# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 3 Existence Results  
In this section, we give the existence of mild solutions of the system {(1),(2),(3)}. To establish our results, we introduce the following hypotheses:  
1. \(f:J\times X\times X\times X\to X\) is continuous, and there exist functions \(\mu_{1},\mu_{2},\mu_{3}\in L[J,R^{+}]\) such that \[||f(t,x_{1},x_{2},x_{3})-f(t,y_{1},y_{2},y_{3})||\leq\mu_{1}(t)||x_{1}-y_{1 }||+\mu_{2}(t)||x_{2}-y_{2}||+\mu_{3}(t)||x_{3}-y_{3}||,\] \[x_{i},y_{i}\in X,\ i=1,2,3.\]
2. \(h,k:J\times J\times X\to X\) is continuous and there exist \(M_{h},\ M_{k}>0\) such that \[||h(t,s,x_{1})-h(t,s,y_{1})|| \leq M_{h}||x_{1}-y_{1}||,\] \[||k(t,s,x_{1})-k(t,s,y_{1})|| \leq M_{k}||x_{1}-y_{1}||,\ \ x_{1},\ y_{1}\in X.\]
3. \(g:PC([0,b],X)\) is continuous and there exists a constant \(G>0\) such that \[||g(x)-g(y)|| \leq G||x-y||,\ \ \forall x,y\in PC([0,b],X)\] \[||g(0)|| \leq k_{1}.\]
4. The function \(I_{k}:X\to X\) are continuous and there exists \(\rho_{k}>0\) such that \[||I_{k}(x)-I_{k}(y)||\leq\rho_{k}||x-y||,\ \ \ x,y\in X,\ \ k=1,2,\cdots,m.\]* The function \(\Omega_{m}(t)\,:J\to\mathbb{R}^{+}\) is defined by \[\Omega_{m}(t)=M_{1}(G+m\rho_{m})+\frac{M_{1}b^{q}}{\Gamma(1+q)}\Big{(}\mu_{1}(t) +\mu_{2}(t)M_{h}b+\mu_{3}(t)M_{k}b\Big{)},\] where \(0<\Omega_{m}(t)<1,\ t\in J\). * The constants \(\Omega_{u}\) and \(\Omega_{m}^{\prime}(t):J\to\mathbb{R}^{+}\) are defined by \[\Omega_{u} = M_{1}K(G+m\rho_{m})+\frac{M_{1}b^{q}K}{\Gamma(1+q)}\Big{(}\mu_{ 1}(t)+\mu_{2}(t)M_{h}b+\mu_{3}(t)M_{k}b\Big{)}\] \[\Omega_{m}^{\prime}(t) = M_{1}(G+m\rho_{m})+\frac{M_{1}b^{q}}{\Gamma(1+q)}\Big{(}\mu_{1}( t)+\mu_{2}(t)M_{h}b+\mu_{3}(t)M_{k}b\Big{)}+\frac{M_{1}b^{q}\Omega_{u}}{ \Gamma(1+q)}\] and \(0<\Omega_{m}^{\prime}(t)<1,\ t\in J\). **Theorem 1**. : _If the hypotheses (H1)-(H5) are satisfied, then the nonlocal fractional impulsive integro-differential Equations (1),(2),(3)) has a unique mild solution \(x\in PC[J,X]\)._  
Proof. : Define an operator \(N\) on \(PC[J,X]\) by  
\[(Nx)(t)=\left\{ \begin{array}{l}\mathcal{T}(t)(x_{0}+g(x))+\int_{0}^{t}(t-s) ^{q-1}\mathcal{S}(t-s)\\ f\Big{(}s,x_{s_{s}}\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau\Big{)}ds,\quad t\in[0,t_{1}],\\ \mathcal{T}(t)(x_{0}+g(x))+\mathcal{T}(t-t_{1})I_{1}(x(t_{1}^{-}))+\int_{0}^{t }(t-s)^{q-1}\mathcal{S}(t-s)\\ f\Big{(}s,x_{s_{s}}\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau\Big{)}ds,,\quad t\in(t_{1},t_{2}],\\ \vdots\\ \mathcal{T}(t)(x_{0}+g(x))+\sum_{k=1}^{m}\mathcal{T}(t-t_{k})I_{k}(x(t_{k}^{- }))+\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)\\ f\Big{(}s,x_{s_{s}}\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_ {\tau})d\tau\Big{)}ds,\quad t\in(t_{m},b].\end{array} \right. \tag{17}\]  
We shall show that \(N\) is well defined on \(PC[J,X]\). For \(0\leq\tau<t\leq t_{1}\), applying (17), we obtain  
\[||(Nx)(t)-(Nx)(\tau)|| \leq ||T(t)-T(\tau)||\ ||x_{0}+g(x)||\] \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\] \[\leq ||T(t)-T(\tau)||\ [||x_{0}||+G||x||+k_{1}]\] \[+\left\|\int_{\tau}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds\right\|\] \[+\left\|\int_{0}^{\tau}(t-s)^{q-1}\left[\mathcal{S}(t-s)-\mathcal{ S}(\tau-s)\right]\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(\times )f\Big{(}s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau\Big{)}ds\right\|\] \[+\left\|\int_{0}^{\tau}\Big{[}(t-s)^{q-1}-(\tau-s)^{q-1}\Big{]} \mathcal{S}(\tau-s)\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(\times )f\Big{(}s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau\Big{)}ds\right\|.\]  
We know that the inequality \(|t^{\sigma}-\tau^{\sigma}|\leq(t-\tau)^{\sigma}\) for \(\sigma\in(0,1]\) and \(0<\tau\leq t\) and Lemma 3, it is obviously that \(||(Nx)(t)-(Nx)(\tau)||\to 0\) as \(t\to\tau\). Thus, \(Nx\in[(0,t_{1}],X]\).