Existence of solutions for multi-point nonlinear differential system equations of fractional orders with integral boundary conditions

Isra Al-Shbeil; Abdelkader Benali; Houari Bouzid; Najla Aloraini; Isra Al-Shbeil; Abdelkader Benali; Houari Bouzid; Najla Aloraini

doi:10.3934/math.2022998

AIMS Mathematics

2022, Volume 7, Issue 10: 18142-18157. doi: 10.3934/math.2022998

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Existence of solutions for multi-point nonlinear differential system equations of fractional orders with integral boundary conditions

1.
Department of Mathematics, Faculty of Sciences, University of Jordan, Amman 11942, Jordan
2.
University of Hassiba Benbouali, Faculty of the Exact Sciences and Computer, Mathematics Department, Laboratory LMA, Chlef 02000, Algeria
3.
Department of Mathematics, Faculty of Science, Qassim University, Qassim, Saudi Arabia

Received: 05 June 2022 Revised: 24 July 2022 Accepted: 27 July 2022 Published: 10 August 2022
MSC : Primary 46C05, Secondary 47A05

In this work, we study existence and uniqueness of solutions for multi-point boundary value problemS of nonlinear fractional differential equations with two fractional derivatives. By using a variety of fixed point theorems, such as Banach's fixed point theorem, Leray-Schauder's nonlinear alternative and Leray-Schauder's degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.
- Riemann-Liouville integral,
- existence of solutions,
- fixed point theorem
Citation: Isra Al-Shbeil, Abdelkader Benali, Houari Bouzid, Najla Aloraini. Existence of solutions for multi-point nonlinear differential system equations of fractional orders with integral boundary conditions[J]. AIMS Mathematics, 2022, 7(10): 18142-18157. doi: 10.3934/math.2022998

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Abstract

In this work, we study existence and uniqueness of solutions for multi-point boundary value problemS of nonlinear fractional differential equations with two fractional derivatives. By using a variety of fixed point theorems, such as Banach's fixed point theorem, Leray-Schauder's nonlinear alternative and Leray-Schauder's degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.

References

[1]	A. Alsaedi, S. K. Ntouyas, B. Ahmad, New existence results for fractional integrodifferential equations with nonlocal integral boundary conditions, Abstr. Appl. Anal., 2015 (2015), 205452. https://doi.org/10.1155/2015/205452 doi: 10.1155/2015/205452
[2]	B. Ahmad, J. J. Nietoa, A. Alsaedi, H. Al-Hutamia, Boundary value problems of nonlinear fractional $q$-difference (integral) equations with two fractional orders and four-point nonlocal integral boundary conditions, Filomat, 28 (2014), 1719–1736. https://doi.org/10.2298/FIL1408719A doi: 10.2298/FIL1408719A
[3]	B. Ahmad, J. J. Nieto, Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions, Int. J. Differ. Equ., 2010 (2010), 649486. https://doi.org/10.1155/2010/649486 doi: 10.1155/2010/649486
[4]	Z. Bai, H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495–505. https://doi.org/10.1016/j.jmaa.2005.02.052 doi: 10.1016/j.jmaa.2005.02.052
[5]	S. Belarbi, Z. Dahmani, Some applications of Banach fixed point and Leray Schauder theorems for fractional boundary value problems, J. Dyn. Syst. Geom. The., 11 (2013), 53–79. https://doi.org/10.1080/1726037X.2013.838448 doi: 10.1080/1726037X.2013.838448
[6]	C. Thaiprayoon, S. K. Ntouyas, J. Tariboon, On the nonlocal Katugampola fractional integral conditions for fractional Langevin equation, Adv. Differ. Equ., 2015 (2015), 374. https://doi.org/10.1186/s13662-015-0712-3 doi: 10.1186/s13662-015-0712-3
[7]	Z. Cui, P. Yu, Z. Mao, Existence of solutions for nonlocal boundary value problems of nonlinear fractional differential equations, Adv. Dyn. Syst. Appl., 7 (2012), 31–40.
[8]	Z. Dahmani, L. Tabharit, Fractional order differential equations involving Caputo derivative, Theory Appl. Math. Comput. Sci., 4 (2014), 40–55.
[9]	L. Gaul, P. Klein, S. Kemple, Damping description involving fractional operators, Mech. Syst. Signal Pr., 5 (1991), 81–88. https://doi.org/10.1016/0888-3270(91)90016-X doi: 10.1016/0888-3270(91)90016-X
[10]	W. G. Glöckle, T. F. Nonnenmacher, A fractional calculus approach to self-semilar protein dynamics, Biophys. J., 68 (1995), 46–53. https://doi.org/10.1016/S0006-3495(95)80157-8 doi: 10.1016/S0006-3495(95)80157-8
[11]	M. Houas, Z. Dahmani, New results for a Caputo boundary value problem, Amer. J. Comput. Appl. Math., 3 (2013), 143–161. https://doi.org/10.5923/j.ajcam.20130303.01 doi: 10.5923/j.ajcam.20130303.01
[12]	M. Houas, Z. Dahmani, M. Benbachir, New results for a boundary value problem for differential equations of arbitrary order, Int. J. Mod. Math. Sci., 7 (2013), 195–211.
[13]	M. Houas, Z. Dahmani, On existence of solutions for fractional differential equations with nonlocal multi-point boundary conditions, Lobachevskii J. Math., 37 (2016), 120–127. https://doi.org/10.1134/S1995080216020050 doi: 10.1134/S1995080216020050
[14]	M. Houas, M. Benbachir, Existence solutions for four point boundary value problems for fractional differential equations, Pure Appl. Math. Lett., 2015 (2015), 37–49.
[15]	F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, In: A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Vol. 378, International Centre for Mechanical Sciences, New York: Springer-Verlag, 1997. https://doi.org/10.1007/978-3-7091-2664-6_7
[16]	R. Metzler, W. Schick, H. G. Kilian, T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103 (1995), 7180–7186. https://doi.org/10.1063/1.470346 doi: 10.1063/1.470346
[17]	N. Nyamoradi, T. Bashiri, S. M. Vaezpour, D. Baleanu, Uniqueness and existence of positive solutions for singular fractional differential equations, Electron. J. Differ. Equ., 130 (2014), 1–13.
[18]	H. Scher, E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B, 12 (1975), 2455–2477. https://doi.org/10.1103/PhysRevB.12.2455 doi: 10.1103/PhysRevB.12.2455
[19]	M. Li, Y. Liu, Existence and uniqueness of positive solutions for a coupled system of nonlinear fractional differential equations, Open J. Appl. Sci., 3 (2013), 53–61. https://doi.org/10.4236/ojapps.2013.31B1011 doi: 10.4236/ojapps.2013.31B1011
[20]	G. T. Wang, B. Ahmad, L. Zhang, A coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain, Abstr. Appl. Anal., 2012 (2012), 248709. https://doi.org/10.1155/2012/248709 doi: 10.1155/2012/248709
[21]	J. Tariboon, S. K. Ntouyas, C. Thaiprayoon, Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions, Adv. Math. Phys., 2014 (2014), 372749. https://doi.org/10.1155/2014/372749 doi: 10.1155/2014/372749
[22]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[23]	G. Wang, W. Liu, C. Ren, Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions, Electron. J. Differ. Equ., 2012 (2012), 1–10.
[24]	Y. H. Zhang, Z. B. Bai, Existence of solutions for nonlinear fractional three-point boundary value problems at resonance, J. Appl. Math. Comput., 36 (2011), 417–440. https://doi.org/10.1007/s12190-010-0411-x doi: 10.1007/s12190-010-0411-x
[25]	W. Yukunthorn, S. K. Ntouyas, J. Tariboon, Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions, Adv. Differ. Equ., 2014 (2014), 315. https://doi.org/10.1186/1687-1847-2014-315 doi: 10.1186/1687-1847-2014-315
[26]	A. Benali, H. Bouzid, M. Haouas, Existence of solutions for Caputo fractional $q$-differential equations, Asia Math., 5 (2021), 143–157. https://doi.org/10.5281/zenodo.4730073 doi: 10.5281/zenodo.4730073
[27]	H. Annaby, Z. Mansour, q-fractional calculus and equations, Lecture Notes in Mathematics, Vol. 2056. Berlin: Springer, 2012.
[28]	R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Math. Proc. Cambridge Philos. Soc., 66 (1969), 365–370. https://doi.org/10.1017/S0305004100045060 doi: 10.1017/S0305004100045060
[29]	P. Rajkovic, S. Marinkovic, M. Stankovic, On $q$-analogues of Caputo derivative and Mittag-Leffler function, Fract. Calc. Appl. Anal., 10 (2007), 359–373.
[30]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000. https://doi.org/10.1142/3779
[31]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1999.
[32]	G. Samko, A. Kilbas, O. Marichev, Fractional integrals and derivatives: Theory and applications, Amsterdam: Gordon and Breach, 1993.
[33]	A. Granas, J. Dugundji, Fixed Point Theory, Springer Verlag, 2003.
[34]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Vol. 204, 2006.
[35]	Z. Baitiche, C. Derbazi, J. Alzabut, M. E. Samei, M. K. A. Kaabar, Z. Siri, Monotone iterative method for langevin equation in terms of $\psi$-Caputo fractional derivative and nonlinear boundary conditions, Fractal Fract., 5 (2021), 81. https://doi.org/10.3390/fractalfract5030081 doi: 10.3390/fractalfract5030081
[36]	J. Alzabut, A. G. M. Selvam, R. A. El-Nabulsi, V. Dhakshinamoorthy, M. E. Samei, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions, Symmetry, 13 (2021), 473. https://doi.org/10.3390/sym13030473 doi: 10.3390/sym13030473
[37]	S. N. Hajiseyedazizi, M. E. Samei, J. Alzabut, Y. Chu, On multi-step methods for singular fractional $q$-integro-differential equations, Open Math., 19 (2021), 1378–1405. https://doi.org/10.1515/math-2021-0093 doi: 10.1515/math-2021-0093
[38]	A. Boutiara, M. Benbachir, J. Alzabut, M. E. Samei, Monotone iterative and upper-lower solutions techniques for solving nonlinear $\psi$-Caputo fractional boundary value problem, Fractal Fract., 5 (2021), 194. https://doi.org/10.3390/fractalfract5040194 doi: 10.3390/fractalfract5040194
[39]	J. Alzabut, B. Mohammadaliee, M. E. Samei, Solutions of two fractional $q$-integro-differential equations under sum and integral boundary value conditions on a time scale, Adv. Differ. Equ., 2020 (2020), 304. https://doi.org/10.1186/s13662-020-02766-y doi: 10.1186/s13662-020-02766-y
[40]	H. Zhang, Y. Cheng, H. Zhang, W. Zhang, J. Cao, Hybrid control design for Mittag-Leffler projective synchronization on FOQVNNs with multiple mixed delays and impulsive effects, Math. Comput. Simulat., 197 (2022), 341357. https://doi.org/10.1016/j.matcom.2022.02.022 doi: 10.1016/j.matcom.2022.02.022
[41]	H. Zhang, J. Cheng, H. Zhang, W. Zhang, J. Cao, Quasi-uniform synchronization of Caputo type fractional neural networks with leakage and discrete delays, Chaos Solitons Fract., 152 (2021), 111432. https://doi.org/10.1016/j.chaos.2021.111432 doi: 10.1016/j.chaos.2021.111432
[42]	C. Wang, H. Zhang, H. Zhang, W. Zhang, Globally projective synchronization for Caputo fractional quaternion-valued neural networks with discrete and distributed delays, AIMS Math., 6 (2021), 14000–14012. https://doi.org/10.3934/math.2021809 doi: 10.3934/math.2021809

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