Global existence for semilinear evolution equations with nonlocal conditions (Q1363598)

The authors apply Schauder's fixed-point theory to differential equations in Banach spaces, \[ x'(t)= Ax(t)+ f(t,x(t)) \] with nonlocal initial conditions of the type \[ x(0)+ g(x)= x_0. \] Here, \(t\in[0, b]\), and some general assumptions on \(g: C([0,b], X)\to X\) and \(f:[0, b]\times X\to X\) are given which guarantee the existence of a solution \(x\in C([0,b], X)\).