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- Let (X, d) be a non-empty complete metric space and let T :X + X. For k EN nd X e X define T" x = T(Tn-1x), with Tºx =
Posted: Mon May 09, 2022 11:32 am
by answerhappygod
- Let X D Be A Non Empty Complete Metric Space And Let T X X For K En Nd X E X Define T X T Tn 1x With Tox 1 (82.46 KiB) Viewed 25 times
- Let (X, d) be a non-empty complete metric space and let T :X + X. For k EN nd X e X define T" x = T(Tn-1x), with Tºx = x. 2. (a) Suppose there exists me N and c € (0,1) such that d(T" x, T'" y)<cd(x,y), X,Y E X. Show that there exists a unique x* e X such that Tr* = x*. Hint: First prove that TM has a unique fixed point. Then show that any ficed point of TM is also a fixed point of T and vice versa. (b) Show further that z* = lim T":20, for any mo EX. Hint: given n e N try writing n = km + p for some k e N, and pe {0, 1, ..., m – 1}, show that In = Tkm (p, then use this to deduce that dxn, x*) → 0 as n +0. n00