A Green function approach for a singularly perturbed vector boundary- value problem
The authors consider the boundary value problem \[ \frac{dx}{dt}=U(t,x,y,\epsilon),\quad \epsilon \frac{dy}{dt}=V(t,x,y,\epsilon)\quad (0\leq t\leq 1), \] \[ H(x(0,\epsilon),x(1,\epsilon),y(0,\epsilon),y(1,\epsilon),\epsilon)=0 \] for small positive \(\epsilon\). Here the solution functions x and y as well as the given data functions U, V, H are vector-valued functions of dimension m, n, m, n, \(m+n\), respectively. If certain assumptions are fulfilled - \(U,V\in {\mathbb{C}}^{N+1}\) is one of them -, then it is proved that there are vector-valued functions \(x^ N\) and \(y^ N\) which solve the boundary-value problem up to \(O(\epsilon^{N+1})\). The main result states that there is exactly one solution x, y of the given problem such that \[ | x(t,\epsilon)-x^ N(t,\epsilon)| \leq const \epsilon^ N,\quad | y(t,\epsilon)-y^ N(t,\epsilon)| \leq const \epsilon^ N \] for \(0\leq t\leq 1\), \(\epsilon_ 0\geq \epsilon >0\). Two examples illustrate the theory. As the authors point out in the introduction, their assumptions are much less restrictive than those of other authors. For example, they permit boundary coupling conditions. Though the five assumptions are rather lengthy due to the generality of the results, the statements as well as the proofs are well-organized. This very interesting paper will stimulate further development in the theory of nonlinear boundary-value problems.
scientific article; zbMATH DE number 4109152
scientific article