Mİlena Velba Pİcs

Local version of the fixed point theorem Banach

Exercise

<(1-k) r.

Let (E, d) a metric space complet.B = B (y, r) is the open ball

center y and radius r in ET is an application of B E

contractante.Si strictly k-d (Ty, y)

a fixed point.

(Hint: try to find a closed ball of center y, stable T)

Proof: Choose 0

Show that the closed ball D = {x: d (x, y) \\ leq r '} is stable T.

Indeed, if x is in D, then:

d (Tx, y) \\ leq d (Tx, Ty) + d (Ty, y) \\ leq kd (x, y) + (1-k) r '\\ leq r'.

Since D is complete, the existence of the fixed point of T follows the principle

Banach contraction. The