Abstract
This paper focuses on the existence and uniqueness of solutions for certain types of fractional differential equations involving Riemann–Liouville derivative. The main results are obtained by some fixed point theorems in Sobolev space.
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Ben, M.A., Nagy, A.M.: Finite-time stability of linear Caputo–Katugampola fractional-order time delay systems. Asian J. Control 22, 297–306 (2020)
Ben Makhlouf, A., Boucenna, D., Hammami, M.A.: Existence and stability results for generalized fractional differential equations. Acta Math. Sci. 40, 141–154 (2020)
Boucenna, D., Ben Makhlouf, A., Naifar, O., Guezane-Lakoud, A., Hammami, M.A.: Linearized stability analysis of Caputo–Katugampola fractional-order nonlinear systems. J. Nonlinear Funct. Anal. 2018, 1–11 (2018)
Boucenna. D., Guezanne-Lakoud, A., Nieto, J.J., Khaldi, R.: On a multipoint fractional boundary value problem with integral conditions. Nonlinear Funct. Anal. (2017)
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)
Guezane-Lakoud, A., Khaldi, R., Boucenna, D., Nieto, J.J.: On a Multipoint fractional boundary value problem in a fractional Sobolev space. Differ. Equ. Dyn. Syst. (2018). https://doi.org/10.1007/s12591-018-0431-9
Idczak, D., Walczak, S.: Fractional Sobolev spaces via Riemann–Liouville derivatives. J. Funct. Spaces Appl. 15, 128043 (2013)
Jmal, A., BenMakhlouf, A., Nagy, A.M., Naifar, O.: Finite-time stability for Caputo–Katugampola fractional-order time-delayed neural networks. Neural Process. Lett. 1, 1 (2019). https://doi.org/10.1007/s11063-019-10060-6
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Math. Stud., 204, Elsevier, Amsterdam (2006)
Magin, R.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 15–377 (2004)
Naifar, O., Ben Makhlouf, A., Hammami, M.A.: Comments on Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simul. 30, 360–361 (2016)
Naifar, O., Ben Makhlouf, A., Hammami, M.A., Chen, L.: Global practical Mittag Leffer stabilization by output feedback for a class of nonlinear fractional order systems. Asian J. Control 20, 599–607 (2017)
Naifar, O., Nagy, A.M., Ben, M.A., Kharrat, M., Hammami, M.A.: Finite time stability of linear fractional order time delay systems. Int. J. Robust Nonlinear Control 29, 180–187 (2019)
Podlubny, I.: Fractional differential equations, Mathematics in Science and Engineering, vol. 198. Academic Press, New York/Londin/Toronto (1999)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (193)
Soczkiewicz, E.: Application of fractional calculus in the theory of viscoelasticity. Mol. Quant. Acoust. 23, 397–404 (2002)
Souahi, A., Ben, M.A., Hammami, M.A.: Stability analysis of conformable fractional-order nonlinear systems. Indagationes Mathematicae 28, 1260–1274 (2017)
Wu, Q.: A new type of the Gronwall–Bellman inequality and its application to fractional stochastic differential equations. Cogent Math. Stat. 4(1), 1279781 (2017)
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Boucenna, D., Boulfoul, A., Chidouh, A. et al. Some results for initial value problem of nonlinear fractional equation in Sobolev space. J. Appl. Math. Comput. 67, 605–621 (2021). https://doi.org/10.1007/s12190-021-01500-5
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DOI: https://doi.org/10.1007/s12190-021-01500-5
Keywords
- Fractional differential equations
- Initial value problem
- Weighted Sobovlev space
- Mittag-Leffler function
- Fixed point theorem