In this paper, a general distributed parameter control problem in Banach spaces with integral cost functional and with given initial and terminal data is considered. An extension of the Dubovitskii-Milyutin method to the case of nonregular operator equality constraints, based on Avakov's generalization of the Lusternik theorem, is presented. This result is applied to obtain an extension of the Extremum Principle for the case of abnormal optimal control problems. Then a version of this problem with nonoperator equality constraints is discussed and the Extremum Principle for this problem is presented.