Results on non local impulsive implicit Caputo-Hadamard fractional differential equations

K. Venkatachalam; M. Sathish Kumar; P. Jayakumar; K. Venkatachalam; M. Sathish Kumar; P. Jayakumar

doi:10.3934/mmc.2024023

Mathematical Modelling and Control

2024, Volume 4, Issue 3: 286-296. doi: 10.3934/mmc.2024023

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Research article

Results on non local impulsive implicit Caputo-Hadamard fractional differential equations

1.
Department of Mathematics, Nandha Engineering College (Autonomous), Erode 638052, Tamil Nadu, India
2.
Department of Mathematics, Paavai Engineering College (Autonomous), Namakkal 637018, Tamil Nadu, India

Received: 28 December 2023 Revised: 27 March 2024 Accepted: 04 June 2024 Published: 02 September 2024

The results for a new modeling integral boundary value problem using Caputo-Hadamard impulsive implicit fractional differential equations with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem, Schaefer's fixed point theorem and the Banach contraction principle serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements.
- fractional differential equation,
- Caputo-Hadamard,
- existence,
- uniqueness,
- fixed point theorems
Citation: K. Venkatachalam, M. Sathish Kumar, P. Jayakumar. Results on non local impulsive implicit Caputo-Hadamard fractional differential equations[J]. Mathematical Modelling and Control, 2024, 4(3): 286-296. doi: 10.3934/mmc.2024023

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Abstract

The results for a new modeling integral boundary value problem using Caputo-Hadamard impulsive implicit fractional differential equations with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem, Schaefer's fixed point theorem and the Banach contraction principle serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, 2006. https://doi.org/10.1016/s0304-0208(06)x8001-5
[2]	I. Podlubny, Fractional differential equations, Academic Press, 1999.
[3]	A. A. Kilbas, Hadamard type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191–1204. https://doi.org/10.4134/JKMS.2001.38.6.1191 doi: 10.4134/JKMS.2001.38.6.1191
[4]	W. Benhamida, J. R. Graef, S. Hamani, Boundary value problems for Hadamard fractional differential equations with nonlocal multi-point boundary conditions, Fractional Differ. Calculus, 8 (2018), 165–176. https://doi.org/10.7153/fdc-2018-08-10 doi: 10.7153/fdc-2018-08-10
[5]	A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Math., 5 (2020), 259–272. https://doi.org/10.3934/math.2020017 doi: 10.3934/math.2020017
[6]	P. Karthikeyan, K. Venkatachalam, Nonlocal multipoint boundary value problems for Caputo-Hadamard fractional integro-differential equations, Appl. Math., 2 (2020), 23–35.
[7]	R. P. Agarwal, A. Alsaedi, A. Alsharrif, B. Ahmed, On nonlinear fractional order boundary value problems with nonlocal multi-point conditions involving Liouville-Caputo derivatives, Differ. Equations Appl., 9 (2017), 147–160. https://doi.org/10.7153/dea-09-12 doi: 10.7153/dea-09-12
[8]	C. Derbazi, H. Hammouche, Caputo-Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory, AIMS Math., 5 (2020), 2694–2709. https://doi.org/10.3934/math.2020174 doi: 10.3934/math.2020174
[9]	M. S. Kumar, M. Deepa, J. Kavitha, V. Sadhasivam, Existence theory of fractional order three-dimensional differential system at resonance, Math. Modell. Control, 3 (2023), 127–138. https://doi.org/10.3934/mmc.2023012 doi: 10.3934/mmc.2023012
[10]	C. V. Bose, R. Udhayakumar, A. M. Elshenhab, M. S. Kumar, J. S. Ro, Discussion on the approximate controllability of Hilfer fractional neutral integro-differential inclusions via almost sectorial operators, Fractal Fract., 6 (2022), 607. https://doi.org/10.3390/fractalfract6100607 doi: 10.3390/fractalfract6100607
[11]	T. Gayathri, M. S. Kumar, V. Sadhasivam, Hille and Nehari type oscillation criteria for conformable fractional differential equations, Iraqi J. Sci., 62 (2021), 578–587. https://doi.org/10.24996/ijs.2021.62.2.23 doi: 10.24996/ijs.2021.62.2.23
[12]	J. Liu, W. Wei, J. Wang, W. Xu, Limit behavior of the solution of Caputo-Hadamard fractional stochastic differential equations, Appl. Math. Lett., 140 (2023), 108586. https://doi.org/10.1016/j.aml.2023.108586 doi: 10.1016/j.aml.2023.108586
[13]	J. Liu, W. Wei, W. Xu, An averaging principle for stochastic fractional differential equations driven by fBm involving impulses, Fractal Fract., 6 (2022), 256. https://doi.org/10.3390/fractalfract6050256 doi: 10.3390/fractalfract6050256
[14]	O. Kahouli, A. B. Makhlouf, L. Mchiri, H. Rguigui, Hyers-Ulam stability for a class of Hadamard fractional Itô-Doob stochastic integral equations, Chaos Solitons Fract., 166 (2022), 112918. https://doi.org/10.1016/j.chaos.2022.112918 doi: 10.1016/j.chaos.2022.112918
[15]	V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Worlds Scientific, 1989.
[16]	A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Scientific, 1995. https://doi.org/10.1142/9789812798664
[17]	M. Benchohra, S. Bouriah, J. R. Graef, Boundary value problems for nonlinear implicit Caputo-Hadamard type fractional differential equations with impulses, Medit. J. Math., 14 (2017), 206. https://doi.org/10.1007/s00009-017-1012-9 doi: 10.1007/s00009-017-1012-9
[18]	M. Benchohra, J. E. Lazreg, Existence results for nonlinear implicit fractional differential equations with impulse, Commun. Appl. Anal., 19 (2015), 413–426. https://doi.org/10.3934/caa.2015.19.413 doi: 10.3934/caa.2015.19.413
[19]	J. Hadamard, Essai sur létude des fonctions donnees par Leur developpement de Taylor, J. Math. Pure Appl., 8 (1892), 101–186. {https://eudml.org/doc/233965}
[20]	P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 270 (2002), 1–15. https://doi.org/10.1016/S0022-247X(02)00066-5 doi: 10.1016/S0022-247X(02)00066-5
[21]	P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Fractional calculus in the mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl., 269 (2002), 1–27. https://doi.org/10.1016/S0022-247X(02)00001-X doi: 10.1016/S0022-247X(02)00001-X
[22]	P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269 (2002), 387–400. https://doi.org/10.1016/S0022-247X(02)00049-5 doi: 10.1016/S0022-247X(02)00049-5
[23]	I. Mahariq, M. Kuzuoğlu, I. H. Tarman, H. Kurt, Photonic nanojet analysis by spectral element method, IEEE Pho. J., 6 (2014), 6802714. https://doi.org/10.1109/JPHOT.2014.2361615 doi: 10.1109/JPHOT.2014.2361615
[24]	I. Mahariq, A. Erciyas, A spectral element method for the solution of magnetostatic fields, Turk. J. Electr. Eng. Comput. Sci., 24 (2017), 2922–2932. https://doi.org/10.3906/elk-1605-6 doi: 10.3906/elk-1605-6
[25]	I. Mahario, On the application of the spectral element method in electromagnetic problems involving domain decomposition, Turk. J. Electr. Eng. Comput. Sci., 25 (2017), 1059–1069. https://doi.org/10.3906/ELK-1511-115 doi: 10.3906/ELK-1511-115
[26]	F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equations, 2012 (2012), 142. https://doi.org/10.1186/1687-1847-2012-142 doi: 10.1186/1687-1847-2012-142
[27]	M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive differential equations and inclusions, Hindawi Publishing Corporation, 2006. https://doi.org/10.1155/9789775945501
[28]	B. Ahmad, G. Wang, A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations, Comput. Math. Appl., 62 (2011), 1341–1349. https://doi.org/10.1016/j.camwa.2011.04.033 doi: 10.1016/j.camwa.2011.04.033
[29]	A. Al-Omari, H. Al-Saadi, $(\omega, \rho)$-BVP solution of impulsive Hadamard fractional differential equations, Mathematics, 11 (2023), 4370. https://doi.org/10.3390/math11204370 doi: 10.3390/math11204370
[30]	N. I. Mahmudov, M. Awadalla, K. Abuassba, Hadamard and Caputo-Hadamard FDE's with three point integral boundary conditions, Nonlinear Anal. Differ. Equations, 5 (2017), 271–282. https://doi.org/10.12988/nade.2017.7916 doi: 10.12988/nade.2017.7916
[31]	Y. Arioua, N. Benhamidouche, Boundary value problem for Caputo-Hadamard fractional differential equations, Sur. Math. Appl., 12 (2017), 103–115.
[32]	W. Benhamida, S. Hamani, J. Henderson, Boundary value problems for Caputo-Hadamard fractional differential equations, Adv. Theory Nonlinear Anal. Appl., 2 (2018), 138–145.
[33]	A. Ardjounia, A. Djoudi, Existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions, Adv. Theory Nonlinear Anal. Appl., 3 (2019), 46–52. https://doi.org/10.31197/atnaa.501118 doi: 10.31197/atnaa.501118
[34]	A. Irguedi, K. Nisse, S. Hamani, Functional impulsive fractional differential equations involving the Caputo-Hadamard derivative and integral boundary conditions, Int. J. Anal. Appl., 21 (2023), 15. https://doi.org/10.28924/2291-8639-21-2023-15 doi: 10.28924/2291-8639-21-2023-15
[35]	J. R. Graef, S. Hamani, Existence results for impulsive caputo-hadamard fractional differential equations with integral boundary conditions, Dyn. Syst. Appl., 32 (2023), 1–16. https://doi.org/10.46719/dsa2023.32.01 doi: 10.46719/dsa2023.32.01
[36]	S. Akhter, Existence of weak solutions for Caputo's fractional derivatives in Banach spaces, GANIT: J. Bang. Math. Soc., 39 (2019), 111–118. https://doi.org/10.3329/ganit.v39i0.44166 doi: 10.3329/ganit.v39i0.44166

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