Quasilinear operators and Hammerstein's equation

We describe the class of operators in a Hilbert space H, introduced by A. I. Perov, which can be represented in the form Ax = D(x)x, where D(x) is a self-conjugate operator satisfying the inequalities B_–D(x) B₊ (B_– and B₊ are fixed self-conjugate operators). As an application we obtain new theorems on the solvability of Hammerstein's equation.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 453–464, October, 1972.