An algorithm for constructing a quasi-regular asymptotic representation for the solution of singularly perturbed linear multi-point boundary value problems with fast and slow variables

The article is devoted to the system epsilondot x=A_{11}(t)x+A_{12}(t)z+f_1(t), dot z=A_{21}(t)x+A_{22}(t)z+f_2(t), where epsilon is a small positive parameter. The points 0=t_1<t_2<dots<t_n=1 are given. A solution x(t,epsilon),z(t,epsilon) of the mentioned system satisfying the multi-point boundary conditions sum^n_{j=1}F_jx(t_j,epsilon)=x^0, z(0,epsilon)=z^0 is to be found. The author gives sufficient conditions when a bounded solution of this problem as epsilonto0 exists. The form of the solution is obtained.