Solvability of Semilinear Operator Equations and Applications to Semilinear Hyperbolic Equations

PROPOSITION 2.1 Let X be a reflexive Banach space compactly embedded in a Hilbert space H, A : D(A) C H -* H be a closed and densely defined linear map and N : H -*■ H be a nonlinear continuous map. Then A + N : D(A) fl X C X -* H is pseudo A-proper w.r.t.T = { Hn H„Pn} for (X, H) with P„Ax = Ax on H„.