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Existence of mild solutions for abstract semilinear evolution equations in Banach spaces. (English) Zbl 1235.34174

Summary: The nonlocal initial value problem,

$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=Ax\left(t\right)+f\left(t,x\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in I=\left[0,1\right],\hfill \\ x\left(0\right)=g\left(x\right),\hfill \end{array}\right\$

where $A$ is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e. ${C}_{0}$-semigroup) $T\left(t\right)$ in Banach space $X$, and $f:I×X\to X$, $g:C\left(\left[0,1\right];X\right)\to X$ are given $X$-valued functions.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 47N20 Applications of operator theory to differential and integral equations
##### References:
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