The solvability of quasilinear equation with monotone  operator

The paper is devoted to the study of the solvability of the operator equation Lx = Fx, where L : X ^ Y - the linear limited noninvertible operator (resonance case) and F : X ^ Y - completely continuous operator; X, Y - Banach spaces. The existence theorem is proved in the conditions of the monotony of an auxiliary operator. To illustrate the developed approach, a solvability of a periodic boundary value problem for the second order differential equation is considered.