A criterion for the solvability of a linear boundary value problem for a system of the second order (Q2740351)

The author deals with the boundary value problem NEWLINE\[NEWLINELx:=\ddot x+A(t)W(x)=\varphi(t),\quad t\in [0,T],\qquad \ell x=\alpha, \tag{1}NEWLINE\]NEWLINE with \(x\in \mathbb R^n\), \(W(x):=\text{colon}(x,\dot x)\), \(A(t)\in C([0,T] \mapsto \mathbb R^{n\times 2n})\), \(\varphi(t)\in C([0,T] \mapsto \mathbb R^n)\), \(\alpha \in \mathbb R^m\), and \(\ell:C([0,T] \mapsto \mathbb R^n) \mapsto \mathbb R^m\) is an \(\mathbb R^m\)-valued linear functional.NEWLINENEWLINENEWLINELet \(X(t)\) be the fundamental matrix of the corresponding homogeneous system \(Lx=0\), \(\bar x(t)\) be a particular solution to the non-homogeneous system \(Lx=\varphi \), and \(D:=\ell X(\cdot)\in \mathbb R^{m\times 2n}\). The critical case is studied when \(\text{rank} D<min(2n,m)\). After having analyzed the algebraic system \(Dc=\alpha -\ell \bar x(\cdot)\) where \(c\in \mathbb R^n\) is an unknown vector, the author establishes necessary and sufficient conditions for the solvability of the boundary value problem (1).NEWLINENEWLINENEWLINEThe results obtained are applied to a two-point boundary value problem.