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Existence and stability for a nonlinear hybrid differential equation of fractional order via regular Mittag-Leffler kernel. (English) Zbl 07782466

Summary: This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag-Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed-point principle, we discuss the existence of mild solutions. Finally, we study the Ulam-Hyers stability of the introduced fractional hybrid problem.
{© 2021 John Wiley & Sons, Ltd.}

MSC:

34A38 Hybrid systems of ordinary differential equations
32A65 Banach algebra techniques applied to functions of several complex variables
26A33 Fractional derivatives and integrals
34K20 Stability theory of functional-differential equations
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[1] HilferR. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000. · Zbl 0998.26002
[2] JiaoZ, ChenY, PodlubnyI. Distributed Order Dynamic Systems, Modeling, Analysis and Simulation. London, Heidelberg, New York: Springer; 2012. · Zbl 1401.93005
[3] KaslikE, SivasundaramS. Nonlinear dynamics and chaos in fractional‐order neural networks. Neural Netw. 2012;32:245‐256. · Zbl 1254.34103
[4] KoellerRC. Applications of fractional calculus to the theory of viscoelasticity. ASME J Appl Mech. 1984;51(2):299‐307. https://doi.org/10.1115/1.3167616 · Zbl 0544.73052 · doi:10.1115/1.3167616
[5] MainardiF. Fractional Calculus and Waves in Linear Viscoelatiity: An Introduction to Mathematical Models. Bologna/Italy: World Scientific Publishing; 2010. · Zbl 1210.26004
[6] AtanganaA. Fractional Operators with Constant and Variable Order with Application to Geo‐hydrology. New York: Academic Press; 2017.
[7] BaleanuD, GuvencZB, MachadoJAT. New Trends in Nanotechnology and Fractional Calculus Applications, XI, 531. 1st Edition. Heidelberg: ISBN Springer; 2010. · Zbl 1196.65021
[8] BaleanuD, EtemadS, RezapourS. A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound Value Probl. 2020;2020:64. https://doi.org/10.1186/s13661-020-01361-0 · Zbl 1495.34006 · doi:10.1186/s13661-020-01361-0
[9] MomaniS, OdibatZ. Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys Lett A. 2006;355(4‐5):271‐279. · Zbl 1378.76084
[10] PintoCMA, CarvalhoARM. A latency fractional order model for HIV dynamics. J Comput Appl Math. 2017;312:240‐256. · Zbl 1352.34076
[11] DiethelmK. Analysis of fractional differential equations: an application‐ oriented exposition using differential operators of Caputo type; 2010. · Zbl 1215.34001
[12] AtanganaA, BaleanuD. New fractional derivative with non‐local and non‐singular kernel. Therm Sci. 2016;20(2):757‐763.
[13] IonescuC, LopesA, CopotaD, MachadoJAT, BatesJHT. The role of fractional calculus in modeling biological phenomena: a review; 2017. · Zbl 1467.92050
[14] SunHG, ZhangY, BaleanuD, ChenW, ChenYQ. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul. 2018;64:213‐231. · Zbl 1509.26005
[15] BaleanuD, GhanbariB, JihadHA, JajarmiA, PirouzHM. Planar system‐masses in an equilateral triangle: numerical study within fractional calculus. CMES‐Comput Model Eng Sci. 2020;124(3):953‐968.
[16] JajarmiA, BaleanuD. A new iterative method for the numerical solution of high‐order nonlinear fractional boundary value problems. Front Phys. 2020;8:220.
[17] SajjadiSS, BaleanuD, JajarmiA, PirouzHM. A new adaptive synchronization and hyperchaos control of a biological snap oscillator. Chaos Solitons Fractals. 2020;109919:138. · Zbl 1490.92005
[18] BaleanuD, JajarmiA, SajjadiSS, JihadHA. The fractional features of a harmonic oscillator with position‐dependent mass. Commun Theor Physics. 2020;72(5):55002.
[19] JajarmiA, BaleanuD. On the fractional optimal control problems with a general derivative operator. Asian J Control. 2021;23:1062‐1071. https://doi.org/10.1002/asjc.2282 · doi:10.1002/asjc.2282
[20] AfshariH, BaleanuD. Applications of some fixed point theorems for fractional differential equations with Mittag‐Leffler kernel. Adv Differ Equ. 2020;2020:140. https://doi.org/10.1186/s13662-020-02592-2 · Zbl 1483.54023 · doi:10.1186/s13662-020-02592-2
[21] HilalK, KajouniA. Boundary value problems for hybrid differential equations with fractional order. Adv Differ Equ. 2015;2015:183. https://doi.org/10.1186/s13662-015-0530-7 · Zbl 1422.34035 · doi:10.1186/s13662-015-0530-7
[22] JaradF, AbdeljawadT, HammouchZ. On a class of ordinary differential equations in the frame of Atagana-Baleanu derivative. Chaos Solitons Fractals. 2018;117:16‐20. · Zbl 1442.34016
[23] KhanH, AbdeljawadT, AslamM, KhanRA, KhanA. Existence of positive solution and Hyers-Ulam stability for a nonlinear singular‐delay‐fractional differential equation. Adv Differ Equ. 2019;2019(1):1‐13. · Zbl 1459.34024
[24] KhanA, KhanH, Gomez‐AguilarJF, AbdeljawadT. Existence and Hyers‐Ulam stability for a nonlinear singular fractional differential equations with Mittag‐Leffler kernel. Chaos, Solitons Fractals. 2019;127:422‐427. · Zbl 1448.34046
[25] RusIA. Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J Math. 2010;26:103‐107. · Zbl 1224.34164
[26] SagarTS, KuccheKD. On nonlinear hybrid fractional differential equations with Atangana‐Baleanu‐Caputo derivative. https://arxiv.org/abs/2007.11034
[27] WangJ, LvL, ZhouY. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron J Qual Theory Diff Equat. 2011;2011(63):1‐10. · Zbl 1340.34034
[28] ZhaoY, SunS, HanZ, LiQ. Theory of fractional hybrid differential equations. Comput Math Appl. 2011;62:1312‐1324. · Zbl 1228.45017
[29] GorenfloR, MainardiF. Fractional calculus: integral and differential equations of fractional order. Wien: Springer Verlag; 1997;223‐276. · Zbl 1438.26010
[30] BaleanuD, FernandezA. On some new properties of fractional derivatives with Mittag‐Leffler kernel. Commun Nonlinear Sci Numer Simul. 2018;59:444‐462. · Zbl 1510.34004
[31] AbdeljawadT. A Lyapunov type inequality for fractional operators with nonsingular Mittag‐Leffler kernel. J Inequalities Appl. 2017;2017:130. https://doi.org/10.1186/s13660-017-1400-5 · Zbl 1368.26003 · doi:10.1186/s13660-017-1400-5
[32] FernandezA. A complex analysis approach to Atangana-Baleanu fractional calculus. Math Methods Appl Sci. 2019;1‐18. https://doi.org/10.1002/mma.5754 · Zbl 1476.33013 · doi:10.1002/mma.5754
[33] DhageBC. On a fixed point theorem in Banach algebras with applications. Appl Math Lett. 2005;18:273‐280. · Zbl 1092.47045
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