Existence of mild solutions for fractional evolution equations with nonlocal initial conditions (Q2872670)

The authors study the existence of mild solutions for a class of semilinear fractional evolution equations with nonlocal initial conditions NEWLINE\[NEWLINE ^{c}D^{q}u(t)+Au(t)=f(t,u(t)), \;\;\;\;t\in J=[0,1], NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0)=\sum^{p}_{k=1}c_{k}u(t_{k})+u_{0} NEWLINE\]NEWLINE in an arbitrary Banach space. They assume that the linear part generates an equicontinuous semigroup, and the nonlinear part satisfies noncompactness measure conditions and appropriate growth conditions, and prove their main result by using Sadovskii's famous fixed point theorem. In the application to partial differential equations, such as parabolic and strongly damped wave equations, the corresponding solution semigroups are analytic. Therefore, the main result in this paper has a broad applicability.