# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 3 Existence Results
Next theorem is based on [PERSON]'s fixed point theorem, let us list the following hypotheses:  
* \(f:J\times X\times X\times X\to X\) is continuous and there exist functions \(c_{1},c_{2},c_{3},c_{4}\in L(J,R^{+})\), such that \[||f(t,x,y,z)||\leq c_{1}(t)+c_{2}(t)||x||+c_{3}(t)||y||+c_{4}(t)||z||,\;t\in J,\;x,y,z\in X.\]
* \(h,k:J\times J\times X\to X\) is continuous and there exist functions \(d_{1},d_{2},d_{3},d_{4}\in C(I,R^{+})\), such that \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\] \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]
* There exist \(\Phi_{k}\in C[J,R^{+}]\), such that \[||I_{k}(x)||\leq\Phi_{k}(t)||x||,\quad x\in X.\]
* For all bounded subsets \(B_{r}\), the set \[\Pi_{h,\phi}(t)=\left\{\mathcal{T}_{\delta}(t)(x_{0}+g(x))+\int_{ 0}^{t-h}(t-s)^{q-1}\mathcal{S}_{\delta}(t-s)F(s)ds\right.\\ +\left.\sum_{k=1}^{m}\mathcal{T}_{\delta}(t-t_{k})I_{k}(x(t_{k}^{- })):\;x\in B_{r}\right\}\] is relatively compact in \(X\) for arbitrary \(h\in(0,t)\) and \(\delta>0\), where \[\mathcal{T}_{\delta}(t)=\int_{\delta}^{\infty}\xi_{\delta}(\theta)\mathcal{T} (t^{q}\theta)d\theta,\;\;\;\mathcal{S}_{\delta}(t)=q\int_{\delta}^{\infty} \theta\xi_{\delta}(\theta)\mathcal{T}(t^{q}\theta)d\theta.\]
* For all bounded subsets \(B_{r}\), the set \[\Pi^{\prime}_{h,\phi}(t)=\left\{\mathcal{T}_{\delta}(t)(x_{0}+g(x))+ \int_{0}^{t-h}(t-s)^{q-1}\mathcal{S}_{\delta}(t-s)F(s)ds\right.\\ +\left.\sum_{k=1}^{m}\mathcal{T}_{\delta}(t-t_{k})I_{k}(x(t_{k}^{- })):\;x\in B_{r}\right\}\] is relatively compact in \(X\) for arbitrary \(h\in(0,t)\) and \(\delta>0\). **Theorem 2**. : _If the hypotheses (H6)-(H10) are satisfied, then the non local fractional impulsive integro-differential Equations (1)-(3) has at least one mild solution \(x\in PC[J,X]\)._  
\[\leq 2c_{1}(s)+2c_{3}(s)\int_{0}^{s}d_{1}(s)ds+2c_{4}(s)\int_{0}^{b}d_{3} (s)ds\] \[+\left(2c_{2}(s)+2c_{3}(s)\int_{0}^{s}d_{2}(s)ds+2c_{4}(s)\int_{0}^{ b}d_{4}(s)ds\right)\ r.\]  
Hence,  
\[(t-s)^{q-1}\left\|f\left(s,x_{n_{s}}\int_{0}^{s}h(s,\tau,x_{n_{ \tau}})d\tau,\int_{0}^{b}k(s,\tau,x_{n_{\tau}})d\tau\right)\right.\] \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]0  
By the Lebesgue dominated convergence theorem, we get  
\[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]1 \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]2  
It is easy to get  
\[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]3  
Thus \(N\) is continuous on \((t_{i},t_{i+1}),\ (i=0,1,2,\cdots,m)\). **Step 2:**\(N\) maps bounded sets into bounded sets in \(PC[J,X]\). From (20) we get  
\[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]4 \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]5 \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]6  
and we know that  
\[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]7 \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]8 \[||h(t,s,x)|| \leq d_{1}(s)+d_{2}(s)||x||\]9 \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]0  
From the above we get,  
\[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]1 \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]2Thus for any \(x\in B_{r}=\{x\in PC[J,X]:||x||_{PC}\leq r\}\),  
\[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]3 \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]4 \[||k(t,s,y)|| \leq d_{3}(s)+d_{4}(s)||y||,\quad x,y\in X.\]5  
Hence \(||(Nx)(t)||\leq\gamma_{1}\), (i.e.,) \(N\) maps bounded sets to bounded sets in \(PC[J,X]\). **Step 3:**\(N(B_{r})\) is equicontinuous with \(B_{r}\) on \((t_{i},t_{i+1}]\)\((i=0,1,2,\cdots,m)\).