The author considers nonlocal Cauchy problems in the abstract setting of evolution equations. Two types of equations are dealt with:
where is the infinitesimal generator of a -semigroup, , the points lie in the interval , and the functions , and satisfy certain conditions. The appropriate concept of a mild solution is introduced and theorems on existence and uniqueness of mild solutions are obtained. Moreover, conditions for the existence of classical solutions are provided. The methods of proof are based upon the general theory of evolution equations and semigroups.