# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 3 Existence Results
Consider the following fractional partial functional mixed differential equations with impulsive conditions of the form\[D_{t}^{q}(z(t,\eta)) =\frac{\partial}{\partial\eta}z(t,\eta)+\Phi\left(t,z(t,\eta-r), \int_{0}^{t}h_{1}(t,v(x,\eta-r))ds,\int_{0}^{b}k_{1}(t,v(x,\eta-r))ds\right),\] \[\text{for }(t,\eta-r)\in[0,T_{0}]\times(0,\pi),\ t\
eq\frac{T_{0}}{2} \tag{35}\] \[z(t,0) =z(t,\pi)=0,\ 0\leq t\leq T_{0}\] (36) \[z(0,\eta) =z_{0}(\eta)+g(z(t,\eta)),\ 0<\eta<\pi\] (37) \[+\lambda\left(G+mM_{1}\Phi_{k}+\frac{b^{q}M_{1}}{\Gamma(1+q)} \int_{0}^{t}\Psi_{2}(s)ds\right)\ ||x(t)||.\]0  
where \(T_{0}>0,\ 0<q<1,\ D_{t}^{q}\) is a Caputo fractional partial derivative of order \(q\in(0,1)\). To write the system (35)-(38) to the form \(\{\)(1),(2),(3)\(\}\), we take  
(i) Let \(X=L^{2}([0,\pi])\) as the state space and \(z(t,\cdot)=\{z(t,\eta)\ :\ 0\leq\eta\leq\pi\}\) as the state;  
(ii) \(A:D(A)\subset X\to X\) is defined as \(Af=f^{{}^{\prime\prime}}\) with domain  
\[+\lambda\left(G+mM_{1}\Phi_{k}+\frac{b^{q}M_{1}}{\Gamma(1+q)} \int_{0}^{t}\Psi_{2}(s)ds\right)\ ||x(t)||.\]1  
Then \(A\) is the infinitesimal generator of a strongly continuous semi-group \(\{T(t)\ :\ t\geq 0\}\) in \(L^{2}[0,\pi]\). Moreover \(T(\cdot)\) is also strongly continuous such that \(||T(t)||\leq M_{1}\) for each \(t\geq 0\). \(A\) can be written as \(Ax=-\sum_{n=1}^{\infty}n^{2}(x,e_{n})e_{n},\ x\in D(A)\), where \(e_{n}(x)=\sqrt{\frac{2}{\pi}}sin(nx),\ n\in N\) is an orthonormal set of eigen functions of \(A\). Furthermore, for \(x\in X\), we get \(T(t)x=\sum_{n=1}^{\infty}exp\left(-n^{2}t\right)(x,e_{n})e_{n}\). We define the operators \(f:J\times X\times X\times X\to X,\ g:j^{2}\times X\to X\), and \(k:J^{2}\times X\to X\) by  
\[+\lambda\left(G+mM_{1}\Phi_{k}+\frac{b^{q}M_{1}}{\Gamma(1+q)} \int_{0}^{t}\Psi_{2}(s)ds\right)\ ||x(t)||.\]2  
Obviously, \(h_{1},k_{1}\) satisfy Lipschitz condition (H2), \(g\) satisfies Lipschitz condition (H3), and \(I_{1}\) satisfies Lipschitz condition (H4). All together satisfy (H1). Also it is easy to verify conditions (H5) and (H6). Thus functions \(\Phi,\ h_{1},\ k_{1},g\) and \(I_{1}\) of the system (35)-(38) satisfies the hypotheses of the Theorems 1 and 2. Thus all the conditions of Theorems 1 and 2 are satisfied. Therefore the system (35)-(38) can be written to the abstract form \(\{\)(1),(2),(3)\(\}\). That phenomenon models Equations (35)-(38). Hence, we conclude that the system (35)-(38) has a mild solution. 2. Consider the following numerical fractional partial functional differential equations with impulsive conditions of the form  
\[+\lambda\left(G+mM_{1}\Phi_{k}+\frac{b^{q}M_{1}}{\Gamma(1+q)} \int_{0}^{t}\Psi_{2}(s)ds\right)\ ||x(t)||.\]3  
\[+\lambda\left(G+mM_{1}\Phi_{k}+\frac{b^{q}M_{1}}{\Gamma(1+q)} \int_{0}^{t}\Psi_{2}(s)ds\right)\ ||x(t)||.\]4  
Above system can be written as an abstract formulation of system \(\{\)(1),(2),(3)\(\}\). **5. Conclusions**  
Here, we have established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using the Banach contraction principle and a fixed point theorem due to Schaefer with generalized Bellman inequality. We have used the distributed characteristic operators to define the mild solution of the system. We took care of all the controversy related to the solution operator (refer to Remark 3), which has not been discussed in the literature. Distributed characteristics for fractional systems are a powerful tool for modeling complex phenomena. More specifically, modeling of the shape memory phenomena with this powerful tool is studied from different perspectives, as well as presenting some physical interpretation. Using the proposed method, we can evaluate different models and predict the constitutive equation in the viscoelastic system; these models are commonly composed of a different combination of springs and dampers elements.