... fn→ f ,asn →∞,inL1(0,T;H). For other properties of the maximal monotone operators and for the main results of existence, uniqueness of the strong and weak solutions for the nonlinear evolutionequations ... Y.Proof. We suppose without loss of generality (for an easy writing) that A and B are single-valued. T he operatorᏮ under the assumptions of this lemma is well-defined in Y. Indeed, for ((un)n,(vn)n) ... W1,∞(0,T;Y),with V(t)∈ D(B),forallt ∈ [0,T],andV verifies the system (S), for all t ∈ [0,T),theextreme condition (EC), for all t ∈ [0,T), and the init ial data (ID).Proof. By Lemmas 4.1 and...