Principal solutions and transformations of linear Hamiltonian systems
Ondřej Došlý · Czech digital mathematics library · 1992
. Sufficient conditions are given which guarantee that the linear transformation converting a given linear Hamiltonian system into another system of the same form transforms principal (antiprincipal) solutions into principal (antiprincipal) solutions. 1. Introduction. Consider a linear Hamiltonian system (1.1) Y 0 = A(t)Y + B(t)Z; Z 0 = \\GammaC (t)Y \\Gamma A T (t)Z; where A; B; C are n \\Theta n matrices of continuous, real valued functions, t 2 I = [a; 1), B; C are symmetric, i. e., B T = B; C T = C; and Y; Z are n \\Theta n matrices. If the matrices B; C are nonnegative definite, it is known that (1.1) is nonoscillatory at 1 (for terminology see Section 2) if and only if the so-called reciprocal system (1.2) U 0 = \\GammaA T (t)U + C(t)V; V 0 = \\GammaB(t)U +A(t)V is nonoscillatory at 1, see [2,5,8,9]. Recently the author established a more general duality in oscillation behaviour of various linear Hamiltonian systems which may be described in the following way. If we se...