Successive approximations to solutions of nonlinear equations with vector parameter in the irregular case

We consider a nonlinear operator equation with a Fredholm operator in the principal part. The nonlinear part of the equation depends on some functionals that are defined on an open set of a normed linear space. We suggest a method of successive asymptotic approximations to branching solutions. We apply this method to study a nonlinear boundary value problem describing the oscillation of a satellite in the plane of its elliptic orbit.