Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces (Q549950)

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.