This paper is concerned with the abstract nonlinear evolution equation \(U'+A(U)=0\), where A is a nonlinear operator in a Hilbert space H. A class of operators is introduced which generalizes the class of monotone operators. This class includes Lipschitz continuous perturbations of monotone operators, but extends well beyond such cases. The class also includes cases in which the solution U is required to remain in some non- convex set in H. Local existence results and regularity results are presented (without proofs). The results generalize previous results of several of the authors. An example is given to illustrate the class of operators considered.