##
**Uniqueness of solution for impulsive fractional functional differential equation.**
*(English)*
Zbl 1398.34115

Summary: In this work, we study linear impulsive fractional functional integro-differential equation of the form:
\[
\begin{gathered} D^\alpha_t y(t)= J^{2-\alpha}_t f(t,y_{\rho(t,y_t)},\;B(y_{\rho(t,y_t)}),\;at\in J=[0,T],\;t\neq t_k,\\ \Delta y(t_k= x_k,\;\Delta y'(t_,)= z_k,\;k=1,2,\dots, m,\\ y(t)= \phi(t),\;y'(t)= \varphi(t),\;t\in[-d,0],\end{gathered}
\]
where \(y'\) denotes the derivative of \(y\) with respect to \(t\) and \(D^\alpha_t\) is Caputo’s derivative of order \(\alpha\in(1,2)\), \(f:J\times PC_0\times PC_0\to X\) is given continuous function and \(PC_0\) is an abstract phase space with \(y_t\) the element of \(PC_0\) defined by \(y_t(\theta)= y(t+\theta)\), \(\theta\in [-d,0]\).

This paper is concerned with existence results.

This paper is concerned with existence results.

### MSC:

34K37 | Functional-differential equations with fractional derivatives |

34K30 | Functional-differential equations in abstract spaces |

34K45 | Functional-differential equations with impulses |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

fractional order differential equation; functional differential equations; impulsive conditions; fixed point theorem### References:

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