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Mathematics 2019, 7(1), 79; https://doi.org/10.3390/math7010079

Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

1
Higher Institute of Engineering, El-Shorouk Academy, P.O. Box 3, El-Shorouk City, Cairo, Egypt
2
Department of Mathematics, Faculty of Science, Alexandria University, P.O. Box 21515, Alexandria, Egypt
3
Department of Mathematics, Faculty of Science, Al-Azhar University, P.O. Box 11511, Cairo, Egypt
*
Author to whom correspondence should be addressed.
Received: 22 November 2018 / Revised: 8 January 2019 / Accepted: 10 January 2019 / Published: 14 January 2019
(This article belongs to the Section Engineering Mathematics)
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Abstract

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We establish the sufficient conditions for the approximate controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. In addition, we prove the exact null controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. Finally, an example is given to illustrate the obtained results. View Full-Text
Keywords: Sobolev type delay nonlinear impulsive fractional integro-differential equations; resolvent operator; approximate controllability; null controllability Sobolev type delay nonlinear impulsive fractional integro-differential equations; resolvent operator; approximate controllability; null controllability
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Ahmed, H.M.; El-Borai, M.M.; El-Owaidy, H.M.; Ghanem, A.S. Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System. Mathematics 2019, 7, 79.

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