldaim and M. Yakout, On some new integral inequalities of Gronwall-Bellman-Pachpatte type, Appl. Math. Comput., 217 (2011), 7887-7899.  doi: 10.1016/j.amc.2011.02.093.

[3]

J.-C. Chang and D. Luor, On some generalized retarded integral inequalities and the qualitative analysis of integral equations, Appl. Math. Comput., 244 (2014), 324-334.  doi: 10.1016/j.amc.2014.06.107.

[4]

Q. FengF. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on time scales, J. Math. Anal. Appl., 382 (2011), 772-784.  doi: 10.1016/j.jmaa.2011.04.077.

[5]

T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20 (1919), 292-296.  doi: 10.2307/1967124.

[6]

J. Gu and F. Meng, Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput., 245 (2014), 235-242.  doi: 10.1016/j.amc.2014.07.056.

[7]

F. Jiang and F. Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math., 205 (2007), 479-486.  doi: 10.1016/j.cam.2006.05.038.

[8]

Z. Li and W.-S. Wang, Some new nonlinear powered Gronwall-Bellman type retarded integral inequalities and their applications, J. Math. Inequal., 13 (2019), 553-564.  doi: 10.7153/jmi-2019-13-36.

[9]

Z. Li and W.-S. Wang, Some nonlinear Gronwall-Bellman type retarded integral inequalities with power and their applications, Appl. Math. Comput., 347 (2019), 839-852.  doi: 10.1016/j.amc.2018.10.019.

[10] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, London, 1998. 
[11]

A. ShakoorI. AliS. Wali and A. Rehman, Some generalizations of retarded nonlinear integral inequalities and its applications, J. Math. Inequal., 14 (2020), 1223-1235.  doi: 10.7153/jmi-2020-14-79.

[12]

Y. Tian and M. Fan, Nonlinear integral inequality with power and its application in delay integro-differential equations, Advances in Difference Equations, 2020 (2020), 142.  doi: 10.1186/s13662-020-02596-y.

[13]

Y. TianM. Fan and F. Meng, A generalization of retarded integral inequalities in two independent variables and their applications, Appl. Math. Comput., 221 (2013), 239-248.  doi: 10.1016/j.amc.2013.06.062.

[14]

W.-S. WangX. Zhou and Z. Guo, Some new retarded nonlinear integral inequalities and their applications in differential-integral equations, Appl. Math. Comput., 218 (2012), 10726-10736.  doi: 10.1016/j.amc.2012.04.046.

[15]

R. Xu and F. Meng, Some new weakly singular integral inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2016 (2016), 78, 16 pp. doi: 10.1186/s13660-016-1015-2.

show all references

References:
[1]

A. Abdeldaim, Nonlinear retarded integral inequalities of Gronwall-Bellman type and applications, J. Math. Inequal., 10 (2016), 285-299.  doi: 10.7153/jmi-10-24.

[2]

A. Abdeldaim and M. Yakout, On some new integral inequalities of Gronwall-Bellman-Pachpatte type, Appl. Math. Comput., 217 (2011), 7887-7899.  doi: 10.1016/j.amc.2011.02.093.

[3]

J.-C. Chang and D. Luor, On some generalized retarded integral inequalities and the qualitative analysis of integral equations, Appl. Math. Comput., 244 (2014), 324-334.  doi: 10.1016/j.amc.2014.06.107.

[4]

Q. FengF. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on time scales, J. Math. Anal. Appl., 382 (2011), 772-784.  doi: 10.1016/j.jmaa.2011.04.077.

[5]

T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, 20 (1919), 292-296.  doi: 10.2307/1967124.

[6]

J. Gu and F. Meng, Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput., 245 (2014), 235-242.  doi: 10.1016/j.amc.2014.07.056.

[7]

F. Jiang and F. Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math., 205 (2007), 479-486.  doi: 10.1016/j.cam.2006.05.038.

[8]

Z. Li and W.-S. Wang, Some new nonlinear powered Gronwall-Bellman type retarded integral inequalities and their applications, J. Math. Inequal., 13 (2019), 553-564.  doi: 10.7153/jmi-2019-13-36.

[9]

Z. Li and W.-S. Wang, Some nonlinear Gronwall-Bellman type retarded integral inequalities with power and their applications, Appl. Math. Comput., 347 (2019), 839-852.  doi: 10.1016/j.amc.2018.10.019.

[10] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, London, 1998. 
[11]

A. ShakoorI. AliS. Wali and A. Rehman, Some generalizations of retarded nonlinear integral inequalities and its applications, J. Math. Inequal., 14 (2020), 1223-1235.  doi: 10.7153/jmi-2020-14-79.

[12]

Y. Tian and M. Fan, Nonlinear integral inequality with power and its application in delay integro-differential equations, Advances in Difference Equations, 2020 (2020), 142.  doi: 10.1186/s13662-020-02596-y.

[13]

Y. TianM. Fan and F. Meng, A generalization of retarded integral inequalities in two independent variables and their applications, Appl. Math. Comput., 221 (2013), 239-248.  doi: 10.1016/j.amc.2013.06.062.

[14]

W.-S. WangX. Zhou and Z. Guo, Some new retarded nonlinear integral inequalities and their applications in differential-integral equations, Appl. Math. Comput., 218 (2012), 10726-10736.  doi: 10.1016/j.amc.2012.04.046.

[15]

R. Xu and F. Meng, Some new weakly singular integral inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2016 (2016), 78, 16 pp. doi: 10.1186/s13660-016-1015-2.

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