Approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay by using Mainardi's function.

Approximate controllability of fractional differential inclusions is studied.We use Bohnenblust-Karlin's fixed point theorem, Mainardi's function.Caputo fractional derivative is employed in term of Riemann-Liouville's derivative.An illustrative example is provided to show the effectiveness of the obtained theory. In this paper, we formulate a new set of sufficient conditions for the approximate controllability of a class of fractional neutral stochastic integro-differential inclusions with infinite delay in Hilbert space. Bohnenblust-Karlin's fixed point theorem, Mainardi's function, fractional calculus and operator semigroups are used to establish the results under the assumption that the corresponding linear system is approximately controllable. In the end, an example is provided to illustrate the applicability of the obtained theoretical results.