# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 3 Existence Results
For any \(x\in B_{r},\ t^{\prime},t^{\prime\prime}\in(t_{i},t_{i+1}]\)\((i=0,1,2,\cdots,m)\), we obtain  
\[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]6 \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]7 \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]8  
after some calculation, we have  
\[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]9 \[||I_{k}(x)||\leq\Phi_{k}(t)||x||,\quad x\in X.\]0 \[||I_{k}(x)||\leq\Phi_{k}(t)||x||,\quad x\in X.\]1 \[||I_{k}(x)||\leq\Phi_{k}(t)||x||,\quad x\in X.\]2  
Using \(\mathcal{T}(t)\) and \(\mathcal{S}(t)\) is uniformly continuous and the well known inequality \(|t^{\prime\sigma}-t^{\prime\prime\sigma}|\leq(t^{\prime\prime}-t^{\prime})^{\sigma}\) for \(\sigma\in(0,1]\) and \(0<t^{\prime}\leq t^{\prime\prime}\)  
\[||I_{k}(x)||\leq\Phi_{k}(t)||x||,\quad x\in X.\]3  
Thus \(N(B_{r})\) is equi-continuous with \(B_{r}\) on \((t_{i},t_{i+1}]\)\((i=0,1,2,\cdots,m)\)  
**Step 4:**\(N\) maps \(B_{r}\) into a precompact set in \(X\). Define \(\Pi=NB_{r}\) and \(\Pi(t)=\left\{(Nx)(t)\ :\ x\in B_{r}\right\}\) for \(t\in J\). Set \(\Pi_{h,\delta}(t)=\left\{(N_{h,\delta}x)(t)\ :\ x\in B_{r}\right\}\) ; where  
\[||I_{k}(x)||\leq\Phi_{k}(t)||x||,\quad x\in X.\]4  
From Lemma 3 (2), (3) and (H10), we can verify that the set \(\Pi(t)\) can be arbitrary approximated by the relatively compact set \(\Pi_{h,\delta}(t)\). Thus, \(N(B_{r})(t)\) is relatively compact in \(X\). **Step 5:** The set \(E=\{x\in PC[J,X]:x=\lambda Nx\text{ }for\text{ }0<\lambda<1\}\) is bounded. Let \(x\in E\), then  
\[x(t)=\left\{ \begin{array}{l}\lambda\mathcal{T}(t)(x_{0}+g(x))+ \lambda\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)\\ (\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau)ds,\quad t\in[0,t_{1}],\\ \lambda\mathcal{T}(t)(x_{0}+g(x))+\lambda\mathcal{T}(t-t_{1})I_{1}(x(t_{1}^{- }))+\lambda\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)\\ (\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau)ds,\quad t\in(t_{1},t_{2}],\\ \vdots\\ \lambda\mathcal{T}(t)(x_{0}+g(x))+\lambda\sum_{k=1}^{m}\mathcal{T}(t-t_{k})I _{k}(x(t_{k}^{-}))+\lambda\int_{0}^{t}(t-s)^{q-1}\mathcal{S}(t-s)\\ (\times)f(s,x_{s},\int_{0}^{s}h(s,\tau,x_{\tau})d\tau,\int_{0}^{b}k(s,\tau,x_{ \tau})d\tau)ds,\quad t\in(t_{m},b].\end{array} \right. \tag{29}\]  
From (30) we know  
\[||x(t)|| \leq \lambda M_{1}\left(||x_{0}||+k_{1}\right)+\frac{b^{q}M_{1}}{ \Gamma(1+q)}\int_{0}^{b}\Psi_{1}(s)ds \tag{30}\] \[+\lambda\left(G+mM_{1}\Phi_{k}+\frac{b^{q}M_{1}}{\Gamma(1+q)} \int_{0}^{t}\Psi_{2}(s)ds\right)\ ||x(t)||.\]  
Obviously there exists \(\lambda\) sufficiently small such that \(\rho=1-M_{1}k_{1}\lambda-\lambda G-\lambda mM_{1}\Phi_{k}>0\) and then we get  
\[||x(t)|| \leq \frac{\lambda M_{1}}{\rho}||x_{0}||+\frac{\lambda b^{q}M_{1}}{ \rho\Gamma(1+q)}\int_{0}^{b}\Psi_{1}(s)ds \tag{32}\] \[+\frac{\lambda b^{q}M_{1}}{\rho\Gamma(1+q)}\int_{0}^{t}\Psi_{2}( s)||x(s)||ds.\]  
Let  
\[Q=\lambda M_{1}\rho||x_{0}||+\frac{\lambda b^{q}M_{1}}{\rho \Gamma(1+q)}\int_{0}^{b}\Psi_{1}(s)ds,\ f(t)=\frac{\lambda b^{q}M_{1}}{\rho \Gamma(1+q)}\int_{0}^{t}\Psi_{2}(s)ds. \tag{33}\]  
It is clear that \(f(t)\) is non negative continuous function on \([0,+\infty)\), generalized Bellman inequality implies that  
\[||x(t)||\leq Qe^{\int_{0}^{t}f(s)ds}\leq Qe^{\int_{0}^{b}f(s)ds}= C_{0}; \tag{34}\]  
where \(C_{0}\) is a constant. Obviously, the set \(E\) is bounded on \((t_{i},t_{i+1}],\ (i=0,1,2,\cdots,m)\). Since \(N\) is continuous and compact. From the [PERSON]'s fixed point theorem, \(N\) has a fixed point which is a \(PC\)-mild solution of \(\{\)(1),(2),(3)\(\}\). This completes the proof. **4. Example**  
1.