THE EXTENDABILITY AND UNIQUENESS OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS

The extendability of solutions of ordinary differential equations is a fundamental and important property since the analysis of stability and boundedness of solutions requi-res extendability. This paper is concerned with the preser-vation of the extendability and uniqueness of solutions under perturbations. In particular conditions on the right hand side of x=f(x)Jrh(t) are exhibited which guarantee the extendability of solutions whenever the solutions of the unperturbed equation x—f(x) extend. This paper continues the author's previous study [1] and also includes the question of uniqueness of the zero solution of perturbed equations satisfying an Osgood condition [5] (See also [2] for recent results on the uniqueness of perturbed systems.) Examples are provided to demonstrate the strength of our results. The interested reader may look in [4], [5], [6] for other results on extendability. 2 * Notation and preliminaries • Let Rd denote d-dimensional space. We represent solutions of the Cauchy problem x — f(t, x), x(Q = χQ by x(t, ί0, xQ). We shall usually be analyzing the equations (U) x = f(x), (P) x = f(x) + hit), where f:R—*R+ is continuous and h:R—+R is Lebesgue measura-ble, and ( * ) J fir) Condition (*) is equivalent to the extendability of all solutions of (U) ([1]). We investigate the class of functions hit) which preserve the extendability of solutions and show by example that this class is, in a meaningful sense, "maximal. " Again if we substitute the word uniqueness for extendability in each of our theorems, the results hold by noting condition (*) is essentially the Osgood condi-tion at infinity. 3 * Results and examples * Before we state our results we will need the following definitions