ON THE GLOBAL ATTRACTIVITY AND ASYMPTOTIC STABILITY FOR AUTONOMOUS SYSTEMS OF DIFFERENTIAL EQUATIONS ON THE PLANE

The autonomous system of differential equations x' = f(x), (x = (x(1), x(2))(T) is an element of R-2, f(x) = (f(1)(x), f(2)(x))(T)), is considered, and sufficient conditions are given for the global attractivity of the unique equilibrium x = 0. This property means that all solutions tend to the origin as t -> infinity. The two cases (a) div f(x) < 0 (x is an element of R-2) and (b) div f(x) >= 0 (x is an element of R-2) are treated, where div f(x) := partial derivative f(1)(x)/partial derivative x(1)+partial derivative f(2)(x)/partial derivative x(2). Earlier results of N. N. Krasovskii and C. Olech about case (a) are improved and generalized to case (b). Three types of assumptions are required: certain stability properties of the origin (local attractivity, stability), boundedness above in some sense for div f(x), and assumptions that If vertical bar f(x)vertical bar is not as small as vertical bar x vertical bar -> infinity. The conditions of the second and third types are connected with each other.