A Characterization of Semilinear Dense Range Operators and Applications

We characterize a broad class of semilinear dense range operators G(H) : W -> Zgiven by the following formula, G(H)w = Gw + H(w), w is an element of W, where Z, W are Hilbert spaces, G is an element of A(W, Z), and H : W -> Z is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator.. to have dense range. Second, under some condition on the nonlinear term H, we prove the following statement: If Rang(G) =.., then Rang(G(H)) = Z and for all z subset of Zthere exists a sequence {w(alpha) subset of Z : 0 < alpha <= 1} given by w(alpha) = G*(alpha I + GG*)(-1) (z - H(w(alpha))), such that lim(alpha -> 0)+{Gu(alpha) + H(mu(alpha))} = z Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: z' = Az + Bu(t) + F(t, z, u(t)), z is an element of Z, u is an element of U, t > 0, where Z, U are Hilbert spaces, A : D(A) subset of Z -> Z is the infinitesimal generator of strongly continuous compact semigroup {T(t)}(t >= 0) in Z, B is an element of L(U, Z), the control function.. belongs to L-2 (0, tau; U), and F : [0, tau] x Z x U -> Z is a suitable function. As a particular case we consider the controlled semilinear heat equation.