> Let (X, d) be a non-empty complete metric space and let T :X + X. For n e N and x e X define Tnx = T(TN-1x), with Tºx

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Let X D Be A Non Empty Complete Metric Space And Let T X X For N E N And X E X Define Tnx T Tn 1x With Tox 1 (114.94 KiB) Viewed 18 times

> Let (X, d) be a non-empty complete metric space and let T :X + X. For n e N and x e X define Tnx = T(TN-1x), with Tºx = x. (a) Suppose there exists m E N and c E (0, 1) such that d(T"x, Tmy) < cd(x, y), X, y E X. Show that there exists a unique x* E X such that Tx* = x*. Hint: First prove that Tm has a unique fixed point. Then show that any fixed point of TM is also a fixed point of T and vice versa. (b) Show further that ** lim Tºxo, n-> 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 Xn = Tkm Xp, then use this to deduce that d(xn, x*) → 0 as n → 0.