Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

Abstract

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

Keywords:

• Stochastic fractional differential equations
• non-instantaneous impulses
• cosine family
• Faedo–Galerkin approximations
• Banach fixed point theorem

2020 AMS Subject Classifications:

• 60H20
• 34A08
• 34G20

Acknowledgments

We are very thankful to the anonymous reviewers and associate editor for providing constructive comments and valuable suggestions to enhance the quality of this manuscript.

Disclosure statement

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

Additional information

Funding

The first author, Shahin Ansari, would like to thank Maulana Azad National Fellowship of India for PhD fellowship [file number 82-27/2019(SA-III), UGC-Ref. No.: 4449/(June 2019)].