On two-point boundary value problems
The authors study two-point boundary value problems of the form \(y'+A(t)y=f(t)\), \(My(a)+Ny(b)=\alpha\) on an interval [a,b], where A is an \(n\times n\) matrix, M and N are \(m\times n\) matrices with rank \([MN]=p\leq \min (m,n)\). Such a problem is said to be invertible if \(m=n\) and the corresponding homogeneous problem \((f=0)\) has only the trivial solution. It is known that in the invertible case the boundary value problem has a unique solution. A characteristic matrix of the boundary value problem is a matrix \(D=MY(a)+NY(b)\), where Y is a fundamental matrix of the homogeneous system \(y'+A(t)y=0\). For the non-invertible case the authors show that the boundary value problem has a unique solution if the columns of D are linearly independent and give a representation for the solution in terms of a Green's matrix.
scientific article; zbMATH DE number 4192148