A nonlinear singular perturbation problem

Let F(u_ε)+ε(u_ε−w)=0 (1) where F is a nonlinear operator in a Hilbert space H, w∈H is an element, and ε>0 is a parameter. Assume that F(y)=0, and F′(y) is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to (1) and for the convergence lim _ε→0‖u_ε−y‖=0. An example of applications is considered. In this example F is a nonlinear integral operator.