A generalized Gronwall inequality via $ \psi $-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system

Weerawat Sudsutad; Jutarat Kongson; Chatthai Thaiprayoon; Nantapat Jarasthitikulchai; Marisa Kaewsuwan; Weerawat Sudsutad; Jutarat Kongson; Chatthai Thaiprayoon; Nantapat Jarasthitikulchai; Marisa Kaewsuwan

doi:10.3934/math.20241191

AIMS Mathematics

2024, Volume 9, Issue 9: 24443-24479. doi: 10.3934/math.20241191

Previous Article Next Article

Research article

A generalized Gronwall inequality via $ \psi $-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system

1.
Theoretical and Applied Data Integration Innovations Group, Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand
2.
Research Group of Theoretical and Computation in Applied Science, Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
3.
Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand

Received: 15 February 2024 Revised: 06 July 2024 Accepted: 01 August 2024 Published: 21 August 2024
MSC : 26A33, 26D10, 33E12, 34A08, 34B10

This paper establishes a novel generalized Gronwall inequality concerning the $ \psi $-Hilfer proportional fractional operators. Before proving the main results, the solution of the linear nonlocal coupled $ \psi $-Hilfer proportional Cauchy-type system with constant coefficients under the Mittag-Leffler kernel is created. The uniqueness result for the proposed coupled system is established using Banach's contraction mapping principle. Furthermore, a variety of the Mittag-Leffler-Ulam-Hyers stability of the solutions for the proposed coupled system is investigated. Finally, a numerical example is given to show the effectiveness and applicability of the obtained results, and graphical simulations in the case of linear systems are shown.
Citation: Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Nantapat Jarasthitikulchai, Marisa Kaewsuwan. A generalized Gronwall inequality via $ \psi $-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system[J]. AIMS Mathematics, 2024, 9(9): 24443-24479. doi: 10.3934/math.20241191

Related Papers:

Abstract

This paper establishes a novel generalized Gronwall inequality concerning the $ \psi $-Hilfer proportional fractional operators. Before proving the main results, the solution of the linear nonlocal coupled $ \psi $-Hilfer proportional Cauchy-type system with constant coefficients under the Mittag-Leffler kernel is created. The uniqueness result for the proposed coupled system is established using Banach's contraction mapping principle. Furthermore, a variety of the Mittag-Leffler-Ulam-Hyers stability of the solutions for the proposed coupled system is investigated. Finally, a numerical example is given to show the effectiveness and applicability of the obtained results, and graphical simulations in the case of linear systems are shown.

References

[1]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Philadelphia: Gordon and Breach Science Publishers, 1993.
[2]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[3]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[4]	V. Lakshmikantham, S. Leela, J. V. Devi, Theory of fractional dynamic systems, Cambridge: Cambridge Scientific Publishers, 2009.
[5]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientific, 2000.
[6]	U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860–865. https://doi.org/10.1016/j.amc.2011.03.062 doi: 10.1016/j.amc.2011.03.062
[7]	U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1–15.
[8]	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
[9]	F. Jarad, M. A. Alqudah, T. Abdeljawad, On more general forms of proportional fractional operators, Open Math., 18 (2020), 167–176. https://doi.org/10.1515/math-2020-0014 doi: 10.1515/math-2020-0014
[10]	F. Jarad, T. Abdeljawad, S. Rashid, Z. Hammouch, More properties of the proportional fractional integrals and derivatives of a function with respect to another function, Adv. Differ. Equ., 2020 (2020), 303. https://doi.org/10.1186/s13662-020-02767-x doi: 10.1186/s13662-020-02767-x
[11]	I. Ahmed, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Differ. Equ., 2020 (2020), 329. https://doi.org/10.1186/s13662-020-02792-w doi: 10.1186/s13662-020-02792-w
[12]	I. Mallah, I. Ahmed, A. Akgul, F. Jarad, S. Alha, On $\psi$-Hilfer generalized proportional fractional operators, AIMS Mathematics, 7 (2022), 82–103. https://doi.org/10.3934/math.2022005 doi: 10.3934/math.2022005
[13]	R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399–408. https://doi.org/10.1016/S0301-0104(02)00670-5 doi: 10.1016/S0301-0104(02)00670-5
[14]	I. Ali, N. A. Malik, Hilfer fractional advection-diffusion equations with power-law initial condition; a numerical study using variational iteration method, Comput. Math. Appl., 68 (2014), 1161–1179. https://doi.org/10.1016/j.camwa.2014.08.021 doi: 10.1016/j.camwa.2014.08.021
[15]	R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application, J. Comput. Appl. Math., 308 (2016), 39–45. https://doi.org/10.1016/j.cam.2016.05.014 doi: 10.1016/j.cam.2016.05.014
[16]	V. M. Bulavatsky, Mathematical modeling of fractional differential filtration dynamics based on models with Hilfer-Prabhakar derivative, Cybern. Syst. Anal., 53 (2017), 204–216. https://doi.org/10.1007/s10559-017-9920-z doi: 10.1007/s10559-017-9920-z
[17]	V. M. Bulavatsky, Mathematical models and problems of fractional-differential dynamics of some relaxation filtration processes, Cybern. Syst. Anal., 54 (2018), 727–736. https://doi.org/10.1007/s10559-018-0074-4 doi: 10.1007/s10559-018-0074-4
[18]	H. Joshi, B. K. Jha, Chaos of calcium diffusion in Parkinson's infectious disease model and treatment mechanism via Hilfer fractional derivative, Math. Model. Numer. Simul. Appl., 1 (2021), 84–94. https://doi.org/10.53391/mmnsa.2021.01.008 doi: 10.53391/mmnsa.2021.01.008
[19]	B. G. Pachpatte, Inequalities for differential and integral equations, New York: Academic Press, 1998.
[20]	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. https://doi.org/10.2307/1967124 doi: 10.2307/1967124
[21]	H. Ye, J. 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
[22]	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
[23]	J. V. D. C. Sousa, E. C. D. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi$-Hilfer operator, Differ. Equ. Appl., 11 (2019), 87–106. https://doi.org/10.7153/dea-2019-11-02 doi: 10.7153/dea-2019-11-02
[24]	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
[25]	J. Alzabut, Y. Adjabi, W. Sudsutad, M. U. Rehman, New generalizations for Gronwall type inequalities involving a $\psi$-fractional operator and their applications, AIMS Mathematics, 6 (2021), 5053–5077. https://doi.org/10.3934/math.2021299 doi: 10.3934/math.2021299
[26]	D. Jiang, C. Bai, On coupled Gronwall inequalities involving a $\psi$-fractional integral operator with its applications, AIMS Mathematics, 7 (2022), 7728–7741. https://doi.org/10.3934/math.2022434 doi: 10.3934/math.2022434
[27]	Q. Wang, S. Zhu, On the generalized Gronwall inequalities involving $\psi$-fractional integral operator with applications, AIMS Mathematics, 7 (2022), 20370–20380. https://doi.org/10.3934/math.20221115 doi: 10.3934/math.20221115
[28]	W. Sudsutad, C. Thaiprayoon, B. Khaminsou, J. Alzabut, J. Kongson, A Gronwall inequality and its applications to the Cauchy-type problem under $\psi$-Hilfer proportional fractional operators, J. Inequal. Appl., 2023 (2023), 20. https://doi.org/10.1186/s13660-023-02929-x doi: 10.1186/s13660-023-02929-x
[29]	S. M. Ulam, A collection of mathematical problems, interscience tracts in pure and applied mathematics, New York, London: Interscience Publishers, 1960.
[30]	D. H. Hyers, G. Isac, T. M. Rassias, Stability of functional equations in several variables, Boston: Birkhäuser, 1998. https://doi.org/10.1007/978-1-4612-1790-9
[31]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.2307/2042795 doi: 10.2307/2042795
[32]	J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63 (2011), 1–10. https://doi.org/10.14232/ejqtde.2011.1.63 doi: 10.14232/ejqtde.2011.1.63
[33]	J. Wang, X. Li, $E_{\alpha}$-Ulam type stability of fractional order ordinary differential equations, J. Appl. Math. Comput., 45 (2014), 449–459. https://doi.org/10.1007/s12190-013-0731-8 doi: 10.1007/s12190-013-0731-8
[34]	Z. Gao, X. Yu, Existence results for BVP of a class of Hilfer fractional differential equations, J. Appl. Math. Comput., 56 (2018), 217–233. https://doi.org/10.1007/s12190-016-1070-3 doi: 10.1007/s12190-016-1070-3
[35]	K. D. Kucche, A. D. Mali, J. V. D. C. Sousa, On the nonlinear $\psi$-Hilfer fractional differential equations, Comput. Appl. Math., 38 (2019), 73. https://doi.org/10.1007/s40314-019-0833-5 doi: 10.1007/s40314-019-0833-5
[36]	K. Liu, J. Wang, D. O'Regan, Ulam-Hyers-Mittag-Leffler stability for $\psi$-Hilfer fractional-order delay differential equations, Adv. Differ. Equ., 2019 (2019), 50. https://doi.org/10.1186/s13662-019-1997-4 doi: 10.1186/s13662-019-1997-4
[37]	M. A. Almalahi, S. K. Panchal, On the theoty of $\psi$-Hilfer nonlocal Cauchy problem, J. Sib. Fed. Univ. Math. Phys., 14 (2021), 159–175. https://doi.org/10.17516/1997-1397-2021-14-2-161-177 doi: 10.17516/1997-1397-2021-14-2-161-177
[38]	M. A. Almalahi1, S. K. Panchal, K. Aldwoah, M. Lotayif, On the explicit solution of $\psi$-Hilfer integro-differential nonlocal Cauchy problem, Progr. Fract. Differ. Appl., 9 (2023), 65–77. http://dx.doi.org/10.18576/pfda/090104 doi: 10.18576/pfda/090104
[39]	R. L. Magin, Fractional calculus in bioengineering, Danbury: Begell House Publishers, 2006.
[40]	H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Fractional calculus and fractional processes with applications to financial economics: Theory and application, London: Academic Press, 2017.
[41]	G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford: Oxford University Press, 2005.
[42]	M. A. Almalahi, M. S. Abdo, S. K. Panchal, Existence and Ulam-Hyers stability results of a coupled system of $\psi$-Hilfer sequential fractional differential equations, Results Appl. Math., 10 (2021), 100142. https://doi.org/10.1016/j.rinam.2021.100142 doi: 10.1016/j.rinam.2021.100142
[43]	M. A. Almalahi, O. Bazighifan, S. K. Panchal, S. S. Askar, G. I. Oros, Analytical study of two nonlinear coupled hybrid systems involving generalized Hilfer fractional operators, Fractal Fract., 5 (2021), 178. https://doi.org/10.3390/fractalfract5040178 doi: 10.3390/fractalfract5040178
[44]	A. Samadi, S. K. Ntouyas, J. Tariboon, On a nonlocal coupled system of Hilfer generalized proportional fractional differential equations, Symmetry, 14 (2022), 738. https://doi.org/10.3390/sym14040738 doi: 10.3390/sym14040738
[45]	B. Ahmad, S. Aljoudi, Investigation of a coupled system of Hilfer-Hadamard fractional differential equations with nonlocal coupled Hadamard fractional integral boundary conditions, Fractal Fract., 7 (2023), 178. https://doi.org/10.3390/fractalfract7020178 doi: 10.3390/fractalfract7020178
[46]	J. R. Wang, M. Feckan, Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. Spec. Top., 222 (2013), 1857–1874. https://doi.org/10.1140/epjst/e2013-01969-9 doi: 10.1140/epjst/e2013-01969-9
[47]	R. Courant, E. J. McShane, Differential and integral calculus, New York: John Wiley & Sons, 1988.
[48]	E. L. Lima, Análise real, Insituto Nacional de Matemática Pura e Aplicada, 2006.
[49]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8

Reader Comments

Your name:*

Email:*
© 2024 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)