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Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces. (English) Zbl 1227.34060
The authors study the existence of solutions to the following problem $$u'(t)=Au(t)+f(t,u(t)),\quad t\in (0,T],\quad u(0) = g(u),$$ where $A$ is the generator of a linear semigroup, $f,g$ satisfy Lipschitz conditions. When $g$ is a constant function this is a Cauchy problem associated with a semilinear evolution equation. If $g$ is an arbitrary Lipschitz function this problem is called “nonlocal” Cauchy problem. The authors use the measure of noncompactness and a fixed point theorem to prove an existence result.

##### MSC:
 34G10 Linear ODE in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
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