# Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces
## 1 Introduction
The advantage of using non local conditions is that they are measurable at more places and these can be incorporated to get better models. The non local Cauchy problem for the abstract evolution differential equation was first studied by [PERSON] [26]. For the importance of non local conditions in different fields, we refer the reader to Refs. [11; 27; 28]. Recently, in Ref. [29], the authors studied the existence of mild solutions for impulsive fractional semi-linear integro-differential equations using the Banach contraction principle and [PERSON]'s fixed point theorem. They have considered the system without delay and without non local condition. Our work generalizes the work done in Ref. [29] with abstract formulation. According to our knowledge this is an untreated article in the literature. Motivated by the above mentioned paper, we study the existence of mild solutions for non local impulsive fractional semi-linear integro-differential equations of the form  
\[{}^{C}D_{0,t}^{q}x(t)=\ ^{C}D_{t}^{q}x(t)=D_{t}^{q}x(t) = Ax(t)+f\Big{(}t,x_{t},\int_{0}^{t}h(t,s,x_{s})ds,\int_{0}^{b}k(t,s,x_{s})ds\Big{)}, \tag{1}\] \[x(0) = x_{0}+g(x)\ \in X,\] (2) \[\Delta x|_{t=t_{k}} = I_{k}(x(t_{k}^{-})),\ \ k=1,2,\cdots,m;\ t\in J=[0,b],\ t\
eq t_{k}, \tag{3}\]  
where \({}^{C}D_{t}^{q}\) is the Caputo fractional derivative of order \(q,0<q<1\), with lower limit zero, the histories \(x_{t}:(-r,0]\to X\) are defined by \(x_{t}(\theta)=x(t+\theta)\) belongs to a Banach space \(X\). \(A:D(A)\subset X\to X\) is the infinitesimal generator of a strongly continuous semi-group \((T(t))_{t\geq 0}\) of a uniformly bounded operator on \(X\), and \(A\) is a bounded linear operator. \(f:J\times X\times X\times X\to X\) is jointly continuous, \(h,k:J\times J\times X\to X\) are continuous, \(I_{k}:X\to X\) are impulsive functions, \(0=t_{0}<t_{1}<\cdots<t_{m}<t_{m+1}=b,\ \Delta x|_{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), \(x(t_{k}^{+})=\lim\limits_{h\to 0^{+}}x(t_{k}+h)\) and \(x(t_{k}^{-})=\lim\limits_{h\to 0^{-}}x(t_{k}+h)\) represent the right and left limits of \(x(t)\) at \(t=t_{k}\), respectively. Our method avoids the compactness conditions on the semi-group \(\{T(t)\}_{t\geq 0}\), and some other hypotheses are more general compared with the previous research papers. System {(1),(2),(3)} described above fits to the mechanical system with impact, the biological phenomenon involving thresholds, the bursting rhythm models and industrial robotics, and many more. In particular, the above mentioned system {(1),(2),(3)} and its piecewise (PC)-mild solution (in Section 2) is helpful to generate the tuning and auto-tuning of fractional order controllers for industry applications. [PERSON] et al. [30] design the fractional order \(PI^{\lambda}D^{\mu}\) controllers to ensuring a robust performance of the controlled system with respect to gain variations and noise, without using the delay part but can be generalized in a more specific way using the delay for a closed loop and the open loop systems. System {(1),(2),(3)} can also be analyzed for a finite time stability test procedure for robotic system where it appears a time delay in fractional control system (refer [31]) but with impulses. In Section 2, we give some preliminary definitions and lemmas that are to be used later to prove our main results. In Section 3, the existence of \(PC\)-mild solutions for system {(1),(2),(3)} with non-local conditions is discussed. The results are obtained by using the Banach contraction principle and [PERSON]'s fixed point theorem. An example is given in Section 4 to illustrate the application of our main results.