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Existence of semilinear differential equations with nonlocal initial conditions. (English) Zbl 1129.34041

The author considers the existence of mild solutions for semilinear Cauchy problems

${u}^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,b\right]\phantom{\rule{4.pt}{0ex}}\text{a.e.,}\phantom{\rule{4.pt}{0ex}}u\left(0\right)=g\left(u\right)+{u}_{0},$

where $A$ is an infinitesimal generator of a strongly continuous semigroup $T\left(t\right)$ of bounded linear operators in a Banach space $X$, $f:\left[0,b\right]×X\to X$, $g\in C\left(\left[0,b\right];X\right)$ are given $X$-valued functionals. Certain assumptions are imposed on the nonlinear terms which allow for using a suitable fixed point theorem in proving the existence result. The map $g$ does not need to be compact in order to reach the existence conclusion. Instead some other weaker condition is assumed.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 47D06 One-parameter semigroups and linear evolution equations
##### Keywords:
compact semigroup; mild solution; nonlocal condition
##### References:
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