Quasilinear operators and Hammerstein's equation

We describe the class of operators in a Hilbert space $\mathrm{H}$, introduced by A. I. Perov, which can be represented in the form $\mathrm{Ax=D(x)x}$, where $\mathrm{D(x)}$ is a self-conjugate operator satisfying the inequalities $\mathrm{B_-\leqslant D(x)\leqslant B_+}$ ($\mathrm{B_-}$ and $\mathrm{B_+}$ are fixed self-conjugate operators). As an application we obtain new theorems on the solvability of Hammerstein's equation.