nonlinear maximal monotone
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Here we survey a few functional methods to existence theory for infinite dimensional stochastic differential equations of the form dX+A(t)X(t)=B(t,X(t))dW(t), X(0)=X_0, where A(t) is a non\-linear maximal monotone operator in a variational couple (V,V'). The emphasis is put on a new approach of the classical existence result of N. Krylov and B. Rozovski on existence for the infinite dimensional stochastic differential equations which is given here via the theory of nonlinear maximal monotone operators in Banach spaces. A variational approach to this problem is also developed.

Conditions are given on two maximal monotone (multivalued) operators A and B which ensure that A + B is also maximal. One condition used is that ∥Bx∥≦k(∥x∥)Ax| +d|(A + B)x| + c(∥x∥) for every x∈D(A)⊆D(B), where 0≦k(r)<1, and c(r)≧0 are nondecreasing functions, and 0≦d≦1 is a constant. Here, for a set C, |C| denotes inf{∥y∥:y∈C}. This extends the well known result which has d = 0 (and is used in the proof here). The second part of the paper uses similar hypotheses to give conditions under which the range of the sum, R(A + B), has the same interior and same closure as the sum of the ranges, R(A) + R(B).