On a Class of Operator Equations

We assume that $E_1$ and $E_2$ are Banach spaces, $a\colon E_1\to E_2$ is a continuous linear surjective operator, $f\colon E_1\to E_2$ is a nonlinear completely continuous operator. In this paper, we study existence problems for the equation $a(x)=f(x)$ and estimate the topological dimension $dim$ of the set of solutions.