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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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