A stability theory for perturbed differential equations

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is X ′ = f ( t , X ) and the associated perturbed differential equation is Y ′ = f ( t , Y ) + g ( t , Y ) . The approach used is to examine the difference between the respective solutions F ( t , t 0 , x 0 ) and G ( t , t 0 , y 0 ) of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the difference G ( t , t 0 , y 0 ) − F ( t , t 0 , x 0 ) in the case where the initial values y 0 and x 0 are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.