Asymptotic equilibrium of ordinary differential systems

A theorem on asymptotic equilibrium is proved for the solutions of the system(1)X ⁿ=f(t,X), x ^t ₀=x_o where f(t,x) is majorized by a funciton g(t,u) which is non-increasing in u. It is of interest to notice that the funcitons f(t,x) and g(t,u) need not be defined for x=0 and u=0 respectively. Such majorant functions occur in gravitational problems and therefore the result is of pracitcal interest.Using this, the asymptotic relatiohship between the solutions of(2)y=A(t)y, y ^t _o=y_oand its nonlinear perturbation(3) X=A(t)x+f(t,x), X^t _o is investigated. This last result includes as a special case two theorems of Hallam2]