On stability and L p solutions of ordinary differential equations

Summary The usual definition of the stability of a solution of a system of ordinary differential equations is extended by introducing two positive control functions. These functions are used to control the rate of growth of the in?tial position of the solution and the rate of growth of the solution. Definitions and results are also given for the corresponding analogues of boundedness, weak boundedness, and uniform properties of the sotions of differential equations. The problem of determining when solutions of certain linear and weakly nonlinear differential equations lie in a modified Lp-space is also considered. This research was supported by the National Science Foundation under grant GP-8921. Entrata in Redazione il 13 maggio 1969.