Q4 (25 points) Let (X, d) be a non-empty complete metric space and let T : X X. Forn e N and x E X define T"x = T(T™-1x)

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Q4 25 Points Let X D Be A Non Empty Complete Metric Space And Let T X X Forn E N And X E X Define T X T T 1x 1 (35.66 KiB) Viewed 16 times

Q4 (25 points) Let (X, d) be a non-empty complete metric space and let T : X X. Forn e N and x E X define T"x = T(T™-1x), with Tºx = x. (a) Suppose there exists me N and c € (0, 1) such that d(T"x,T"y) Scd(x, y), x,y e X. Show that there exists a unique x* E X such that Tx" = x*. Hint: First prove that I 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), for any xo e X Hint: given n € N try writing n = km + p for some k E No and p € (0,1,...,m - 1}, show that x, = Tkm Xp, then use this to deduce that d(x,, *") → 0 as n00. + Drag and drop an image or PDF file or click to browse...