Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions

In this work, we study the existence of mild solutions and strict solutions of semi linear functional evolution equations with non local conditions, where the linear part is non-autonomous and generates a linear evolution system. The fraction power theory and α-norm are used to discuss the problems so that the obtained results can be applied to the equations in which the nonlinear terms involve spatial derivatives. In particular, the compactness condition or Lipschitz condition for the function g in the non local conditions appearing in various literatures is not required here. An example is presented to show the applications of the obtained results.