Non-instantaneous impulsive Hilfer–Katugampola fractional stochastic differential equations with fractional Brownian motion and Poisson jumps

Abstract

The existence of solutions of non-instantaneous impulsive Hilfer–Katugampola fractional differential equations of order $1/2<\alpha <1$ and parameter $0\le \text{β}\le 1$ with fractional Brownian motion (fBm) and Poisson jumps is investigated in this paper. The required results are obtained based on fractional calculus, stochastic analysis, semigroups, and the fixed point theorem. In the end of the paper, an example is provided to illustrate the applicability of the theoretical results.

Keywords:

• Hilfer–Katugampola fractional derivative
• non-instantaneous impulsive stochastic differential equations
• fractional Brownian motion
• Poisson jumps

Mathematics Subject Classifications:

• 34A08
• 26A33
• 34A12
• 60H10

Acknowledgements

The authors would like to thank the referees and the editor for their important comments and suggestions which helped to significantly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

A. M. Sayed Ahmed

A.M. Sayed Ahmed is a lecturer of pure mathematics, Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria University, Egypt. His current research interests include differential equations, integral equations, stochastic differential equations and fractional differential equations.

Hamdy M. Ahmed

Hamdy M. Ahmed is Professor of engineering mathematics, affiliated to the Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt. His recent research interests include fractional stochastic differential equations, controllability of fractional differential equations and exact solution of nonlinear partial differential equations.