Conjugate points of vector-matrix differential equations

Abstract

The system of equations \[ ∑ k = 0 n ( − 1 ) n − k ( P k ( x ) y ( n − k ) ( x ) ) ( n − k ) = 0 ( 0 ⩽ x > ∞ ) \sum \limits _{k = 0}^n {{{( - 1)}^{n - k}}{{\left ( {{P_k}(x){y^{(n - k)}}(x)} \right )}^{(n - k)}}} = 0\quad (0 \leqslant x > \infty ) \] is considered where the coefficients are real, continuous, symmetric matrices, y is a vector, and P 0 ( x ) {P_0}(x) is positive definite. It is shown that the well-known quadratic functional criterion for existence of conjugate points for this system can be further utilized to extend results of the associated scalar equation to the vector-matrix case, and in some cases the scalar results are also improved. The existence and nonexistence criteria for conjugate points of this system are stated in terms of integral conditions on the eigenvalues or norms of the coefficient matrices.