 Solution

ID: past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2017/2/a/i/solution

T : S \to S

on a nonempty complete metric space has a unique fixed point. Explicitly, if

d (T u, T v) \leq q d (u, v)

for all

u, v \in S

and one fixed

0 \leq q < 1

, then the iterates

u_{n + 1} = T u_{n}

converge to that fixed point from every

u_{0} \in S

. Both self-mapping and completeness are needed; the set need not be a vector space.

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