Singular perturbation of nonlinear vector boundary value problem

Abstract

In this paper we study the perturbed boundary value problem of the form

$$\begin{gathered} dx/dt = f(x,y,t;\varepsilon ),\varepsilon dy/dt = g(x,y,t;\varepsilon ), \hfill \\ a_1 (\varepsilon )x(0,\varepsilon ) + a_2 (\varepsilon )y(0,\varepsilon ) = a(\varepsilon ) \hfill \\ b_1 (\varepsilon )x(1,\varepsilon ) + \varepsilon b_2 (\varepsilon )y(1,\varepsilon ) = \beta (\varepsilon ) \hfill \\ \end{gathered}$$

in which x, f, β∈E^m, y, g, a∈Eⁿ, 0<ε≪1 and a₁(ε), a₂(ε), b₁(ε) and b₂(ε) are matrices of the appropriate size. Under the condition that g_γ(t) is nonsingular and other suitable restrictions, the existence of the solution is proved, the asymptotic expansion of solution of order n is constructed, and the remainder term is estimated.