Error estimates of a numerical method for a class of nonlinear evolution equations

Let H be a real Hilbert space and let X, Y be two orthogonal subspaces of H such that H = X ⨁ Y. Let T and gt; 0, A ϵ ℒ(H) and B : [0, T] x X x Y → H be a nonlinear operator. We are interested in the following evolution problem: