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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, $\begin{cases} x'(t)=Ax(t)+f(t,x(t)), \quad t\in I=[0,1],\\ x(0)=g(x),\end{cases}$ where $$A$$ is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators (i.e. $$C_0$$-semigroup) $$T(t)$$ in Banach space $$X$$, and $$f:I\times X\to X$$, $$g:C([0,1];X)\to X$$ are given $$X$$-valued functions.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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