Page 1 of 1
Let (X, d) be a non-empty complete metric space and let T : X → X. Forn e N and x e X define Tºx = T(TN-1x), with Tºx =
Posted: Mon May 09, 2022 11:24 am
by answerhappygod
- Let X D Be A Non Empty Complete Metric Space And Let T X X Forn E N And X E X Define Tox T Tn 1x With Tox 1 (108.37 KiB) Viewed 26 times
Let (X, d) be a non-empty complete metric space and let T : X → X. Forn e N and x e X define Tºx = T(TN-1x), with Tºx = x. n (a) Suppose there exists m E N and c E (0, 1) such that dTMx, Tmy) < cd(x, y), m т x, y E X. * Show that there exists a unique x* e X such that Tx* = x*. = ow that any fixed point of TM is also a fixed Hint: First prove that TM has a unique fixed point. Then point of T and vice versa. (b) Show further that x* = lim Tºxo, = no for any xo E X. Hint: given n E N try writing n = km + p for some k E No and pe {0, 1, ... ,m – 1}, show that Тkmx "xp, then use this to deduce that d(xn, x*) → 0 as n → 0. Xn =