Existence theorems for $ \Psi $-fractional hybrid systems with periodic boundary conditions

Iyad Suwan; Mohammed S. Abdo; Thabet Abdeljawad; Mohammed M. Matar; Abdellatif Boutiara; Mohammed A. Almalahi; Iyad Suwan; Mohammed S. Abdo; Thabet Abdeljawad; Mohammed M. Matar; Abdellatif Boutiara; Mohammed A. Almalahi

doi:10.3934/math.2022010

AIMS Mathematics

2022, Volume 7, Issue 1: 171-186. doi: 10.3934/math.2022010

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Research article

Existence theorems for $ \Psi $-fractional hybrid systems with periodic boundary conditions

1.
Department of Mathematics and Statistics, The Arab American University, P.O.Box 240, 13 Zababdeh, Jenin, Palestine
2.
Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen
3.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics, Al-Azhar University-Gaza, Gaza, State of Palestine
6.
Laboratory of Mathematics And Applied Sciences, University of Ghardaia 47000, Algeria
7.
Department of Mathematics, Hajjah University, Hajjah, Yemen

Received: 17 August 2021 Accepted: 28 September 2021 Published: 09 October 2021
MSC : 34A08, 26A33, 34A34

This research paper deals with two novel varieties of boundary value problems for nonlinear hybrid fractional differential equations involving generalized fractional derivatives known as the $ \Psi $-Caputo fractional operators. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function $ \Psi $. The existence results to the proposed systems are obtained by using Dhage's fixed point theorem. Two pertinent examples are provided to confirm the feasibility of the obtained results. Our presented results generate many special cases with respect to different values of a $ \Psi $ function.
- hybrid fractional differential equations,
- existence,
- fixed point theorem
Citation: Iyad Suwan, Mohammed S. Abdo, Thabet Abdeljawad, Mohammed M. Matar, Abdellatif Boutiara, Mohammed A. Almalahi. Existence theorems for $ \Psi $-fractional hybrid systems with periodic boundary conditions[J]. AIMS Mathematics, 2022, 7(1): 171-186. doi: 10.3934/math.2022010

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Abstract

This research paper deals with two novel varieties of boundary value problems for nonlinear hybrid fractional differential equations involving generalized fractional derivatives known as the $ \Psi $-Caputo fractional operators. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function $ \Psi $. The existence results to the proposed systems are obtained by using Dhage's fixed point theorem. Two pertinent examples are provided to confirm the feasibility of the obtained results. Our presented results generate many special cases with respect to different values of a $ \Psi $ function.

References

[1]	A. A. Kilbas, H. M. Shrivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. doi: 10.1016/s0304-0208(06)x8001-5.
[2]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, 2010. doi: 10.1142/p614.
[3]	J. Hadamdard, Essai sur l'etude des fonctions données par leur développement de Taylor, J. Math. Pure. Appl., 8 (1892), 101–186.
[4]	R. Hilfer, Applications of Fractional Calculus in Physics, Singapore: World Scientific, 35 (2000), 87–130. doi: 10.1142/9789812817747_0008.
[5]	F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142. doi: 10.1186/1687-1847-2012-142. doi: 10.1186/1687-1847-2012-142
[6]	F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 247. doi: 10.1186/s13662-017-1306-z. doi: 10.1186/s13662-017-1306-z
[7]	U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860–865. doi: 10.1016/j.amc.2011.03.062. doi: 10.1016/j.amc.2011.03.062
[8]	M. Caputo, M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85. doi: 10.12785/pfda/010201. doi: 10.12785/pfda/010201
[9]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. doi: 10.2298/TSCI160111018A. doi: 10.2298/TSCI160111018A
[10]	A. Atangana, Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system, Chaos Soliton. Fract., 102 (2017), 396–406. doi: 10.1016/j.chaos.2017.04.027. doi: 10.1016/j.chaos.2017.04.027
[11]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci., 44 (2017), 460–481. doi: 10.1016/j.cnsns.2016.09.006. doi: 10.1016/j.cnsns.2016.09.006
[12]	J. V. C. Sousa, C. E. Oliveira, On the $\Psi $-Hilfer fractional derivative, Commun. Nonlinear Sci., 60 (2018), 72–91. doi: 10.1016/j.cnsns.2018.01.005. doi: 10.1016/j.cnsns.2018.01.005
[13]	F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Cont. Dyn-S., 13 (2020), 709. doi: 10.3934/dcdss.2020039. doi: 10.3934/dcdss.2020039
[14]	M. S. Abdo, Further results on the existence of solutions for generalized fractional quadratic functional integral equations, J. Math. Anal. Model., 1 (2020), 33–46. doi: 10.48185/jmam.v1i1.2. doi: 10.48185/jmam.v1i1.2
[15]	H. A. Wahash, M. S. Abdo, A. M. Saeed, S. K. Panchal, Singular fractional differential equations with $\Psi $-Caputo operator and modified Picard's iterative method, Appl. Math. E-Notes, 20 (2020), 215–229.
[16]	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. doi: 10.1016/j.rinam.2021.100142. doi: 10.1016/j.rinam.2021.100142
[17]	N. Adjimi, A. Boutiara, M. S. Abdo, M. Benbachir, Existence results for nonlinear neutral generalized Caputo fractional differential equations, J. Pseudo-Differ. Oper., 12 (2021), 25. doi: 10.1007/s11868-021-00400-3. doi: 10.1007/s11868-021-00400-3
[18]	H. A. Wahash, M. S. Abdo, S. K. Panchal, Existence and stability of a nonlinear fractional differential equation involving a $ \Psi $-Caputo operator, Adv. Theor. Nonlinear Anal. Appl., 4 (2020), 266–278. doi: 10.31197/atnaa.664534. doi: 10.31197/atnaa.664534
[19]	Z. Baitiche, C. Derbazi, M. M. Matar, Ulam stability for nonlinear-Langevin fractional differential equations involving two fractional orders in the $\Psi $-Caputo sense, Appl. Anal., 2021, 1–16. doi: 10.1080/00036811.2021.1873300.
[20]	Y. Zhao, On the existence for a class of periodic boundary value problems of nonlinear fractional hybrid differential equations, Appl. Math. Lett., 121 (2021), 107368. doi: 10.1016/j.aml.2021.107368. doi: 10.1016/j.aml.2021.107368
[21]	Y. Zhao, S. Suna, Z. Hana, M. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput., 217 (2011), 6950–6958. doi: 10.1016/j.amc.2011.01.103. doi: 10.1016/j.amc.2011.01.103
[22]	Y. Zhao, X. Hou, Y. Sun, Z. Bai, Solvability for some class of multi-order nonlinear fractional systems, Adv. Differ. Equ., 2019 (2019), 23. doi: 10.1186/s13662-019-1970-2. doi: 10.1186/s13662-019-1970-2
[23]	M. Almalahi, S. Panchal, Existence and $\delta$-Approximate solution of implicit fractional pantograph equations in the frame of Hilfer-Katugampola operator, J. Fract. Calc. Nonlinear Sys., 2 (2021), 1–17. doi: 10.48185/jfcns.v2i1.59. doi: 10.48185/jfcns.v2i1.59
[24]	Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl., 62 (2011), 1312–1324. doi: 10.1016/j.camwa.2011.03.041. doi: 10.1016/j.camwa.2011.03.041
[25]	B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal- Hybri., 4 (2010), 414–424. doi: 10.1016/j.nahs.2009.10.005. doi: 10.1016/j.nahs.2009.10.005
[26]	B. Dhage, N. Jadhav, Basic results in the theory of hybrid differential equations with linear perturbations of second type, Tamkang J. Math., 44 (2013), 171–186. doi: 10.5556/j.tkjm.44.2013.1086. doi: 10.5556/j.tkjm.44.2013.1086
[27]	M. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., 3 (2014), 389386. doi: 10.1155/2014/389386. doi: 10.1155/2014/389386
[28]	M. S. Abdo, T. Abdeljawad, K. Shah, S. M. Ali, On nonlinear coupled evolution system with nonlocal subsidiary conditions under fractal-fractional order derivative, Math. Method Appl. Sci., 44 (2021), 6581–6600. doi: 10.1002/mma.7210. doi: 10.1002/mma.7210
[29]	G. Nazir, K. Shah, T. Abdeljawad, H. Khalil, R. A. Khan, Using a prior estimate method to investigate sequential hybrid fractional differential equations, Fractals, 28 (2020), 2040004. doi: 10.1142/S0218348X20400046. doi: 10.1142/S0218348X20400046
[30]	A. Ali, K. Shah, R. A. Khan, Existence of solution to a coupled system of hybrid fractional differential equations, Bull. Math. Anal. Appl., 9 (2017), 9–18.
[31]	M. B. Zada, K. Shah, R. A. Khan, Existence theory to a coupled system of higher order fractional hybrid differential equations by topological degree theory, Int. J. Appl. Comput. Math., 4 (2018), 102. doi: 10.1007/s40819-018-0534-6. doi: 10.1007/s40819-018-0534-6
[32]	B. Ahmad, S. K. Ntouyas, J. Tariboon, A nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations, Acta Math. Sci., 36 (2016), 1631–1640. doi:10.1016/S0252-9602(16)30095-9. doi: 10.1016/S0252-9602(16)30095-9
[33]	S. Etemad, S. Rezapour, M. E. Samei, On fractional hybrid and non-hybrid multi-term integro-differential inclusions with three-point integral hybrid boundary conditions, Adv. Differ. Equ., 2020 (2020), 161. doi: 10.1186/s13662-020-02627-8. doi: 10.1186/s13662-020-02627-8
[34]	S. B. Chikh, A. Amara, S. Etemad, S. Rezapour, On Ulam-Hyers-Rassias stability of a generalized Caputo type multi-order boundary value problem with four-point mixed integro-derivative conditions, Adv. Differ. Equ., 2020 (2020), 680. doi: 10.1186/s13662-020-03139-1. doi: 10.1186/s13662-020-03139-1
[35]	A. Amara, S. Etemad, S. Rezapour, Approximate solutions for a fractional hybrid initial value problem via the Caputo conformable derivative, Adv. Differ. Equ., 2020 (2020), 608. doi: 10.1186/s13662-020-03072-3. doi: 10.1186/s13662-020-03072-3
[36]	N. Mahmudov, M. M. Matar, Existence of mild solution for hybrid differential equations with arbitrary fractional order, TWMS J. Pure Appl. Math., 8 (2017), 160–169.
[37]	M. M. Matar, Existence of solution for fractional neutral hybrid differential equations with finite delay, Rocky Mt. J. Math., 50 (2020), 2141–2148. doi: 10.1216/rmj.2020.50.2141. doi: 10.1216/rmj.2020.50.2141
[38]	M. M. Matar, Qualitative properties of solution for hybrid nonlinear fractional differential equations, Afr. Mat., 30 (2019), 1169–1179. doi: 10.1007/s13370-019-00710-2. doi: 10.1007/s13370-019-00710-2
[39]	M. M. Matar, Approximate controllability of fractional nonlinear hybrid differential systems via resolvent operators, J. Math., 2019 (2019), 7 pages. doi: 10.1155/2019/8603878. doi: 10.1155/2019/8603878
[40]	B. C. Dhage, A fixed point theorem in Banach algebras involv-ing three operators with applications, Kyungpook Math. J., 44 (2004), 145–155.
[41]	B. C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett., 18 (2005), 273–280. doi: 10.1016/j.aml.2003.10.014. doi: 10.1016/j.aml.2003.10.014

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