On the Fixed Point Extension Results in the Differential Systems of Ordinary Differential Equations On the Fixed Point Extension Results in the Differential Systems of Ordinary Differential Equations

Abstract

In this work, a fixed point x(t), t ∈ [a,b], a ≤ b ≤ +∞ of differential system is said to be extendable to t = b if there exists another fixed point xttaccb()[],ô€€,,ô€€ of the system (1.1) below and xtxttab()=()[),ô€€, so that given the system
x’ = f(t,x); f: J × M → Rn
We aim at using the established Peano’s theorem on existence of the fixed point plus Picard–Lindelof theorem on uniqueness of same fixed point to extend the ordinary differential equations whose local existence is ensured by the above in a domain of open connected set producing the result that if D is a domain of R × Rn so that F: D → Rn is continuous and suppose that (t0,x0) is a point D where if the system has a fixed point 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()−=
exist as finite vectors and if the point (a, x(a+)),(b, x(b–)) is in D, then the fixed point x(t) is extendable to the point t = a(t = b). Stronger results establishing this fact are in the last section of this work.