# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 3 Existence Results
For \(t_{1}<\tau<t\leq t_{2}\), we have  
\[||(Nx)(t)-(Nx)(\tau)|| \leq ||\mathcal{T}(t)-\mathcal{T}(\tau)||\ \left[||x_{0}||+G||x||+k\right]\] \[+||\mathcal{T}(t-t_{1})-\mathcal{T}(\tau-t_{1})||\ ||I_{1}(x(t_{1}^{-}))||\] \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]0 \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]1  
\[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]2 \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]3 \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]4 \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]5  
It is easy to get, as \(t\to\tau\), the right hand side of the above inequality tends to zero. Thus, we can deduce that \(Nx\in C[(t_{1},t_{2}],X]\). By repeating the same procedure, we can also obtain that \(Nx\in C[(t_{2},t_{3}],X],\cdots,Nx\in C[(t_{m},b],X]\). That is, \(Nx\in PC[J,X]\). Take \(t\in(0,t_{1}]\), then  
\[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]6 \[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]7  
So we deduce that  
\[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]8  
For each \(t\in(t_{1},t_{2}]\), using hypotheses, and (18), we have  
\[||\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)f\Big{(}s,x_{s},\int_{0} ^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{\tau})d\tau\Big{)}ds||\]9 \[\leq ||T(t)-T(\tau)||\ [||x_{0}||+G||x||+k_{1}]\]0  
In general, for each \(t\in(t_{i},t_{i+1}]\), using (H5)  
\[\leq ||T(t)-T(\tau)||\ [||x_{0}||+G||x||+k_{1}]\]1 \[\leq ||T(t)-T(\tau)||\ [||x_{0}||+G||x||+k_{1}]\]2 \[\leq ||T(t)-T(\tau)||\ [||x_{0}||+G||x||+k_{1}]\]3  
From the assumption (H5) and in the view of the contraction mapping principle, \(N\) has a unique fixed point \(x\in PC[J,X]\), that is \[x(t)=\left\{ \begin{array}{c}\mathcal{T}(t)(x_{0}+g(x))+\int_{0}^{t}(t-s)^{q-1} \mathcal{S}(t-s)\\ (\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau)ds,\quad t\in[0,t_{1}],\\ \mathcal{T}(t)(x_{0}+g(x))+\mathcal{T}(t-t_{1})I_{1}(x(t_{1}^{-}))+\int_{0}^{t }(t-s)^{q-1}\mathcal{S}(t-s)\\ (\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau)ds,,\quad t\in(t_{1},t_{2}],\\ \vdots\\ \mathcal{T}(t)(x_{0}+g(x))+\sum_{k=1}^{m}\mathcal{T}(t-t_{k})I_{k}(x(t_{k}^{- }))+\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)\\ (\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau)ds,\quad t\in(t_{m},b],\end{array} \right. \tag{19}\]  
is a \(PC\)-mild solution of Equations (1)-(3).