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Nonlocal problems for integrodifferential equations. (English) Zbl 1163.45010

The paper deals with the nonlocal Cauchy problem for the nonlinear integrodifferential equation

${u}^{\text{'}}\left(t\right)=Au\left(t\right)+{\int }_{0}^{t}B\left(t-s\right)u\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds+f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0\le t\le T,\phantom{\rule{2.em}{0ex}}\left(1\right)$
$u\left(0\right)={u}_{0}+g\left(u\right),\phantom{\rule{2.em}{0ex}}\left(2\right)$

in a Banach space $X$, where $A:D\left(A\right)\subset X\to X$ is a densely defined, closed linear operator that generates a ${C}_{0}$-semigroup $\left\{T\left(t\right),\phantom{\rule{4pt}{0ex}}t\in \left[0,T\right]\right\}$, $\left\{B\left(t\right),\phantom{\rule{4pt}{0ex}}t\in \left[0,T\right]\right\}$ is a family of continuous linear operators from $D\left(A\right)$ into $X$, the function $f:\left[0,T\right]×X\to X$ is continuous and the operator $g:C\left(\left[0,T\right]×X\right)\to X$ is compact, which satisfy some additional assumptions. The authors prove that the resolvent operator $R\left(t\right)$ of equation $\left(1\right)$ is continuous in the uniform operator topology, for $t>0$, and then they establish the existence of mild solutions of the problem (1)–(2), by using Schaefer’s fixed point theorem.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations