On the hybrid fractional semilinear evolution equations

Document Type : Research Paper


Laboratory LMACS, Sultan Moulay Slimane University, BP 523, Beni Mellal, 23000, Morocco



In this manuscript, we study the existence of mild solutions to initial value problems for hybrid fractional semi-linear evolution equations. On the other hand, we prove four different types of Ulam-Hyers stability results for mild solutions. The existence of mild solutions is proved by the Dhage fixed point theorem. Finally, an example is given to illustrate our results.


[1] P. Chen, Y. Li, Q. Chen, and R. Feng, On the initial value problem of fractional evolution equations with noncompact semigroup, Comput. Math. Appl. 67 (2014), no. 5, 1108–1115.
[2] B.C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (2005), no. 3, 273–280.
[3] M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal. 2004 (2004), no. 3, 197–211.
[4] M. M. El-Borai, On some fractional evolution equations with nonlocal conditions, Int. J. Pure Appl. Math. 24 (2005), no. 3, 405–413.
[5] K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.
[6] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Amsterdam, Elsevier Science B V, 2006.
[7] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, Waves Stab. Contin. Media, World Scientific, Singapore, 1994, pp. 246–251.
[8] F. Mainardi, P. Paraddisi, and R. Gorenflo, Probability distributions generated by fractional diffusion equations, J. Kertesz, I. Kondor (eds.) Econophysics: An Emerging Science. Kluwer Academic, Dordrecht, 2000.
[9] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
[10] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam, 1998.
[11] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[12] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Dordrecht, Springer, 2007.
[13] C.C. Travis and G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hung. 32 (1978), no. 1-2, 75–96.
[14] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), 1063–1077.
[15] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal.: Real World Appl. 11 (2010), no. 5, 4465–4475.

Articles in Press, Corrected Proof
Available Online from 04 February 2024
  • Receive Date: 05 August 2022
  • Revise Date: 21 June 2023
  • Accept Date: 24 June 2023