# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 2 Preliminaries
)\) are called characteristic solution operators and given by_  
\[\mathcal{T}(t)=\int_{0}^{\infty}\xi_{q}(\theta)T(t^{q}\theta)d\theta,\ \ \ \mathcal{S}(t)=q\int_{0}^{\infty}\theta\xi_{q}(\theta)T(t^{q}\theta)d\theta, \tag{10}\]  
_and for \(\theta\in(0,\infty)\),_  
\[\xi_{q}(\theta)=\frac{1}{q}\theta^{-1-\frac{1}{q}}\varpi_{q}(\theta^{-\frac{ 1}{q}})\geq 0,\ \ \varpi_{q}(\theta)=\frac{1}{\pi}\sum_{n=1}^{\infty}(-1)^{n-1}\theta^{-qn-1} \frac{\Gamma(nq+1)}{n!}sin(n\pi q), \tag{11}\]  
_where \(\xi_{q}\) is a probability density function defined on \((0,\infty)\), that is_  
\[\xi_{q}(\theta)\geq 0,\ \ \theta\in(0,\infty)\ \ \text{and}\ \ \int_{0}^{\infty}\xi_{q}(\theta)d\theta=1. \tag{12}\]  
**Remark 3**. : _Controversy on the solution operator, Definition 5, based on Definitions 3 and 4:_  
1. _In this paper, we emphasize that we use the generalized Caputo derivative with the lower bound at zero for Equation (_1_). However, we have not chosen the classical Caputo derivative and have not changed it in each sub-interval for the Equation (_1_), where the impulses start at the lower bound_ \(t_{k}\)_. Obviously, we mean keeping a different one, in each of the impulses the lower bound is at zero. Moreover, Definition_ 5 _is more reasonable since the generalized Caputo derivative in Equation (_1_) should be fixed at the lower bound at zero once we set the initial time at zero. So we do not expect to change the lower bound again and again in the definition of Caputo derivative for the same equation._
2. _We use Definition_ 4 _(generalized Caputo derivative), where the integrable function_ \(f\) _can be discontinuous. Definition_ 4 _is more general with respect to Remark_ 2 _(_1_) _(relationship between strong and weak Caputo derivatives). So the result would be wrong if we have used a strong Caputo derivative._
3. _Finally, we would like to mention the recently published paper written by [PERSON] and [PERSON]_ _[_34_]__, where the formula of solutions for semi-linear impulsive fractional Cauchy problems (see (_20_) in Ref._ _[_34_]__) coincided with ours (see Definition_ 5_), if one imposes that the semi-linear term and the impulsive term have the same expression in the given interval._  
**Remark 4**. : _Problems associated with impulsive effects and hereditary properties are modeled by impulsive delay differential equations. So we use impulsive finite delay system (1),(2),(3)) and the solution is a piecewise continuous with discontinuities at impulses time. So, here the mild solution is called PC-mild solution. (As we know a function \(x\) is continuous is said to be a mild solution. A function \(x\) which is differential almost everywhere on \([0,T]\) is called a strong solution. Clearly, every strong solution is a mild solution, since differentiability implies continuity)._  
**Definition 6** ([35]). : _Let \(X\) be a Banach space, a one parameter family \(T(t),\ 0\leq t<+\infty\), of bounded linear operators from \(X\) to \(X\) is a semi-group of bounded linear operators on \(X\) if_  
1. \(T(0)=I\)_, (here 1 is the identity operator on_ \(X\)_);_
2. \(T(t+s)=T(t)T(s)\) _for every_ \(t,s\geq 0\)_, (the semi-group property)._  
_A semi-group of bounded linear operator, \(T(t)\), is uniformly continuous if \(\lim\limits_{t\downarrow 0}||T(t)-I||=0\)._  
**Lemma 1** ([35]). : _Linear operator \(A\) is the infinitesimal generator of a uniformly continuous semi-group if and only if \(A\) is the bounded linear operator._  
**Lemma 2** ([36]). : _([PERSON]'s fixed point theorem) Let \(X\) be a Banach space and \(F:X\to X\) be a completely continuous operator. If the set_  
\[E(F)=\left\{x\in X\ :\ x=\lambda Fx\text{ for some }0\leq\lambda\leq 1\right\}. \tag{13}\]  
_is bounded, then \(F\) has at least a fixed point._  
**Lemma 3** ([33]). : _The operator \(\mathcal{T}(t)\) and \(\mathcal{S}(t)\) have the following properties:_  
1.