contraction

# contraction - Exercise 9.17(6(Contraction mapping Patrick...

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Exercise 9 . 17(6) (Contraction mapping) Patrick De Leenheer February 18, 2005 Proposition 1. Let f : I I be continuous on the closed interval I and assume that there exists 0 < α < 1 such that: x, y I : | f ( x ) - f ( y ) | ≤ α | x - y | (1) Then f is continuous. Pick x 1 I and construct the sequence { x n } as follows: x n = f ( x n - 1 ) , n > 1 . Then there is some x * I such that x n x * as n → ∞ . Moreover, f ( x * ) = x * (that is, f has a Fxed point in I ). Proof. Fix x, y I and ² > 0, and choose δ = ²/α . Then if | x - y | < δ , it follows from (1) that | f ( x ) - f ( y ) | ≤ α | x - y | and thus that | f ( x ) - f ( y ) | < αδ = ² . This implies that f is continuous on I . Pick x 1 I and construct the sequence { x n } as outlined above. We will show that { x n } is a Cauchy sequence. If we do that, then it follows that x n x * as n → ∞ for some x * . Since I is closed, there must hold that x * I . Now, since f is continuous on
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