Fredholm Boundary-Value Problems with Parameter in Sobolev Spaces

Abstract

For systems of linear differential equations of order $r ∈ ℕ$, we study the most general class of inhomogeneous boundary-value problems whose solutions belong to the Sobolev space $W_p^{n + r} ([a, b],ℂ^m)$, where $m, n + 1 ∈ ℕ$ and $p ∈ [1,∞)$. We show that these problems are Fredholm problems and establish the conditions under which these problems have unique solutions continuous with respect to the parameter in the norm of this Sobolev space.