L9: Lebesgue Number and Contraction Mapping

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L9 Lebesgue Number and Contraction Mapping

• Required reading
• Lecture and lecture notes,
• Munkers (Ch26)
• Contents
• The Lebesgue number lemma
• Uniform continuity Theorem
• (Banach) Contraction Mapping Theorem

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Compactness!

• Extreme value theorem
• Theorem Let f X R where f continuous and X
  compact. Then
• PS5
• Uniform Continuity Theorem
• Contraction Mapping Fixed Point Theorem
• Extension of CMT to shrinking maps
• Today Useful lemmas
• Lebesgue number lemma
• Hints for CMT and shrinking map

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Distance from x to A X

• Let (X,d) and A X
• Distance from x X to A?
• Continuity?

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Continuity of d
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Lebesque Number Lemma

• Theorem Let be an open covering of
  (X,d) and X compact. such that
  

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Lebesque Number Lemma Proof
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Uniform Continuity Theorem Hint

• D f is continuous if
• D f is uniformly continuous if
• PS5
• Theorem f X Y is continuous and X is compact.
  Then

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Fixed points

• D let fX X. x X is a fixed point of f,
  if
• Competitive or Nash equilibrium
• Value function in macro
• Solution to the system of equations
• Fundamental questions
• Does equilibrium/solution exist?
• It is unique?

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Contraction Mapping Theorem (Steps)

• D fX X is a contraction if
• D (X,d) is complete if every Cauchy sequence
  converges
• Theorem X is non-empty and complete, f X X a
  contraction. Then
• Map of Madison put on the floor
  

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Contraction Mapping Theorem

• Theorem X is non-empty and complete, f X X a
  contraction. Then
• Example X0,1 f(x)0.50.4x
• Examples X (0,1), f(x) 10.4x, shrinking map

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Contraction Mapping Theorem (Steps) see wikipedia

• Defiene sequence
• Step 1. Auxiliary result
• Step 2. Show that is Cauchy
• Step 3. X complete converges to x
  
• Step 4. Suppose

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Hint PS5

• We know that
• Let (X,d) and fX X continuous
• F defined as is
  continuous