Michal FEČKAN, Choukri DERBAZI, Zidane BAITICHE

Some new uniqueness and Ulam stability results for a class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative using the Φ-fractional Bielecki-type norm

In this research article, a novel Φ-fractional Bielecki-type norm introduced by Sousa and Oliveira [23] is used to obtain results on uniqueness and Ulam stability of solutions for a new class of multiterms fractional differential equations in the framework of generalized Caputo fractional derivative. The uniqueness results are obtained by employing Banach’ and Perov’s fixed point theorems. While the Φ-fractional Gronwall type inequality and the concept of the matrices converging to zero are implemented to examine different types of stabilities in the sense of Ulam–Hyers (UH) of the given problems. Finally, two illustrative examples are provided to demonstrate the validity of our theoretical findings.

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[1] Abbas S, Benchohra M, Samet B, Zhou Y. Coupled implicit Caputo fractional q-difference systems. Advances in Difference Equations 2019; 2019: 527.
[2] Almeida R. A Caputo fractional derivative of a function with respect to another function. Communications in Nonlinear Science and Numerical Simulation 2017; 44: 460-481.
[3] Almeida R, Bastos NRO, Monteiro MTT. Modeling some real phenomena by fractional differential equations. Mathematical Methods in the Applied Science 2016; 39 (16): 4846-4855.
[4] Baitiche Z, Derbazi C, Matar, MM. Ulam-stability results for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the ψ–Caputo sense. Applicable Analysis 2021; 2021. doi: 10.1080/00036811.2021.1873300
[5] Derbazi C, Baitiche Z. Coupled systems of ψ-Caputo differential equations with initial conditions in Banach spaces. Mediterranean Journal of Mathematics 2020; 17 (5): 1-13.
[6] Derbazi C, Baitiche Z, Benchohra M, N’Guérékata G. Existence, uniqueness, and Mittag–Leffler–Ulam stability results for Cauchy problem involving ψ-Caputo derivative in Banach and Fréchet spaces. International Journal of Differential Equations 2020; 2020. doi: 10.1155/2020/6383916
[7] Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV. Mittag–Leffler Functions, Related Topics and Applications. Springer, New York 2014.
[8] Hilfer R. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.
[9] Jarad F, Abdeljawad T. Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems. Series S 2020; 13 (3): 709-722.
[10] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Sudies, Vol. 204. Amsterdam, Netherlands: Elsevier Science B.V., 2006.
[11] Liu K, Fečkan M, Wang J. Hyers–Ulam stability and existence of solutions to the generalized Liouville–Caputo fractional differential equations. Symmetry 2020; 2(6): 955. doi: 10.3390/sym12060955.
[12] Peng S, Wang J. Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivatives. Electronic Journal of Qualitative Theory of Differential Equations. 2015; 2015: 52.
[13] Petre IR, Petruşel A. Krasnoselskii’s theorem in generalized Banach spaces and applications. Electronic Journal of Qualitative Theory of Differential Equations 2012; 2012: 85.
[14] Perov AI. On the Cauchy problem for a system of ordinary differential equations. Pviblizhen. Met. Reshen. Differ. Uvavn. 1964; 2: 115-134. (in Russian).
[15] Precup R. The role of matrices that are convergent to zero in the study of semilinear operator systems. Mathematical and Computer Modelling 2009; 49: 703-708.
[16] Precup R, Viorel A. Existence results for systems of nonlinear evolution equations. International Journal of Pure and Applied Mathematics 2008; 47(2): 199-206.
[17] Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering. New York, NY, USA: Academic Press, 1999.
[18] Rus IA. Generalized contractions and applications. Cluj University Press, Cluj-Napoca, 2001.
[19] Rus IA. Ulam stability of ordinary differential equations. Studia. Universitatis Babeş-Bolyai. Mathematica 2009; 54 (4): 125-133.
[20] Vanterler da Costa Sousa J. Existence results and continuity dependence of solutions for fractional equations. Differential Equations & Applications 2020; 12 (4): 377-396.
[21] Vanterler da C. Sousa J, Capelas de Oliveira E. On the ψ-Hilfer fractional derivative. Communications in Nonlinear Science and Numerical Simulation 2018; 60: 72-91.
[22] Vanterler da C. Sousa, J, Capelas de Oliveira E. A Gronwall inequality and the Cauchy-type problem by means of Φ–Hilfer operator. Differential Equations & Applications 2019; 11(1): 87-106.
[23] Vanterler da C. Sousa J, Capelas de Oliveira E. Existence, uniqueness, estimation and continuous dependence of the solutions of a nonlinear integral and an integrodifferential equations of fractional order. 2018; arXiv:1806.01441.
[24] Urs C. Coupled fixed point theorems and applications to periodic boundary value problems. Miskolc Mathematical Notes 2013; 14 (1): 323-333.
[25] Varga RS. Matrix iterative analysis, second revised and expanded edition, Springer Series in Computational Mathematics, 27, Springer-Verlag, Berlin, 2000.
[26] Wang J, Fečkan M, Zhou Y. Presentation of solutions of impulsive fractional Langevin equations and existence results. The European Physical Journal Special Topics 2013; 222 (8): 1857-1874
[27] Wang J, Lv L, Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electronic Journal of Qualitative Theory of Differential Equations 2011; 2011: 63.
[28] Wang J, Li X. Eα -Ulam type stability of fractional order ordinary differential equations. Journal of Applied Mathematics and Computing 2014; 45: 449-459.
[29] Wang J, Zhang Y. Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations. Optimization 2014; 63 (8): 1181-1190.