Nonlocal Cauchy problems governed by compact operator families (Q1879741)

Let \(A\) be the infinitesimal generator of a compact semigroup of linear operators on a Banach space \(X\). The authors establish the existence of mild solutions to the nonlocal Cauchy problem \[ u'(t)=Au(t)+f(t,u(t)), \quad t\in[t_0,t_0+T], \quad u(t_0)+g(u)=u_0, \] under some conditions on \(f\) and \(g\), where \(f:[t_0,t_0+T]\times X\to X\) and \(g:C([t_0,t_0+T];X)\to X\) are given functions. They assume a Lipschitz condition on \(f\) with respect to \(u\), but they do not require any compactness assumption on \(g\), opposed to \textit{S. Aizicovici} and \textit{M. McKibben} [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649--668 (2000; Zbl 0954.34055)] and \textit{L. Byszewski} and \textit{H. Akca} [Nonlinear Anal., Theory Methods Appl. 34, No. 1, 65--72 (1998; Zbl 0934.34068)], where the authors assume a compactness property for \(g\), but do not require any Lipschitz condition on \(f\).