SOLVED: Question 3. (5 marks) Let C([a, b]; R) be the space of all continuous functions on [a, b], where 0 < a < b, with the metric ||f - g|| = maxa≤t≤b |f(t) - g(t)|. For each f ∈ C([a, b]; R), define the map F(f) byF(f)(c) = ∫[a,b] f(t) dt, for all x ∈ [a, b]. (65 ≤ a ≤ b)Prove that there is a unique fixed point of F in the space C([a, b]; R); i.e. there is a unique f ∈ C([a, b]; R) such that f = F(f). (You can assume that (C([a, b]; R), ||.||) is a complete metric space).
SOLVED: Question 3. (5 marks) Let C([a, b]; R) be the space of all continuous functions on [a, b], where 0 < a < b, with the metric ||f - g|| = maxa≤t≤b |f(t) - g(t)|. For each f ∈
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