On the generalized  Gronwall inequalities involving $ \psi $-fractional integral operator with applications

Qi Wang; Shumin Zhu; Qi Wang; Shumin Zhu

doi:10.3934/math.20221115

AIMS Mathematics

2022, Volume 7, Issue 11: 20370-20380. doi: 10.3934/math.20221115

Previous Article Next Article

Research article

On the generalized Gronwall inequalities involving $ \psi $-fractional integral operator with applications

Qi Wang ^,,
Shumin Zhu

School of Mathematical Sciences, Anhui University, Hefei, 230601, China

Received: 28 July 2022 Revised: 03 September 2022 Accepted: 09 September 2022 Published: 19 September 2022
MSC : 34A08

In this paper, a Gronwall inequality involving $ \psi $-fractional integral operator is obtained as a generalization of ^[23]. An example is listed to show the applications.
- generalized Gronwall inequality,
- $ \psi $-fractional integral operator,
- existence results
Citation: Qi Wang, Shumin Zhu. On the generalized Gronwall inequalities involving $ \psi $-fractional integral operator with applications[J]. AIMS Mathematics, 2022, 7(11): 20370-20380. doi: 10.3934/math.20221115

Related Papers:

Abstract

In this paper, a Gronwall inequality involving $ \psi $-fractional integral operator is obtained as a generalization of ^[23]. An example is listed to show the applications.

References

[1]	D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Vol. 840, Springer Berlin, Heidelberg, 1981. https://doi.org/10.1007/BFb0089647
[2]	H. P. Ye, J. M. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075–1081. https://doi.org/10.1016/j.jmaa.2006.05.061 doi: 10.1016/j.jmaa.2006.05.061
[3]	M. P. Lazarević, A. M. Spasić, Finite-time stability analysis of fractional order time-delay systems: Gronwall'sapproach, Math. Comp. Model., 49 (2009), 475–481. https://doi.org/10.1016/j.mcm.2008.09.011 doi: 10.1016/j.mcm.2008.09.011
[4]	A. Al-Jaser, K. M. Furati, Singular fractional integro-differential inequalities and applications, J. Inequal. Appl., 2011 (2011), 110. https://doi.org/10.1186/1029-242X-2011-110 doi: 10.1186/1029-242X-2011-110
[5]	J. R. Wang, Y. Zhou, Existence of mild solutions for fractional delay evolution systems, Appl. Math. Comput., 218 (2011), 357–367. https://doi.org/10.1016/j.amc.2011.05.071 doi: 10.1016/j.amc.2011.05.071
[6]	J. R. Wang, L. L. Lv, Y. Zhou, Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces, J. Appl. Math. Comput., 38 (2012), 209–224. https://doi.org/10.1007/s12190-011-0474-3 doi: 10.1007/s12190-011-0474-3
[7]	Y. Jalilian, R. Jalilian, Existence of solutions of delay fractional differential equations, Mediterr. J. Math., 10 (2013), 1731–1747. https://doi.org/10.1007/s00009-013-0281-1 doi: 10.1007/s00009-013-0281-1
[8]	D. N. Chalishajar, K. Karthikeyan, Eexistence and uniqueness results for bounadary value problems of higher order fractional integro-differential equations involving Gronwall's inequality in Banach spaces, Acta Math. Sci., 33 (2013), 758–772. https://doi.org/10.1016/S0252-9602(13)60036-3 doi: 10.1016/S0252-9602(13)60036-3
[9]	W. S. Wang, Estimation of unknown function of a class of integral inequalities and applications in fractional integral equaations, Appl. Math. Comput., 268 (2015), 1029–1037. https://doi.org/10.1016/j.amc.2015.07.015 doi: 10.1016/j.amc.2015.07.015
[10]	Q. Wang, D. C. Lu, Y. Y. Fang, Stability analysis of impulsive fractional differential systems with delay, Appl. Math. Lett., 40 (2015), 1–6. https://doi.org/10.1016/j.aml.2014.08.017 doi: 10.1016/j.aml.2014.08.017
[11]	Y. Jalilian, Fractional integr inequalities and their applications to fractional differential equations, Acta Math. Sci., 36 (2016), 1317–1330. https://doi.org/10.1016/S0252-9602(16)30071-6 doi: 10.1016/S0252-9602(16)30071-6
[12]	Z. Zhang, Z. Wei, A generalized Gronwall inequality and its application to a fractional neutral evolution inclusions, J. Inequal. Appl., 2016 (2016), 45. https://doi.org/10.1186/s13660-016-0991-6 doi: 10.1186/s13660-016-0991-6
[13]	J. L. Sheng, W. Jiang, Existence and uniqueness of the solution of fractional damped dynamical systems, Adv. Differ. Equ., 2017 (2017), 16. https://doi.org/10.1186/s13662-016-1049-2 doi: 10.1186/s13662-016-1049-2
[14]	V. N. Phat, N. T. Thanh, New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach, Appl. Math. Lett., 83 (2018), 169–175. https://doi.org/10.1016/j.aml.2018.03.023 doi: 10.1016/j.aml.2018.03.023
[15]	A. Ekinci, M. E. Ozdemir, Some new integral inequalities via Riemann-Liouville integral operators, Appl. Comput. Math., 18 (2019), 288–295.
[16]	J. R. L. Webb, Weakly singular Gronwall inequalities and applications to fractional differential equations, J. Math. Anal. Appl., 471 (2019), 692–711. https://doi.org/10.1016/j.jmaa.2018.11.004 doi: 10.1016/j.jmaa.2018.11.004
[17]	R. Almeida, A Gronwall inequality for a general Caputo fractional operator, Math. Inequal. Appl., 20 (2017), 1089–1105. https://doi.org/10.7153/mia-2017-20-70 doi: 10.7153/mia-2017-20-70
[18]	K. S. Nisar, G. Rahman, J. Choi, S. Mubeen, M. Arshad, Certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives and their applications, East Asian Math. J., 34 (2018), 249–263. https://doi.org/10.7858/eamj.2018.018 doi: 10.7858/eamj.2018.018
[19]	Y. Adjabi, F. Jarad, T. Abdeljawad, On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31 (2017), 5457–5473. https://doi.org/10.2298/FIL1717457A doi: 10.2298/FIL1717457A
[20]	Ş. Kızıl, M. A. Ardıç, Inequalities for strongly convex functions via Atangana-Baleanu integral operators, Turkish J. Sci., 6 (2021), 96–109.
[21]	J. Alzabut, T. Abdeljawad, F. Jarad, W. Sudsutad, A Gronwall inequality via the generalized proportional fractional derivative with applications, J. Inequal. Appl., 2019 (2019), 101. https://doi.org/10.1186/s13660-019-2052-4 doi: 10.1186/s13660-019-2052-4
[22]	J. V. D. C. Sousa, E. C. D. Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[23]	J. V. D. C. Sousa, E. C. D. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi$-Hilfer operator, arXiv, 2017. https://doi.org/10.48550/arXiv.1709.03634
[24]	J. Ren, C. B. Zhai, Stability analysis of generalized neutral fractional differential systems with time delays, Appl. Math. Lett., 116 (2021), 106987. https://doi.org/10.1016/j.aml.2020.106987 doi: 10.1016/j.aml.2020.106987
[25]	D. H. Jiang, C. Z. Bai, On coupled Gronwall inequalities involving a $\psi$-fractional integral operator with its applications, AIMS Math., 7 (2022), 7728–7741. https://doi.org/10.3934/math.2022434 doi: 10.3934/math.2022434
[26]	H. Kalsoom, M. A. Alı, M. Abbas, H. Budak, G. Murtaza, Generalized quantum Montgomery identity and Ostrowski type inequalities for preinvex functions, TWMS J. Pure Appl. Math., 13 (2022), 72–90.
[27]	A. Salim, J. E. Lazreg, B. Ahmad, M. Benchohra, J. J. Nieto, A study on $k$-generalized $\psi$-Hilfer derivative operator, Vietnam J. Math., 2022. https://doi.org/10.1007/s10013-022-00561-8 doi: 10.1007/s10013-022-00561-8
[28]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[29]	S. I. Butt, M. Nadeem, G. Farid, On Caputo fractional derivatives via exponential s-convex functions, Turk. J. Sci., 5 (2020), 140–146.
[30]	S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, AIMS Math., 6 (2021), 4507–4525. https://doi.org/10.3934/math.2021267 doi: 10.3934/math.2021267
[31]	V. Kiryakova, A brief story about the operators of generalized fractional calculus, Fract. Calc. Appl. Anal., 11 (2008), 203–220.
[32]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Methods Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[33]	A. Seemab, M. U. Rehman, J. Alzabut, A. Hamdi, On the existence of positive solutions for generalized fractional boundary value problems, Bound. Value Probl., 2019 (2019), 186. https://doi.org/10.1186/s13661-019-01300-8 doi: 10.1186/s13661-019-01300-8
[34]	F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Cont. Dyn. Syst. Ser. S, 13 (2019), 709–722. https://doi.org/10.3934/dcdss.2020039 doi: 10.3934/dcdss.2020039
[35]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer-Verlag, 2003. https://doi.org/10.1007/978-0-387-21593-8
[36]	R. Wong, Asymptotic approximations of integrals, Philadelphia: Society for Industrial and Applied Mathematics, 2001.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)