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Existence of solutions to nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces. (English) Zbl 0957.34058

The authors deal with the nonlocal Cauchy problem

${\left(Bu\left(t\right)\right)}^{\text{'}}+Au\left(t\right)=f\left(t,u\left(t\right)\right)+{\int }_{0}^{t}g\left(t,s,u\left(s\right)\right)ds,\phantom{\rule{4pt}{0ex}}0
$u\left(0\right)+\sum _{k=1}^{p}{c}_{k}u\left({t}_{k}\right)={u}_{0},\phantom{\rule{2.em}{0ex}}\left(2\right)$

where $A$ and $B$ are closed linear operators in a Banach space $X$ with $D\left(B\right)\subset D\left(A\right)$ and the compact ${B}^{-1}$, $0\le {t}_{1}<{t}_{2}<\cdots <{t}_{p}\le a$, ${u}_{0}\in X$, and $f:\left[{t}_{0},{t}_{0}+a\right]×X\to X$, $g:\left\{\left(s,t\right):0\le s\le t\le a\right\}×X\to X$ are given functions. The main results are the existence of mild (under assumptions about the boundedness of $f$ and $g\right)$ and unique strong (under assumptions about the boundedness of $f,g$, Lipschitzian continuity of $f\left(·,u\right)$ with respect to $u$ and Lipschitzian continuity of $g\left(t,·,·\right)$ with respect to $t\right)$ solutions to problem (1), (2) based on the Schauder fixed-point principle. As an example the following problem

$\frac{\partial }{\partial t}\left(z\left(t,x\right)-{z}_{xx}\left(t,x\right)\right)-{z}_{xx}\left(t,x\right)=\mu \left(t,z\left(t,x\right)\right)+{\int }_{0}^{t}\eta \left(t,s,z\left(s,x\right)\right)ds,\phantom{\rule{4pt}{0ex}}0\le x\le \pi ,\phantom{\rule{4pt}{0ex}}0
$z\left(t,0\right)=z\left(t,\pi \right)=0,\phantom{\rule{1.em}{0ex}}z\left(0,x\right)+\sum _{k=1}^{p}z\left({t}_{k},x\right)={z}_{0}\left(x\right),$

is considered.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34K05 General theory of functional-differential equations 45J05 Integro-ordinary differential equations 47J35 Nonlinear evolution equations 35K90 Abstract parabolic equations
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