On Extension Theorems in Ordinary Differential Systems of Equations

Abstract

Given the differential system x′ = f (t, x); f: J × M → Rn, we establish the Peano’s theorem on existence of the solution plus Picard Lindelof theorem on uniqueness of the solution. Using the two, we then worked on the extendibility of the solutions whose local existence is ensured by the above in a domain of open connected set producing the following results; If D is a domain of R × Rn and F: D → Rn is continuous and suppose that (t0, x0) is a point D and if the system has a solution x(t) defined on a finite interval (a, b) with t ∈ (a, b) and x(t0) = x0, then whenever f is bounded and D, the limits
xa
xtta+()=+lim()
xb
xttb−()=−lim()
Exists as finite vectors and if the point (a, x(a+)),(b, x(b-)) is in D then x(t) is extendable to the point t = a (t = b). In addition to this are other results as could be seen in major sections of this work.