  Let
  F(u_\ve)+\ve(u_\ve-w)=0 \eqno{(1)}
  where $F$ is a nonlinear operator in a Hilbert space $H$, $w\in H$ is an
element, and $\ve>0$ 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 \eqref{e1.1} and for the convergence $\lim_{\ve\to
0}\|u_\ve-y\|=0$. An example of applications is considered. In this example $F$
is a nonlinear integral operator.
