# 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

## Main Article Content

## Abstract

In this work, a fixed point x(t), t âˆˆ [a,b], a â‰¤ b â‰¤ +âˆž 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

xâ€™ = f(t,x); f: J Ã— M â†’ Rn

We aim at using the established Peanoâ€™s theorem on existence of the fixed point plus Picardâ€“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 Ã— Rn so that F: D â†’ 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 âˆˆ(a,b) and x(t0) = x0, then whenever f is bounded and D, the limits

xa

xtta()lim()+=+

xb

xttb()lim()âˆ’=

exist as finite vectors and if the point (a, x(a+)),(b, x(bâ€“)) 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.

## Article Details

*Asian Journal of Mathematical Sciences(AJMS)*,

*3*(2). Retrieved from http://ajms.in/index.php/ajms/article/view/203

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