Existence results for hybrid differential equation with generalized fractional derivative

Year 2020, Issue: 2, 55 - 60, 12.11.2020

Elsayed Elsayed

https://doi.org/10.47087/mjm.482567

Abstract

This article deals with some existence results for fractional order hybrid differential equations involving Katugampola fractional derivative in Caputo sense. Here the Katugampola fractional derivative is generalization of two familiar fractional derivatives, namely, the Riemann-Liouville and the Hadamard fractional derivatives into a single form. Our investigations are based upon Dhage fixed point theorem, which is used to obtain the sufficient conditions for existence of at least one solution for the proposed problem.

Keywords

Dhage fixed point theorem, Katugampola fractional derivative, Hybrid differential equation;

References

• [1] R. Hilfer, Applications of fractional Calculus in Physics, World scientific, Singapore, 1999.[2] K. Hilal, A. Kajouni, Boundary value problems for hybrid differential equations with fractional order, Advances in Difference Equations (2015), 2015:183.[3] K. Hilal, A. Kajouni, Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions, International Journal of Differential Equations, (2016), 9 pages.[4] M. A. E. Herzallah, D. Baleanu, On Fractional Order Hybrid Differential Equations, Abstract and Applied Analysis, 2014, 7 pages.[5] A. A. Kilbas, H.M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier,2006.[6] U.N. Katugampola, New approach to a genaralized fractional integral, Applied Mathematics and Computation, 218 (3) (2011) 860-865. https://doi.org/10.1016/j.amc.2011.03.062[7] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations Bulletin of Mathematical Analysis and Applications, arXiv:1411.5229, v1 (2014). https://arxiv.org/abs/1411.5229.[8] U.N. Katugampola, New fractional integral unifying six existing fractional integrals, epint arxiv: 1612.08596, 6 pages.[9] D. Vivek, K. Kanagarajan, S. Harikrishnan, Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Application, 2017,(Accepted article-ID 00370).[10] D. Vivek, K. Kanagarajan, S. Harikrishnan, Existence results for implicit differential equations with generalized fractional derivative, Journal of Nonlinear Analysis and Application, 2017,(Accepted article-ID 00371).[11] Y. Zhao, S. Sun, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Computers and Mathematics with Applications, 62, (2011), 1312-1324.