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Zbl 1209.34095
Wang, Jinrong; Wei, W.
A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. (English)
[J] Result. Math. 58, No. 3-4, 379-397 (2010). ISSN 1422-6383; ISSN 0378-6218/e

The paper deals with the nonlinear integrodifferential impulsive equation with nonlocal conditions $$\cases x'(t)=Ax(t)+f\left(t,x(t),\int_0^tk(t,s,x(s))\,ds\right),\quad t\in J=[0,b],\ t\not=t_i,\\ x(0)=g(x)+x_0,\\ \Delta x(t_i)=I_i(x(t_i)),\quad i=1,2,\dots,p,\ 0=t_0<t_1<\dots<t_p<t_{p+1}=b,\endcases\tag P$$ where $A:D(A)\subset X\to X$ is the generator of a strongly continuous semigroup $\{T(t),\ t\ge 0\}$ 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 $(P)$.
[Rodica Luca Tudorache (Iaşi)]

MSC 2000:: *34K30 Functional-differential equations in abstract spaces
45J05 Integro-ordinary differential equations
34K45 Equations with impulses
47N20 Appl. of operator theory to differential and integral equations

Keywords: integrodifferential equations; nonlocal conditions; impulsive conditions; fixed point theorem

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