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compactness and q=1 is enough. Picard-Lindelof. TeX |
Correction of a typo in the definition of the counterexample |
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:<math>d(x^*, x_n) \le \frac{q^n}{1-q} d(x_1,x_0)</math>
Note that the requirement d(''Tx'', ''Ty'') < d(''x'', ''y'') for all ''x'' and ''y'' is in general not enough to ensure the existence of a fixed point, as is shown by the map ''T'' : [1,∞) → [1,∞) with ''T''(''x'') =
When using the theorem in practice, the most difficult part is typically to define ''X'' properly so that ''T'' actually maps elements from ''X'' to ''X'', i.e. that ''Tx'' is always an element of ''X''.
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