 Solution

ID: past-exam-of-the-fixed-point-methods-for-nonlinear-partial-differential-equations-course-of-the-university-of-oxford/2008/1/a/solution

d (T x, T y) < d (x, y) whenever x \neq = y

. “Strongly contractive” means a contraction mapping with one uniform constant

q \in [0, 1)

satisfying

d (T x, T y) \leq q d (x, y) for all x, y \in M

If distinct

p, q

were both fixed points, strict decrease would give

d (p, q) = d (Tp, Tq) < d (p, q)

, a contradiction. Thus there is at most one fixed point. This proves uniqueness if a fixed point exists; the pointwise strict inequality alone does not assert existence.

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