- Q4 25 Points Let X D Be A Non Empty Complete Metric Space And Let T X X For N E N And I E X Define T T T 12 W 1 (39.65 KiB) Viewed 31 times
Q4 (25 points) Let (X, d) be a non-empty complete metric space and let T:X + X. For n e N and I e X define T"=T(T-12), with Tºr=. (a) Suppose there exists m € N and CE (0,1) such that dT"2,Tmy)<cd(2,y), y eX. Show that there exists a unique r* e X such that Tz* = 2*. 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 T" 10, 72-00 for any Do EX = Hint: given n € N try writing n=km + p for some k e N, and pe {0, 1,...,m-1}, show that In = Timp then use this to deduce that dIn, 2*) as n +0.