On total preservation of solvability for a controlled Hammerstein type equation with non-isotone and non-majorized operator

We prove some nontrivial corollaries of the Schauder theorem. We use these corollaries to prove a theorem concerning the total preservation of solvability for a controlled functional operator equation of the Hammerstein type with non-isotone and non-majorized operator component of the right-hand side. We illustrate an application of the abstract theory by the example of the Dirichlet problem associated with a semilinear elliptic equation of the stationary diffusion-reaction type.