Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives

Shayma A. Murad; Zanyar A. Ameen; Shayma A. Murad; Zanyar A. Ameen

doi:10.3934/math.2022357

AIMS Mathematics

2022, Volume 7, Issue 4: 6404-6419. doi: 10.3934/math.2022357

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Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives

Shayma A. Murad ,
Zanyar A. Ameen ^,

Department of Mathematics, College of Science, University of Duhok, Duhok 42001, IRAQ

Received: 07 October 2021 Revised: 17 November 2021 Accepted: 05 December 2021 Published: 20 January 2022
MSC : 26A33, 34A08, 34A12, 34B27, 34D20, 34K20

In this paper, we study the existence, uniqueness, and stability theorems of solutions for a differential equation of mixed Caputo-Riemann fractional derivatives with integral initial conditions in a Banach space. Our analysis is based on an application of the Shauder fixed point theorem with Ulam-Hyers and Ulam-Hyers-Rassias theorems. A couple of examples are presented to illustrate the obtained results.
- fractional differential equation,
- Riemann and Caputo fractional derivatives,
- fixed point theorem,
- stability analysis
Citation: Shayma A. Murad, Zanyar A. Ameen. Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives[J]. AIMS Mathematics, 2022, 7(4): 6404-6419. doi: 10.3934/math.2022357

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Abstract

In this paper, we study the existence, uniqueness, and stability theorems of solutions for a differential equation of mixed Caputo-Riemann fractional derivatives with integral initial conditions in a Banach space. Our analysis is based on an application of the Shauder fixed point theorem with Ulam-Hyers and Ulam-Hyers-Rassias theorems. A couple of examples are presented to illustrate the obtained results.

References

[1]	M. I. Abbas, Existence and uniqueness of solution for a boundary value problem of fractional order involving two Caputo's fractional derivatives, Adv. Differ. Equ., 2015 (2015), 252. https://doi.org/10.1186/s13662-015-0581-9 doi: 10.1186/s13662-015-0581-9
[2]	D. N. Abdulqader, S. A. Murad, Existence and uniqueness results for certain fractional boundary value problems, J. Duhok Univ., 22 (2019), 76–88. https://doi.org/10.26682/sjuod.2019.22.2.9 doi: 10.26682/sjuod.2019.22.2.9
[3]	J. G. Abulahad, S. A. Murad, Existence, uniqueness and stability theorems for certain functional fractional initial value problem, Al-Rafidain J. Comput. Sci. Math., 8 (2011), 59–70. https://doi.org/10.33899/csmj.2011.163608 doi: 10.33899/csmj.2011.163608
[4]	J. G. Abulahad, S. A. Murad, Global existence and uniqueness theorems of certain fractional boundary value problem, J. Duhok Univ., 12 (2009), 150–161.
[5]	B. Ahmad, J. J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces, 2013 (2013), 149659. https://doi.org/10.1155/2013/149659 doi: 10.1155/2013/149659
[6]	B. Ahmad, J. J. Nieto, Sequential fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 64 (2012), 3046–3052. https://doi.org/10.1016/j.camwa.2012.02.036 doi: 10.1016/j.camwa.2012.02.036
[7]	N. Alghamdi, B. Ahmad, S. K. Ntouyas, A. Alsaedi, Sequential fractional differential equations with nonlocal boundary conditions on an arbitrary interval, Adv. Differ. Equ., 2017 (2017), https://doi.org/10.1186/s13662-017-1303-2
[8]	M. H. Aqlan, A. Alsaedi, B. Ahmad, J. J. Nieto, Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions, Open Math., 14 (2016), 723–735. https://doi.org/10.1515/math-2016-0064 doi: 10.1515/math-2016-0064
[9]	J. H. Barrett, Differential equations of non-integer order, Can. J. Math., 6 (1954), 529–541. https://doi.org/10.4153/CJM-1954-058-2 doi: 10.4153/CJM-1954-058-2
[10]	T. Blaszczyk, M. Ciesielski, Numerical solution of Euler-Lagrange equation with Caputo derivatives, Adv. Appl. Math. Mech., 9 (2017), 173–185. https://doi.org/10.4208/aamm.2015.m970 doi: 10.4208/aamm.2015.m970
[11]	R. I. Butt, T. Abdeljawad, M. ur Rehman, Stability analysis by fixed point theorems for a class of non-linear Caputo nabla fractional difference equation, Adv. Differ. Equ., 2020 (2020), 209. https://doi.org/10.1186/s13662-020-02674-1 doi: 10.1186/s13662-020-02674-1
[12]	G. E. Chatzarakis, M. Deepa, N. Nagajothi, V. Sadhasivam, Oscillatory properties of a certain class of mixed fractional differential equations, Appl. Math. Inf. Sci., 14 (2020), 123–131. http://doi.org/10.18576/amis/140116 doi: 10.18576/amis/140116
[13]	C. R. Chen, M. Bohner, B. G. Jia, Ulam-Hyers stability of Caputo fractional difference equations, Math. Meth. Appl. Sci., 42 (2019), 7461–7470. https://doi.org/10.1002/mma.5869 doi: 10.1002/mma.5869
[14]	Q. Dai, R. M. Gao, Z. Li, C. J. Wang, Stability of Ulam–Hyers and Ulam–Hyers–Rassias for a class of fractional differential equations, Adv. Differ. Equ., 2020 (2020), 103. https://doi.org/10.1186/s13662-020-02558-4 doi: 10.1186/s13662-020-02558-4
[15]	A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
[16]	R. Herrmann, Fractional calculus: An introduction for physicists, Singapor: World Scientific Publication Company, 2011. https://doi.org/10.1142/8072
[17]	M. Hu, L. L. Wang, Existence of solutions for a nonlinear fractional differential equation with integral boundary condition, Int. J. Math. Comput. Sci., 5 (2011), 55–58. https://doi.org/10.5281/zenodo.1335374 doi: 10.5281/zenodo.1335374
[18]	R. W. Ibrahim, Ulam stability of boundary value problem, Kragujev. J. Math., 37 (2013), 287–297.
[19]	H. A. Jalab, R. W. Ibrahim, S. A. Murad, S. B. Hadid, Exact and numerical solution for fractional differential equation based on neural network, Proc. Pakistan Aca. Sci., 49 (2012), 199–208.
[20]	S. A. Jose, A. Tom, M. S. Ali, S. Abinaya, W. Sudsutad, Existence, uniqueness and stability results of semilinear functional special random impulsive differential equations, Dyn. Cont. Discrete Impulsive Syst. Series A: Math. Anal., 28 (2021), 269–293.
[21]	S. A. Jose, W. Yukunthornx, J. E. N. Valdes, H. Leiva, Some existence, uniqueness and stability results of nonlocal random impulsive integro-differential equations, Appl. Math. E-Notes, 20 (2020), 481–492.
[22]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[23]	K. Liu, M. Feckan, J. R. Wang, Hyers–Ulam stability and existence of solutions to the generalized Liouville–Caputo fractional differential equations, Symmetry, 12 (2020), 955. https://doi.org/10.3390/sym12060955 doi: 10.3390/sym12060955
[24]	K. Liu, J. R. Wang, Y. Zhou, D. O'Regan, Hyers–Ulam stability and existence of solutions for fractional differential equations with Mittag–Leffler kernel, Chaos Soliton. Fract., 132 (2020), 109534. https://doi.org/10.1016/j.chaos.2019.109534 doi: 10.1016/j.chaos.2019.109534
[25]	L. Lv, J. Wang, W. Wei, Existence and uniqueness results for fractional differential equations with boundary value conditions, Opusc. Math., 31 (2011), 629–643. http://doi.org/10.7494/OpMath.2011.31.4.629 doi: 10.7494/OpMath.2011.31.4.629
[26]	P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., 102 (2015), 631–642. http://doi.org/10.12732/ijpam.v102i4.4 doi: 10.12732/ijpam.v102i4.4
[27]	S. A. Murad, R. W. Ibrahim, S. B. Hadid, Existence and uniqueness for solution of differential equation with mixture of integer and fractional derivative, Pak. Acad. Sci., 49 (2012), 33–37.
[28]	S. A. Murad, S. B. Hadid, Existence and uniqueness theorem for fractional differential equation with integral boundary condition, J. Fract. Calc. Appl., 3 (2012), 1–9.
[29]	S. A. Murad, H. J. Zekri, S. Hadid, Existence and uniqueness theorem of fractional mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions, Int. J. Differ. Equ., 2011 (2011), 304570. https://doi.org/10.1155/2011/304570 doi: 10.1155/2011/304570
[30]	S. A. Murad, A. S. Rafeeq, Existence of solutions of integro-fractional differential equations when $\alpha \in(2, 3]$ through fixed point theorem, J. Math. Comput. Sci., 11 (2021), 6392–6402. https://doi.org/10.28919/jmcs/6272 doi: 10.28919/jmcs/6272
[31]	S. I. Muslih, D. Baleanu, Fractional Euler–Lagrange equations of motion in fractional Space, J. Vib. Control, 13 (2007), 1209–1216. https://doi.org/10.1177/1077546307077473 doi: 10.1177/1077546307077473
[32]	I. Podlubny, Fractional differential equation, mathematics in science and engineering, San Diego: Academic Press, 1999.
[33]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107.
[34]	S. Y. Song, Y. J. Cui, Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance, Bound. Value Probl., 2020 (2020), 23. https://doi.org/10.1186/s13661-020-01332-5 doi: 10.1186/s13661-020-01332-5
[35]	J. V. da C. Sousa, L. S. Tavares, E. C. de Oliveira, Existence and uniqueness of mild and strong solutions for fractional evaluation equation, Palest. J. Math., 10 (2021), 592–600.
[36]	E. Zeidler, Nonlinear analysis and its applications I: Fixed point theorems, New York: Springer-Verlag, 1986.

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