A small-parameter problem from the theory of boundary-value problems with a small nonlinearity

We consider a system of nonlinear equations with a small (in a certain sense) nonlinearity. The existence of a solution is investigated in the presence of a perturbation parameter of small absolute value, and the perturbed solution is required to approach the original solution of the limiting linear system. The sufficient conditions are presented in geometrical and analytical forms. The problem considered in this article arises in connection with nonlinear boundary-value problems for systems of differential equations with a small (in a certain sense) nonlinearity. © 2001 Plenum Publishing Corporation.