Let (X, d) be a non-empty complete metric space and let T : X → X. For n e N and Xe X define T"x = T(T™-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 Xe X Define T X T T 1x With Tox 1 (42 KiB) Viewed 24 times

Let (X, d) be a non-empty complete metric space and let T : X → X. For n e N and Xe X define T"x = T(T™-1x), with Tºx = x. = (a) Suppose there exists me N and c E (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 Tx* = x*. a 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 x" = lim Tºx, 11-09 for any Xo EX Hint: given n E N try writing n= km + p for some k E No and pe {0, 1,...,m - 1}, show that xn then use this to deduce that d(x,x) 0 as n → 00. Tkm, "xpi -