# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 2 Preliminaries  
Symbols:  
1. X: Banach Space;
2. \(PC[J,X]\): piece-wise continuous function from \([J,X]\);
3. \(A(t)\): infinitesimal generator of a strongly continuous semigroup \(T(t)\);
4. \(I^{\alpha}f(t)\): fractional integral of order \(\alpha\) for a function \(f\);
5. \({}^{(R-L)}D^{\alpha}\): Riemann-Liouville (R-L) fractional derivative of order \(\alpha>0,\ n-1<\alpha<n,n\in N\);
6. \({}^{\mathbb{C}}D_{t}^{\beta}\): Caputo fractional derivative of order \(q,\ 0<q<1\);
7. \({}^{\mathbb{C}}D_{s}^{\alpha}f(t)\): strong Caputo derivative of order \(\alpha>0\). Let us consider the set of functions \(PC[J,X]=\{x:J\to X|x\in C[(t_{k},t_{k+1}),X]\) and there exists \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\), \(k=0,1,2,\cdots,m\) and \(x(t_{k}^{-})=x(t_{k})\}\), endowed with the norm \(||x||_{PC}=\sup\limits_{t\in J}||x(t)||\). It is easy to know that \((PC[J,X],||.||_{PC})\) is a Banach space. Throughout this paper, let \(A\) be the infinitesimal generator of a \(C_{0}\) semi-group \((T(t))_{t\geq 0}\) of a uniformly bounded operators on \(X\) and let \(L_{B}(X)\) be the Banach space of all linear and bounded operators on \(X\). For a \(C_{0}\) semi-group \((T(t))_{t\geq 0}\), we set \(M_{1}=\sup\limits_{t\in J}||T(t)||_{L_{B}(X)}\). For each positive constant \(r\), set \(B_{r}=\{x\in PC[J,X]:||x||\leq r\}\). Let us recall the following known definitions. For more details, see Ref. [32]. Definition 1: The fractional integral of order \(\alpha\) with the lower limit zero for a function \(f\) is defined as  
\[I^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}\frac{f(s)}{(t-s)^{1-\alpha }}ds,\quad t>0,\quad\alpha>0, \tag{4}\]  
provided the right hand-side is point-wise defined on \([0,\infty)\), where \(\Gamma(\cdot)\) is the gamma function, which is defined by \(\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t}dt\). Definition 2: The Riemann-Liouville (R-L) fractional derivative of order \(\alpha>0,\ n-1<\alpha<n,\ n\in N\), is defined as  
\[{}^{(R-L)}D_{0+}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\left(\frac{d}{dt} \right)^{n}\int_{0}^{t}(t-s)^{n-\alpha-1}f(s)ds, \tag{5}\]_where the function \(f(t)\) has absolutely continuous derivative up to order \((n-1)\)._  
**Definition 3**. : _The (strong or classical) Caputo derivative of order \(\alpha\) for a function \(f\in L^{1}([0,\infty),R)\) given on the interval \([0,\infty)\) is defined by (if it exists)_  
\[{}^{C}D_{s}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}\frac{f^{(n)}( s)}{(t-s)^{\alpha+1-n}}ds,\quad t>0,\quad n-1<\alpha\leq n,\ n\in N. \tag{6}\]  
**Remark 1**. :
1. _R-L and Caputo are just two different operators that are related to each other in a quite simple way. Quite a few details about this are given in the book by [PERSON]_ _  
**Definition 5** ([33]). : _By a PC-mild solution of the Equations (1),(2),(3)) we mean that a function \(x\in PC[J,X]\) which satisfies the following integral equation:_  
\[x(t)=\left\{ \begin{array}{c}\mathcal{T}(t)(x_{0}+g(x))+\int_{0}^{t}(t-s)^{q-1} \mathcal{S}(t-s)\\ \qquad\qquad(\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{ b}k(s,\tau,x_{\tau})d\tau)ds,\quad t\in[0,t_{1}],\\ \mathcal{T}(t)(x_{0}+g(x))+\mathcal{T}(t-t_{1})I_{1}(x(t_{1}^{-}))+\int_{0}^{t }(t-s)^{q-1}\mathcal{S}(t-s)\\ \qquad\qquad(\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{ b}k(s,\tau,x_{\tau})d\tau)ds,\quad t\in(t_{1},t_{2}],\\ \vdots\\ \mathcal{T}(t)(x_{0}+g(x))+\sum_{k=1}^{m}\mathcal{T}(t-t_{k})I_{k}(x(t_{k}^{ -}))+\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)\\ \qquad\qquad(\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{ b}k(s,\tau,x_{\tau})d\tau)ds,\quad t\in(t_{m},b],\end{array} \right. \tag{9}\]  
_where \(\mathcal{T}(. )\) and \(\mathcal{S}(.