Ranges of nonlinear asymptotically linear operators

This paper deals with the solvability of the equation A(u) ? S(u) = f, where A is a continuous self-adjoint operator defined on a real Hilbert space H with values in H, the null-space of A is nontrivial, and N is a nonlinear completely continuous perturbation. Sufficient, and necessary-sufficient conditions are given for the equation to be solvable. Abstract theorems are applied to solving boundary value problems for partial differential equations.