Markov systems of vector-valued functions and disconjugacy (Q914910)

The aim of the author is to generalize the classical concept of disconjugacy, which is defined as to be related to linear differential operators, to certain abstract function spaces. In order to this end he considers a subspace \({\mathcal X}\) of \(C(J\to {\mathbb{R}}^ k)\) where \(J=(a,b)\) is a finite or infinite interval on \({\mathbb{R}}\), and a class of partitions \(\tau =(\tau_ 1,...,\tau_ k)\) of points in J, say \({\mathcal F}\). Then he says that the space \({\mathcal X}\) is \({\mathcal F}\)-disconjugate on a subset \(I\subset J\) if there exists at least one \(\tau\in {\mathcal F}\) such that \(\tau\) is a partition of n points in I and the equation \(x[\tau]=c\) has a unique solution \(x\in {\mathcal X}\) for any \(c\in {\mathbb{R}}^ n\). Here x[\(\tau\) ], which is the value of x at \(\tau\), is a column vector defined in block form by \(x[\tau]=(x_ 1[\tau],...,x_ k[\tau])^ T\) with \(x_ i[\tau]=(x_ i(s_ 1),...,x_ i(s_ q))^ T\) if \(\tau_ i=(s_ 1,...,s_ q)\) \((s_ j\) are all distinct). The paper is primarily concerned with the cases when I is either a single point in J or a subinterval of J. The main result of the paper is the criteria for the disconjugacy in the generalized sense. They are stated in two theorems which involve several new concepts such as ``generalized wronskian'', ``generalized sygnum functions'', ``\({\mathcal F}_{\pm}\)- Markov systems'' etc. The paper includes some simple illustrative examples which give the criteria for incompatibility of certain classes of linear homogeneous boundary value problems for ordinary differential equations.