Yuliya Gorb Compactness (cont.), Contraction Mapping Theorem Week 04 September 18–22, 2017 Week 04 Compactness (cont.), Contraction Mapping Theorem Yuliya Gorb Applicable Analysis I Compactness (cont.) Theorem (Hausdorff) 1 If a subset K of a metric space X is precompact then for any ε> 0 there exists a finite ε–net for K 2 If X is complete and for any ε> 0 there exists a finite ε–net for K ⊂ X then K is precompact Corollary 1: If K is precompact then it is bounded Corollary 2: A subset K of a complete metric space X is precompact iff for any ε> 0 there exists a precompact ε–net for K Corollary 3: A compact space X is separable Week 04 Compactness (cont.), Contraction Mapping Theorem Yuliya Gorb Applicable Analysis I Compactness (cont.) Definition (cover) A collection {Gα| α ∈ A} is a cover of M ⊂ X iff M ⊂ [ Gα α∈A Definition (open cover) If every Gα open then {Gα| α ∈ A} is called an open cover of M Theorem (alternative def’n of compact set) A subset K of a metric space X is compact iff every open cover of K has a finite subcover Theorem Let F be a set of all closed subsets of the closed set F . Consider a set F of all F collections {Fα| α ∈A}⊂ such that for any {Fα1 ,..., Fαn }⊂F: n α [ F i 6= ∅ (here A is some indexing set, not necessarily countable). i=1 Then F is compact if and only if for any {Fα}⊂F: [ Fα 6= ∅ α∈A Week 04 Compactness (cont.), Contraction Mapping Theorem Yuliya Gorb Applicable Analysis I Contraction Mapping Definition (contraction mapping) Let (X , ρ) be a metric space, and f : X → X . The mapping f is a contraction iff ∃ k ∈ R : 0 ≤ k ≤ 1 s.t. ρ(f (x), f (y)) ≤ kρ(x, y) ∀x, y ∈ X Clearly, contraction mapping is uniformly and Lipschitz continuous Sometimes, a map satisfying above def’n with k = 1 is called a contraction, while with k < 1 is a strict contraction Definition (fixed point of a mapping) If f : X → X , then a point x ∈ X such that f (x)= x is called a fixed point of f Week 04 Compactness (cont.), Contraction Mapping Theorem Yuliya Gorb Applicable Analysis I Contraction Mapping Theorem Theorem (contraction mapping theorem) Let (X , ρ) be a complete metric space, and f : X → X be a strict contraction. Then there exists one and only one fixed point x0 ∈ X of f (x), i.e. f (x0)= x0. Moreover, if x is any point in X and xn is defined inductively by x1 = f (x), x2 = f (x1), ... xn = f (xn−1), then xn → x0 as n →∞ Note: Condition of k < 1 is needed for uniqueness of the fixed point Corollary Let (X , ρ) bea complete metric space, and f : X → X be a function (not necessarily continuous). If for some p ∈ N the function f p is a contraction, then f has a unique fixed point Week 04 Compactness (cont.), Contraction Mapping Theorem Yuliya Gorb Applicable Analysis I References Hunter/Nachtergaele “Applied Analysis” pp. 22–27, 62–78 Naylor/Sell “Lineat Operator Theory...” pp. 142–153, 125–133 Week 04 Compactness (cont.), Contraction Mapping Theorem.