We study Hammerstein operator equations of the form $$ x-KFx=f \eqno (1.1) $$ where $K$ is linear and $F$ is a nonlinear map. We first study Eq.~(1.1) in the operator form using the (pseudo) $A$-proper mapping approach and the Brouwer degree theory. Then we apply the obtained results to Hammerstein integral equations.