Existence for an implicit nonlinear differential equation

The implicit Cauchy problem, d/dt (Bu(t))+Au(t)qqf(t), a.e. tqq(0, T), (1.1) Bu(0) = Bu0, (1.2) is studied in a real Hilbert space H under the main assumption that the operator A+λ0B is nondegenerate for some λ0≥0. Four assumptions are imposed: V, H are real Hilbert spaces with the norms denoted &middot and |&middot|, respectively, V′ denotes the dual of V and VqqHqqV′ algebraically and topologically; BqqL(V,V′), (Bu, u)≥0 for all u∈V and B is symmetric; A:V&rarrV′ is nonlinear, maximal monotone and there exists λ0≥0 such that λ0B+A is coercive; and D(A) = V and the realization of A into L2(0, T;V)×L2(0, T; V′), again denoted A is bounded on subsets of the domain. Two theorems are obtained in this context.