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Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. (English) Zbl 1178.34071
Summary: We study the existence of mild solutions for the following system in a general Banach space $X$ (with norm $\|\cdot\|$): $$\cases \frac{d}{dt}[x(t)-F(t,x(h_1(t)))]\in Ax(t)+\int^t_0K(t,s)G(s,x(h_2(s)))\,ds,\\ t\in J-\{t_1,\dots,t_m\},\quad \text{where }J=[0,a],\\ \Delta x|_{t_k}=I_k(x(t^-_k)),\quad k=1,\dots,m,\\ x(0)=g(x)\in X.\endcases\tag1.1$$ Here $A$ is the infinitesimal generator of a compact analytic semigroup $T(t($, $t>0$, $G$ is a multi-valued map and $\Delta x|_{t=t_k}=x(t^+_k)-x(t^-_k)$, where $x(t^-_k)$ and $x(t^+_k)$ represent the left and right limits of $x(t)$ and $t=t_k$, respectively. Let $K:D\to R$, $D=\{(t,s)\in J\times J:t\ge s\}$ and $F,G,g,I_k$ $(k=1,\dots,m)$ and $h_1, h_2$ are given functions. By using a fixed point theorem for multi-valued maps due to Dhage, a main existence theorem is established. Finally, we present an example to illustrate this main theorem.

##### MSC:
 34G25 Evolution inclusions 47N20 Applications of operator theory to differential and integral equations
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##### References:
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