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A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. (English) Zbl 1209.34095
The paper deals with the nonlinear integrodifferential impulsive equation with nonlocal conditions
$\begin{cases} x'(t)=Ax(t)+f\left(t,x(t),\int_0^tk(t,s,x(s))\,ds\right),\quad t\in J=[0,b],\;t\not=t_i,\\ x(0)=g(x)+x_0,\\ \Delta x(t_i)=I_i(x(t_i)),\quad i=1,2,\dots,p,\;0=t_0<t_1<\dots<t_p<t_{p+1}=b,\end{cases}\tag{P}$
where $$A:D(A)\subset X\to X$$ is the generator of a strongly continuous semigroup $$\{T(t),\;t\geq 0\}$$ on a Banach space $$X$$, $$f:J\times X\times X\to X$$, $$k:J\times J\times X\to X$$, $$g:PC(J,X)\to X$$ and $$I_i:X\to X$$, $$i=1,2,\dots, p$$ are given functions which satisfy some suitable assumptions, $$\Delta x(t_i)=x(t_i^+)-x(t_i^-)$$, $$x(t_i^+)=\lim_{h\to 0^+}x(t_i+h)$$ and $$x(t_i^-)=\lim_{h\to 0^-}x(t_i+h)$$ are respectively the right and left limits of $$x(t)$$ at $$t=t_i$$. By using a generalization of the Ascoli-Arzela theorem and some fixed point theorems such as Schaefer’s fixed point theorem and Krasnosel’skii’s fixed point theorem, the authors study the existence and uniqueness of $$PC$$-mild solutions for problem $$(P)$$.

##### MSC:
 34K30 Functional-differential equations in abstract spaces 45J05 Integro-ordinary differential equations 34K45 Functional-differential equations with impulses 47N20 Applications of operator theory to differential and integral equations
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