Generalized Green’s Matrices for Linear Differential Systems
Howard Chitwood · SIAM Journal on Mathematical Analysis · 1973
This paper investigates the $n \times n$ matrix differential equation $Y' = AY$ together with boundary conditions of the form $\int_a^b {dF(t)Y(t) = 0} $, where F is an $n \times n$ matrix whose elements are of bounded variation. It is known that if the above boundary problem is incompatible then the nonhomogeneous boundary problem $Y' = AY + R$, $\int_a^b {dF(t)Y(t) = 0} $ has a unique solution; here it is shown that if the homogeneous problem is compatible, then the Moore–Penrose generalized inverse of a matrix can be employed to obtain conditions which ensure the existence of a solution to the nonhomogeneous problem. A generalized Green’s matrix is constructed and its properties studied. An adjoint system is defined and properties relating it to the given system and the generalized Green’s matrix are explored. A principal generalized Green’s matrix is defined and properties analogous to those for the classical case are developed.