On abstract Cauchy problems in the frame of a generalized Caputo type derivative

Year 2023, Volume: 7 Issue: 1, 1 - 28, 31.03.2023

Soumıa Bourchi , Fahd Jarad , Yassine Adjabı , Thabet Abdeljawad , Ibrahim Mahariq

https://doi.org/10.31197/atnaa.1147950

Abstract

In this paper, we consider a class of abstract Cauchy problems in the framework of a generalized Caputo type fractional. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results

Keywords

Caputo type generalized fractional operators, abstract Cauchy problem, existence, uniqueness, continuous dependence on parameters, stability, uniformly continuous semigroups, fixed point theorems, Mittag-Leffler type function.

References

• [1] S.G. Samko, A.A. Kilbas,O.I. Marichev: Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, (1993).
• [2] V. Kiryakova: Generalized Fractional Calculus and Applications, Longman & Wiley, Harlow, New York. (1994).
• [3] R. Gorenflo, F. Mainardi: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, 223-276. Springer, New York. (1996).
• [4] N. Heymans, I. Podlubny: Physical interpretation of initial conditions for fractional differential equations with Riemann- Liouville fractional derivatives. Rheologica Acta, 45(2006), 765-772.
• [5] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo: Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam. (2006).
• [6] R. Hilfer: Applications of Fractional Calculus in Physics, World Scientific, Singapore. (2000).
• [7] M. Caputo, M. Fabrizio: A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. and Appl. 1(2)(2015),1-13.
• [8] A. Atangana , D. Baleanu: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci. Vol. 20, No. 2 (2016), 763-769.
• [9] U. N. Katugampola: New Approach to a generalized fractional integral. Appl. Math. Comput. 218(3),860-865. (2011).
• [10] U. N. Katugampola: A new approach to generalized fractional derivatives. Bull. Math. Anal. and App., Vol. 6 Issue 4 (2014), 1-15.
• [11] F.Jarad, T. Abdeljawad, D. Baleanu: On the generalized fractional derivatives and their Caputo modi?cation. Nonlinear Sci. Appl., 10, 2607-2619. (2017).
• [12] Y. Adjabi, F. Jarad, D. Baleanu, T. Abdeljawad: On Cauchy problems with Caputo-Hadamard fractional derivatives, J. Comput. Anal. Appl., 21(2016), 661-681.
• [13] F. Jarad, D. Baleanu, A. Abdeljawad: Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ. 142(2012). 2012.
• [14] D.R.Anderson, D.J.Ulness: Properties of the Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys., 56, Article No. 063502,18 pages, (2015).
• [15] R. Almeida: What is the best fractional derivative to fit data? Applicable Analysis and Discrete Mathematics (2017) Volume 11, Issue 2, 358-368.
• [16] A. Ebaid, B. Masaedeh, E. El-Zahar: A new fractional model for the falling body problem. Chin. Phys. Lett. 2017,34,020201.
• [17] E. Bas,.R. Ozarslan: Real world applications of fractional models by Atangana?Baleanu fractional derivative. Chaos Solitons Fractals (2018),116,121-125.
• [18] E. Bas, B. Acay, R. Ozarslan: Fractional models with singular and non-singular kernels for energy efficient buildings. Chaos (2019),29,023110.
• [19] E. Bas, R.Ozarslan, D. Baleanu, A. Ercan: Comparative simulations for solutions of fractional Sturm-Liouville problems with non-singular operators. Adv. Differ. Equ. (2018), 2018,350.
• [20] E. Bas: The Inverse Nodal problem for the fractional diffusion equation. Acta Sci. Technol.(2015),37, 251-257.
• [21] K.B. Oldham and J. Spanier, The fractional calculus, Mathematics in Science and Engineering, vol. 111, Academic Press, New York, (1974).
• [22] R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27(1983),201-210.
• [23] M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento, 1(1971),161-198.
• [24] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Syst. Signal Process., 5(1991),81-88. [25] I. Podlubny: Fractional?order systems and fractional-order controllers, Tech. Report UEF-03-94, Institute for Experimental Physics, Slovak Academy of Sciences, (1994).
• [26] T. Poinot, J.C. Trigeassou, Identification of fractional systems using an output-error technique, Nonl. Dynamics, 38(2004), 133-154.
• [27] V. F.Morales-Delgado, J.FGómez-Aguilar, M.A.Taneco-Hernandez: Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense. AEU Int. J. Electron. Commun. (2018),85,108-117.
• [28] V. Lakshmikanthan, S. Leela: Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, (2009). [29] G. Jumarie: Laplace's transform of fractional order via the Mittag-Leffler function and modiffied Riemann-Liouville deriva- tive. Appl Math Lett.2009;22(11) : 1659-1664.
• [30] A. Bultheel, S.H. Martinez: Recent developments in the theory of the fractional Fourier and linear canonical transforms. B Belg Math Soc-Sim. 2006;13(5) : 971-1005.
• [31] K. Diethelm: The Analysis of Fractional Differential Equations: an application-oriented exposition using di?erential oper- ators of Caputo type. Berlin, Heidelberg: Springer. (2010).
• [32] F. Jarad, T. Abdeljawad: Generalized fractional derivatives and Laplace transform. Discrete and Continuous Dynamical Systems-S. 2019 : 1775-1786. [33] F. Jarad, T. Abdeljawad: A modified Laplace transform for certain generalized fractional operators. Results Nonlinear Anal.2018;2018(2) : 8898.
• [34] F.S. Silva, D.M Moreira, M.A. Moret: Conformable Laplace Transform of Fractional Differential Equations. Axioms. 2018;7(3),.55.
• [35] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Inc. (1983).
• [36] J. M. Ball: Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Amer. Math. Soc.63(1977), 370-373.
• [37] M.M. El-Borai: Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons and Fractals 14(2002),433-440.
• [38] E. Hernández, D. O'Regan, K. Balachandran: On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis, 73, 3462-3471. (2010).
• [39] l.E. Hille: A Note on Cauchy's Problem, Ann. Soc. polonaise math., 25,56-68(1952); 'Une généralisation du problème de Cauchy'. Ann. Inst. Fourier, 4, 31-48. (1953).
• [40] M. Li, Q. Zh: On spectral inclusions and approximation of α-times resolvent families. Semigroup Forum 69,356-368. (2004).
• [41] C. Chen, M. Li: On fractional resolvent operator functions. Semigroup Forum 80, 121-142. (2010).
• [42] L. Kexue, P. Jigen: Fractional Abstract Cauchy Problems , Integr. Equ. Oper. Theory 70(2011),333-361.
• [43] M. Japund, D. Rajter-Ciric: Generalized Uniformly Continuous Solution Operators and Inhomogeous Fractional Evolution Equations xith Variable Coefficients. Electronic Journal of Differential Equations, Vol. 2017(2017), No. 293, 1-24.
• [44] Y. Li: Regularity of mild Solutions for fractional abstract Cauchy problem with order α ∈ (1,2), Z. Angew. Math. Phys. 66(2015), 3283-3298.
• [45] A.V. Glushak, T.A. Manaenkova: Direct and Inverse Problems for an Abstract Di?erential Equation Contain- ing Hadamard Fractional Derivatives, Di?erential Equations, (2011), Vol. 47, No. 9, 1307-1317.
• [46] E.G. Bajlekova: Fractional Evolution Equations in Banach Spaces. Technische Universiteit Eindhoven. (2001).
• [47] M. I. Abbas: On the existence of mild solutions for a class of fractional differential equations with nonlocal conditions in the α-norm, Studia Scientiarum Mathematicarum Hungarica, 51(2),141-154. (2014).
• [48] P. Bedi, A. Kumar, T. Abdeljawad, A. Khan, JF Gómez-Aguilar: Mild solutions of coupled hybrid fractional order system with Caputo-Hadamard derivatives, Fractals. Vol. 29, No. 06,2150158, (2021)
• [49] H.R. Henríquez, J.G. Mesquita, J.C. Pozo: Existence of solutions of the abstract Cauchy problem of fractional order Journal of Functional Analysis 281(4),109028,(2021).
• [50] J.V.C. Sousa, K.D. Kucche, E.C. de Oliveira: Stability of mild solutions of the fractional nonlinear abstract Cauchy problem. Electronic Research Archive 30(1),272-288,(2022).
• [51] J. Bravo, C. Lizama: The Abstract Cauchy Problem with Caputo?Fabrizio Fractional Derivative. Mathematics (2022), 10(19),3540.
• [52] G. Ascione: Abstract Cauchy problems for the generalized fractional calculus. Nonlinear Analysis 209(2021)112339.
• [53] E. Bazhlekova: Fractional Evolution Equations in Banach Spaces, University Press Facilities, Eindhoven University of Technology. (2001).
• [54] R.H. Martin Jr: Nonlinear operators and di?erential equations in Banach spaces, Wiley-Interscience, New York, (1976).
• [55] R. Almeida: A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, Vol. 44, March 2017, 460-481.
• [56] F. Jarad, T. Abdeljawad: Variational principles in the frame of certain generalized fractional derivatives, Discrete & Continuous Dynamical Systems-S, Vol. 13, No.3, March 2020, 695-708.
• [57] R.G. Anatoly, A. Kilbas, F. Mainardi, S.V. Rogosin: , Mittag-Leffler Functions, Related Topics and Applications, book Springer-Verlag Berlin Heidelberg March (2014).
• [58] T.H. Gronwall: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations Annals of Mathematics. 20(2)(1919), 293-296.
• [59] A. Granas, J. Dugundji: Fixed Point Theory, Springer-verlag, New York. (2003).
• [60] S.M. Ulam: A Collection of mathematical problems, Interscience, New York. (1968).
• [61] D.H. Hyers: On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27(1941), 222-224.
• [62] T. M. Rassias: On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematica 62 (1), 23- 130. (2000).
• [63] Y. Adjabi, F. Jarad,T. Abdeljawad: On generalized fractional operators and a Gronwall type inequality with applications, Filomat, 31(2017), 5457-5473.