Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup

• We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems.
• Because classical solutions of the impulsive fractional differential equations do not satisfy the definition of a mild solution given by the author of [10] and the semi-group property T(t+s)=T(t)T(s)T(t+s)=T(t)T(s) for the system is not used correctly.
• Up until now the existence of mild solution for an impulsive fractional functional integro differential equations with non-compact semi-group in Banach spaces has not been considered in the literature.

In this paper, we study local and global existence of mild solution for an impulsive fractional functional integro differential equation with non-compact semi-group in Banach spaces. We establish a general framework to find the mild solutions for impulsive fractional integro-differential equations, which will provide an effective way to deal with such problems. The theorems proved in this paper improve and extend some related conclusions on this topic. Finally, two applications are given to illustrate that our results are valuable.