Ulam's stability of conformable neutral fractional differential equations
Résumé
This article is concerned with the existence and uniqueness of solutions of a nonlinear neutral conformable fractional differential system with infinite delay, involving conformable fractional derivative. Additionally, we study the Ulam--Hyres stability, Ulam--Hyres--Mittag--Leffler stability, Ulam--Hyres--Mittag--Leffler--Rassias stability for the solutions of considered system using Picard operator. For application of the theory, we add an example at the end.
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Références
Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math., 279, 57-66 (2015) https://doi.org/10.1016/j.cam.2014.10.016 DOI: https://doi.org/10.1016/j.cam.2014.10.016
Ahmad, B., Ntouyas, S. K., Alsaedi, A., Alnahdi, M., Existence theory for fractional-order neutral boundary value problems, Frac. Differ. Calc., 8, 111-126 (2018) https://doi.org/10.7153/fdc-2018-08-07 DOI: https://doi.org/10.7153/fdc-2018-08-07
Ahmad, A., Zada, A., Wang, X., Existence, uniqueness and stability of implicit switched coupled fractional differential equations of ψ-Hilfer type, Int. J. Nonlin. Sci. Num., 21 , no. 3-4, 327-337 (2020). https://doi.org/10.1515/ijnsns-2018-0371 DOI: https://doi.org/10.1515/ijnsns-2018-0371
Ahmad, M., Zada, A., Alzabut, J., Stability analysis of a nonlinear coupled implicit switched singular fractional differential system with p−Laplacian, Adv. Differ. Equ., 2019:436, 1-22 (2019), https://doi.org/10.1186/s13662-019-2367-y DOI: https://doi.org/10.1186/s13662-019-2367-y
Ahmad, M., Zada, A., Alzabut, J., Hyers-Ulam stability of a coupled system of fractional differential equations of Hilfer-Hadamard type, Demonstr. Math., 52, 283-295 (2019) https://doi.org/10.1515/dema-2019-0024 DOI: https://doi.org/10.1515/dema-2019-0024
Baleanu, D., Machado, J., Luo, A., Fractional Dynamics and Control, Springer., 2012 https://doi.org/10.1007/978-1-4614-0457-6 DOI: https://doi.org/10.1007/978-1-4614-0457-6
Balachandran, K., Park, J. Y., Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Analysis: Hybrid Systems, 3, 363-367 (2009) https://doi.org/10.1016/j.nahs.2009.01.014 DOI: https://doi.org/10.1016/j.nahs.2009.01.014
Bayour, B., Torres, D. F. M., Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math., 312, 127-133 (2017) https://doi.org/10.1016/j.cam.2016.01.014 DOI: https://doi.org/10.1016/j.cam.2016.01.014
Chen, W. H., Zheng, W. X., Delay-dependent robust stabilization for uncertain neutral systems with distributed delays, Automatica, 43, 95-104 (2007) https://doi.org/10.1016/j.automatica.2006.07.019 DOI: https://doi.org/10.1016/j.automatica.2006.07.019
Diethelm, K., The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics., 2010 https://doi.org/10.1007/978-3-642-14574-2 DOI: https://doi.org/10.1007/978-3-642-14574-2
Dong, X., Bai, Z., Zhang, W., Positive solutions for nonlinear eigenvalue problems with conformable fractional differential derivatives, J. Shandong Univ. Sci. Technol. Nat. Sci., 35, 85-90 (2016)
Eslami, M., Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations, Appl. Math. Comput., 285, 141-148 (2016) https://doi.org/10.1016/j.amc.2016.03.032 DOI: https://doi.org/10.1016/j.amc.2016.03.032
Ekici, M., Mirzazadeh, M., Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives, Optik, 127, 10659-19669 (2016) https://doi.org/10.1016/j.ijleo.2016.08.076 DOI: https://doi.org/10.1016/j.ijleo.2016.08.076
Granas, A., Dugundji, J., Fixed point theory, Springer-Verlag, New York, 2003 https://doi.org/10.1007/978-0-387-21593-8 DOI: https://doi.org/10.1007/978-0-387-21593-8
Khalil, R., Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70 (2014) https://doi.org/10.1016/j.cam.2014.01.002 DOI: https://doi.org/10.1016/j.cam.2014.01.002
Lakshmikantham, V., Leela, S., Devi, J. V., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers., 2009.
Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley., 1993
Podlubny, I., Fractional Differential Equations, Academic Press., 1999
Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y., A new collection of real world applications of fractional calculus in science and engineering, Commun Nonlinear Sci Numer Simulat., 64, 213-231 (2018) https://doi.org/10.1016/j.cnsns.2018.04.019 DOI: https://doi.org/10.1016/j.cnsns.2018.04.019
Riaz, U., Zada, A., Ali, Z., Ahmad, M., Xu, J., Fu, Z., Analysis of Nonlinear Coupled Systems of Impulsive Fractional Differential Equations with Hadamard Derivatives, Math. Probl. Eng., 2019, 1-20 (2019) https://doi.org/10.1155/2019/5093572 DOI: https://doi.org/10.1155/2019/5093572
Rizwan, R., Zada, A., Wang, X., Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses, Adv. Differ. Equ., 2019 85 (2019) https://doi.org/10.1186/s13662-019-1955-1 DOI: https://doi.org/10.1186/s13662-019-1955-1
Wang, J. R., Zada, A., Waheed, H., Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem, Math. Meth. App. Sci., 42, 6706-6732 (2019) https://doi.org/10.1002/mma.5773 DOI: https://doi.org/10.1002/mma.5773
Zada, A., Ali, S., Li, T., Analysis of a new class of impulsive implicit sequential fractional differential equations, Int. J. Nonlin. Sci. Num., (2020). DOI: 10.1515/ijnsns-2019-0030 https://doi.org/10.1515/ijnsns-2019-0030 DOI: https://doi.org/10.1515/ijnsns-2019-0030
Zada, A., Alzabut, J., Waheed, H., Popa, I. L., Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions, Adv. Differ. Equ., 2020, 64 (2020) https://doi.org/10.1186/s13662-020-2534-1 DOI: https://doi.org/10.1186/s13662-020-2534-1
Zhou, Y., Basic Theory of Fractional Differential Equations, World Scientific., 2014 https://doi.org/10.1142/9069 DOI: https://doi.org/10.1142/9069
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