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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

$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=Ax\left(t\right)+f\left(t,x\left(t\right),{\int }_{0}^{t}k\left(t,s,x\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}ds\right),\phantom{\rule{1.em}{0ex}}t\in J=\left[0,b\right],\phantom{\rule{4pt}{0ex}}t\ne {t}_{i},\hfill \\ x\left(0\right)=g\left(x\right)+{x}_{0},\hfill \\ {\Delta }x\left({t}_{i}\right)={I}_{i}\left(x\left({t}_{i}\right)\right),\phantom{\rule{1.em}{0ex}}i=1,2,\cdots ,p,\phantom{\rule{4pt}{0ex}}0={t}_{0}<{t}_{1}<\cdots <{t}_{p}<{t}_{p+1}=b,\hfill \end{array}\right\\phantom{\rule{2.em}{0ex}}\left(\mathrm{P}\right)$

where $A:D\left(A\right)\subset X\to X$ is the generator of a strongly continuous semigroup $\left\{T\left(t\right),\phantom{\rule{4pt}{0ex}}t\ge 0\right\}$ on a Banach space $X$, $f:J×X×X\to X$, $k:J×J×X\to X$, $g:PC\left(J,X\right)\to X$ and ${I}_{i}:X\to X$, $i=1,2,\cdots ,p$ are given functions which satisfy some suitable assumptions, ${\Delta }x\left({t}_{i}\right)=x\left({t}_{i}^{+}\right)-x\left({t}_{i}^{-}\right)$, $x\left({t}_{i}^{+}\right)={lim}_{h\to {0}^{+}}x\left({t}_{i}+h\right)$ and $x\left({t}_{i}^{-}\right)={lim}_{h\to {0}^{-}}x\left({t}_{i}+h\right)$ are respectively the right and left limits of $x\left(t\right)$ 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 $\left(P\right)$.

##### 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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