Is it true that d(tx, ty)

Question

Is it true that d(tx, ty) < d(x, y) for all x ≠ y on a complete space ⇒ t has a fixed point?

Accepted Answer

No. T(x) = x + 1/x on [1, ∞) Distances strictly decrease but no contraction constant k < 1 exists; no fixed point.

d(Tx, Ty) < d(x, y) for all x ≠ y on a complete space ⇒ T has a fixed point