Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application

Thabet Abdeljawad; Pshtiwan Othman Mohammed; Hari Mohan Srivastava; Eman Al-Sarairah; Artion Kashuri; Kamsing Nonlaopon; Thabet Abdeljawad; Pshtiwan Othman Mohammed; Hari Mohan Srivastava; Eman Al-Sarairah; Artion Kashuri; Kamsing Nonlaopon

doi:10.3934/math.2023177

AIMS Mathematics

2023, Volume 8, Issue 2: 3469-3483. doi: 10.3934/math.2023177

Previous Article Next Article

Research article

Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application

1.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
2.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
3.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq
4.
Department of Mathematics and Statistics, University of Victoria, BC V8W 3R4, Canada
5.
Department of Mathematics and Informatics, Azerbaijan University, AZ1007 Baku, Azerbaijan
6.
Center for Converging Humanities, Kyung Hee University, Dongdaemun-gu, Seoul 02447, Republic of Korea
7.
Department of mathematics, Khalifa University, P.O. Box 127788, Abu Dhabi, United Arab Emirates
8.
Department of mathematics, Al-Hussein Bin Talal University, P.O. Box 33011, Ma'an, Jordan
9.
Department of Mathematics, Faculty of Technical and Natural Sciences, University "Ismail Qemali", 9400 Vlora, Albania
10.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 17 September 2022 Revised: 06 November 2022 Accepted: 14 November 2022 Published: 22 November 2022
MSC : 26A33, 34A12, 34K20

In this article, we discuss conditions that are sufficient for the existence of solutions for some $ {\psi} $-Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions. By applying Krasnoselskii's and Banach's fixed point theorems, we investigate the existence and uniqueness of these solutions. Moreover, we have proved its boundedness of the method. We extend some earlier results by introducing and including the $ {\psi} $-Hilfer fractional derivative, nonlinear integral terms and non-instantaneous impulsive conditions. Finally, we offer an application to explain the consistency of our theoretical results.
- $ {\psi} $-Hilfer fractional derivative,
- non-instantaneous impulsive,
- fixed point theory
Citation: Thabet Abdeljawad, Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Eman Al-Sarairah, Artion Kashuri, Kamsing Nonlaopon. Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application[J]. AIMS Mathematics, 2023, 8(2): 3469-3483. doi: 10.3934/math.2023177

Related Papers:

Abstract

In this article, we discuss conditions that are sufficient for the existence of solutions for some $ {\psi} $-Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions. By applying Krasnoselskii's and Banach's fixed point theorems, we investigate the existence and uniqueness of these solutions. Moreover, we have proved its boundedness of the method. We extend some earlier results by introducing and including the $ {\psi} $-Hilfer fractional derivative, nonlinear integral terms and non-instantaneous impulsive conditions. Finally, we offer an application to explain the consistency of our theoretical results.

References

[1]	R. Hilfer, Applications of fractional calculus in physics, Singapore: World Scientiﬁc, 2000, 87–130. http://doi.org/10.1142/9789812817747_0002
[2]	S. Asawasamrit, Y. Thadang, S. K. Ntouyas, J. Tariboon, Non-instantaneous impulsive boundary value problems containing Caputo fractional derivative of a function with respect to another function and Riemann–Stieltjes fractional integral boundary conditions, Axioms, 10 (2021), 130. https://doi.org/10.3390/axioms10030130 doi: 10.3390/axioms10030130
[3]	M. S. Abdo, S. K. Pancha, Fractional integro-differential equations involving ${\psi}$-Hilfer fractional derivative, Adv. Appl. Math. Mech., 11 (2019), 338–359. http://doi.org/10.4208/aamm.OA-2018-0143 doi: 10.4208/aamm.OA-2018-0143
[4]	M. R. Ali, A. R. Hadhoud, H. M. Srivastava, Solution of fractional Volterra–Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method, Adv. Differ. Equ., 2019 (2019), 115. https://doi.org/10.1186/s13662-019-2044-1 doi: 10.1186/s13662-019-2044-1
[5]	A. Anguraj, P. Karthikeyan, M. Rivero, J. J. Trujillo, On new existence results for fractional integro-differential equations with impulsive and integral conditions, Comput. Math. Appl., 66 (2014), 2587–2594. https://doi.org/10.1016/j.camwa.2013.01.034 doi: 10.1016/j.camwa.2013.01.034
[6]	R. Agarwal, S. Hristova, D. O'Regan, Non-instantaneous impulses in Caputo fractional differential equations, Fract. Calc. Appl. Anal., 20 (2017), 595–622. https://doi.org/10.1515/fca-2017-0032 doi: 10.1515/fca-2017-0032
[7]	M. S. Abdo, S. K. Panchal, A. M. Saeed, Fractional boundary value problem with ${\psi}$-Caputo fractional derivative, Proc. Indian Acad. Sci., Math. Sci., 129 (2019), 65. https://doi.org/10.1007/s12044-019-0514-8 doi: 10.1007/s12044-019-0514-8
[8]	S. Kailasavalli, M. MallikaArjunan, P. Karthikeyan, Existence of solutions for fractional boundary value problems involving integro-differential equations in Banach spaces, Nonlinear Stud., 22 (2015), 341–358.
[9]	P. Karthikeyan, K. Venkatachalam, S. Abbas, Existence results for fractional impulsive integro differential equations with integral conditions of Katugampola type, Acta Math. Univ. Comenianae, 90 (2021), 421–436.
[10]	C. Nuchpong, S. K. Ntouyas, J. Tariboon, Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions, Open Math., 18 (2020), 1879–1894. https://doi.org/10.1515/math-2020-0122 doi: 10.1515/math-2020-0122
[11]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[12]	I. Podlubny, Fractional differential equations, San Diego: Acadamic Press, 1999.
[13]	H. M. Srivastava, Fractional-order derivatives and integrals: Introductory overview and recent developments, Kyungpook Math. J., 60 (2020), 73–116. https://doi.org/10.5666/KMJ.2020.60.1.73 doi: 10.5666/KMJ.2020.60.1.73
[14]	H. M. Srivastava, Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations, J. Nonlinear Convex Anal., 22 (2021), 1501–1520.
[15]	H. M. Srivastava, An introductory overview of fractional-calculus operators based upon the Fox-Wright and related higher transcendental functions, J. Adv. Engrg. Comput., 5 (2021), 135–166. http://doi.org/10.55579/jaec.202153.340 doi: 10.55579/jaec.202153.340
[16]	M. I. Abbas, Non-instantaneous impulsive fractional integro-differential equations with proportional fractional derivatives with respect to another function, Math. Methods Appl. Sci., 44 (2021), 10432–10447. https://doi.org/10.1002/mma.7419 doi: 10.1002/mma.7419
[17]	K. Aissani, M. Benchohra, N. Benkhettou, On fractional integro-differential equations with state-dependent delay and non-instantaneous impulses, Cubo, 21 (2019), 61–75. http://doi.org/10.4067/S0719-06462019000100061 doi: 10.4067/S0719-06462019000100061
[18]	V. Gupta, J. Dabas, Nonlinear fractional boundary value problem with not-instantaneous impulse, AIMS Mathematics, 2 (2020), 365–376. http://doi.org/10.3934/Math.2017.2.365 doi: 10.3934/Math.2017.2.365
[19]	A. G. Ibrahim, A. A. Elmandouh, Existence and stability of solutions of ${\psi}$-Hilfer fractional functional differential inclusions with non-instantaneous impulses, AIMS Mathematics, 6 (2021), 10802–10832. http://doi.org/10.3934/math.2021628 doi: 10.3934/math.2021628
[20]	C. Long, J. Xie, G. Chen, D. Luo, Integral boundary value problem for fractional order differential equations with non-instantaneous impulses, Int. J. Math. Anal., Ruse, 14 (2020), 251–266. https://doi.org/10.12988/ijma.2020.912110 doi: 10.12988/ijma.2020.912110
[21]	E. Hernandez, D. O'Regan, On a new class of abstract impulsive differential equation, Proc. Am. Math. Soc., 141 (2013), 1641–1649. http://doi.org/10.1090/S0002-9939-2012-11613-2 doi: 10.1090/S0002-9939-2012-11613-2
[22]	R. Terzieva, Some phenomena for non-instantaneous impulsive differential equations, Int. J. Pure Appl. Math., 119 (2018), 483–490. http://doi.org/10.12732/ijpam.v119i3.8 doi: 10.12732/ijpam.v119i3.8
[23]	A. Salim, M. Benchohra, J. R. Graef, J. E. Lazreg, Boundary value problem for fractional order generalized Hilfer-type fractional derivative with non-instantaneous impulses, Fractal Fract., 5 (2021), 1–21. https://doi.org/10.3390/fractalfract5010001 doi: 10.3390/fractalfract5010001
[24]	D. Yang, J. R. Wang, Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations, J. Appl. Math. Comput., 55 (2017), 59–78. http://doi.org/10.1007/s12190-016-1025-8 doi: 10.1007/s12190-016-1025-8
[25]	A. Zada, S. Ali, Y. Li, Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equ., 2017 (2017), 317. http://doi.org/10.1186/s13662-017-1376-y doi: 10.1186/s13662-017-1376-y
[26]	A. Zada, S. Ali, Stability of integral Caputo type boundary value problem with non instantaneous impulses, Int. J. Appl. Comput. Math., 5 (2019), 55. https://doi.org/10.1007/s40819-019-0640-0 doi: 10.1007/s40819-019-0640-0
[27]	A. Zada, N. Ali, U. Riaz, Ulam's stability of multi-point implicit boundary value problems with non-instantaneous impulses, Boll. Unione Mat. Ital., 13 (2020), 305–328. https://doi.org/10.1007/s40574-020-00219-8 doi: 10.1007/s40574-020-00219-8
[28]	S. Asawasamrit, A. Kijjathanakorn, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull. Korean Math. Soc., 55 (2018), 1639–1657. https://doi.org/10.4134/BKMS.b170887 doi: 10.4134/BKMS.b170887
[29]	N. I. Mahmudov, S. Emin, Fractional-order boundary value problems with Katugampola fractional integral conditions, Adv. Differ. Equ., 2018 (2018), 81. https://doi.org/10.1186/s13662-018-1538-6 doi: 10.1186/s13662-018-1538-6
[30]	J. V. da Costa Sousa, E. Capelas de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ${\psi}$-Hilfer operator, Differ. Equ. Appl., 11 (2017), 87–106. http://doi.org/10.7153/dea-2019-11-02 doi: 10.7153/dea-2019-11-02
[31]	N. Phuangthong, S. K. Ntouyas, J. Tariboon, K. Nonlaopon, Nonlocal sequential boundary value problems for Hilfer type fractional integro-differential equations and inclusions, Mathematics, 9 (2021), 615. https://doi.org/10.3390/math9060615 doi: 10.3390/math9060615
[32]	S. Sitho, S. K. Ntouyas, A. Samadi, J. Tariboon, Boundary value problems for ${\psi}$-Hilfer type sequential fractional differential equations and inclusions with integral multi-point boundary conditions, Mathematics, 9 (2021), 1001. https://doi.org/10.3390/math9091001 doi: 10.3390/math9091001
[33]	W. Sudsutad, C. Thaiprayoon, S. K. Ntouyas, Existence and stability results for ${\psi}$-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions, AIMS Mathematics, 6 (2021), 4119–4141. http://doi.org/10.3934/math.2021244 doi: 10.3934/math.2021244
[34]	R. Subashini, C. Ravichandran, K. Jothimani, H. M. Baskonus, Existence results of Hilfer integro-differential equations with fractional order, AIMS Mathematics, 13 (2020), 911–923. http://doi.org/10.3934/dcdss.2020053 doi: 10.3934/dcdss.2020053
[35]	Y. Wang, S. Liang, Q. Wang, Existence results for fractional differential equations with integral and multipoint boundary conditions, Bound. Value Probl., 2018 (2018), 4. https://doi.org/10.1186/s13661-017-0924-4 doi: 10.1186/s13661-017-0924-4
[36]	X. Yu, Existence and $\beta$-Ulam-Hyers stability for a class of fractional differential equations with non-instantaneous impulses, Adv. Differ. Equ., 2015 (2015), 104. http://doi.org/10.1186/s13662-015-0415-9 doi: 10.1186/s13662-015-0415-9
[37]	X. Zhang, P. Agarwal, Z. Liu, X. Zhang, W. Ding, A. Ciancio, On the fractional differential equations with not instantaneous impulses, Open Physics, 14 (2016), 676–684. https://doi.org/10.1515/phys-2016-0076 doi: 10.1515/phys-2016-0076
[38]	K. D. Kucchea, J. P. Kharadea, J. V. da Costa Sousa, On the nonlinear impulsive ${\psi}$-Hilfer fractional differential equations, Math. Model. Anal., 25 (2020), 642–660. https://doi.org/10.3846/mma.2020.11445 doi: 10.3846/mma.2020.11445
[39]	H. M. Srivastava, J. V. da Costa Sousa, Multiplicity of solutions for fractional-order differential equations via the $\kappa(x)$-Laplacian operator and the Genus theory, Fractal Fract., 6 (2022), 481. https://doi.org/10.3390/fractalfract6090481 doi: 10.3390/fractalfract6090481
[40]	A. Zada, J. Alzabut, H. Waheed, I. L. Popa, Ulam-Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions, Adv. Differ. Equ., 2020 (2020), 64. https://doi.org/10.1186/s13662-020-2534-1 doi: 10.1186/s13662-020-2534-1
[41]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107.
[42]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer-Verlag, 2005.
[43]	M. Rashid, A. Kalsoom, A. Ghaffar, M. Inc, N. Sene, A multiple fixed point result for $(\theta, \phi, \psi)$-type contractions in the partially ordered $s$-distance spaces with an application, J. Funct. Spaces, 2022 (2022), 6202981. https://doi.org/10.1155/2022/6202981 doi: 10.1155/2022/6202981
[44]	N. Sene, Fundamental results about the fractional integro-differential equation described with Caputo derivative, J. Funct. Spaces, 2022 (2022), 9174488. https://doi.org/10.1155/2022/9174488 doi: 10.1155/2022/9174488
[45]	C. Dineshkumar, K. S. Nisar, R. Udhayakumar, V. Vijayakumar, A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions, Asian J. Control, 24 (2022), 2378–2394. https://doi.org/10.1002/asjc.2650 doi: 10.1002/asjc.2650
[46]	K. Kavitha, K. S. Nisar, A. Shukla, V. Vijayakumar, S. Rezapour, A discussion concerning the existence results for the Sobolev-type Hilfer fractional delay integro-differential systems, Adv. Differ. Equ., 2021 (2021), 467. https://doi.org/10.1186/s13662-021-03624-1 doi: 10.1186/s13662-021-03624-1

Reader Comments

Your name:*

Email:*
© 2023 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)