Noninstantaneous impulsive and nonlocal Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps

Hamdy M. Ahmed 1 , Mahmoud M. El-Borai 2 , and Mohamed E. Ramadan 3 , 4
  • 1 Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt
  • 2 Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt
  • 3 Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt
  • 4 Department of Mathematics, Faculty of Science, Islamic University in Madinah, Medina, Saudi Arabia
Hamdy M. Ahmed
  • Corresponding author
  • Higher Institute of Engineering, El-Shorouk Academy, El-Shorouk City, Cairo, Egypt
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, Mahmoud M. El-Borai and Mohamed E. Ramadan
  • Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt
  • Department of Mathematics, Faculty of Science, Islamic University in Madinah, Medina, Saudi Arabia
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Abstract

In this paper, we introduce the mild solution for a new class of noninstantaneous and nonlocal impulsive Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps. The existence of the mild solution is derived for the considered system by using fractional calculus, stochastic analysis and Sadovskii’s fixed point theorem. Finally, an example is also given to show the applicability of our obtained theory.

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