The Fixed Point Equation (FPE): S[ψ]=ψ on the Linear Algebra A ψ element in some Linear Algebra A

Transcription

1 The Fixed Point Equation (FPE): S[ψ]=ψ on the Linear Algebra A ψ element in some Linear Algebra A S: A A a Map on A 0 is the null element of A The elements ψ of A which satisfy the FPE are called the Fixed Points of the map S The set Y has the Fixed Point Property for a (suitably selected) class of maps Every map in the class has a FP. Typical Example: The Topological Space Y has the Fixed Point Property Every Continuous map has a Fixed Point

2 The Real Function S(x) = x 2 3x + 4 has 2 as FP: S(2)=2 1. The FP is the point (x,s(x)) on the line y = x the graph of S has a point in common with the line y = x. 2. Τhe sequence of iterations ψ n =S n ψ converges to the FP

3 Attractive fixed points The fixed point iteration x n+1 = cos x n with initial value x 1 = -1.

4 Not all fixed points are attractive: for example, x = 0 is a fixed point of the function S(x) = 2x, but iteration of this function for any value other than zero rapidly diverges. There may be Attracting sets of points but not Attracting FP: Ex. x n+1 =S[x n ]=3,1 x n (1- x n ) Limit Cycle, Periodic Attractor with period 2

5 The τ-periodic points are the Fixed points of the map S τ, τ=1,2,3,... There may be no Attracting sets: Ex. xn+1 =S[xn]= 4 xn (1- xn)

6 The resulting iterations diverge Not all functions have fixed points: Example: The Real Function S(x) = x + 1 has no fixed points, since x is never equal to x + 1 for any real number. The graph and the line are a pair of parallel lines.

7 Fixed Point Theorems FPT specify conditions which guarantee existence, uniqueness and construction of FP of Maps FPT are amongst the most generally useful in mathematics. Usually FPT are stated as follows: Given the FPE: S[ψ]=ψ on the Linear Algebra A: 1) Find a Topology on A such that the FPE has a solution ξ in some Domain 2) Find if the FP ξ is unique 3) Find if the FP ξ is Attractor = Attractive FP and the Region of Accessibility=Domain (Basin) of Attraction Α(ξ) Α(ξ)={ψ in A lim n S n ψ = ξ } Τhe Recursion Relation (Αναδρομικη Σχεση) of 1 st order: S[ψ n ]=ψ n+1, n=0,1,2,... is the Iteration Formula (Επαναληπτικος Τυπος) defined by the Map S: A A 4)Εstimate the accuracy of each step of the approximation

8 FPT in Finite dim VS FPT1 Closed Intervals 1)The Continuous function S: [-1,1] [-1,1] has at least one FP ξ in [-1,1]. Proof Apply Weierstrass Mean Value Theorem to the Continuous Function g(y) = S(y) y for which g(-1) 0, g(1) 0. There is a point ξ in [-1,1], such that: g(y) =0. [-1,1] is a Fixed Point TS 2)The closed interval [0,1] is a Fixed Point TS Proof Let S:[0,1] [0,1] be a continuous map. If S(0) = 0 or S(1) = 1, then our map has a fixed point at 0 or 1. If not, then S(0) > 0 and S(1) < 1. Thus the function g(x) = S(x) x is a continuous real valued function which is positive at x = 0 and negative at x = 1. By the intermediate value theorem, there is some point ξ with g(ξ) = 0, ξ is a fixed point of S. 3)The open interval does not have the fixed point property. Proof The map S(x) = x 2 has no fixed point on the interval (0,1).

9 FPT2 Closed Disc The Continuous function S: has at least one FP ξ in, ={(x,y) R 2 : x 2 +y 2 1} the closed unit disc. is a Fixed Point TS Proof [Courant, Robins 1941, What is Mathematics, Oxford, p 241]

10 FPT3 Brower Closed Sphere on Euclidean Space The Continuous function S: has at least one FP ξ in, = { y R N : y 1} the closed unit sphere in R N. is a Fixed Point TS Proof The case N = 3 Bohl P. 1904, Über die Bewegung eines mechanischen Systems in die Nähe einer Gleichgewichtslage, Journal Reine Angewandte Mathematik 127, Brower L. 1911, Über die Abbildung von Mannigfaltigkeiten, Math. Ann. 71, The general case : Hadamard J.1910, Note sur quelques applications de l indice de Kronecker in Jules Tannery: Introduction à la théorie des fonctions d une variable (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp Brouwer L. 1911, Über Abbildungen von Mannigfaltigkeiten, Math. Annalen 71, pp Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals.

11 Methods to construct (approximations to) fixed points guaranteed by Brouwer's theorem: Karamadian S (ed.), Fixed points. Algorithms and applications, Academic Press, Istrăţescu V.I. 1981, Fixed point theory, Reidel Many Proofs have been proposed The properties involved (continuity, being a fixed point) are invariant under homeomorphisms, therefore the theorem equally applies if the domain is not the closed unit ball itself but some set homeomorphic to it (closed, bounded, connected, without holes) Non-Constructive Proof 0.5 Constructive Proof 0.5 Remark Brower FPT on the open unit disk is false. Example: The map S(x,y) = ( x+ 1 y2 2 ) y maps every point of the open unit disk in R 2 to another point of the open unit disk slightly to the right of the given one. Aσκ {0.3}

12 Game Theory began with the idea that mixed-strategy equilibria in two-person zero-sum games exist and its proof by Von Neumann using Brouwer's Fixed-Point Theorem on continuous functions into compact convex sets Von Neumann J., Morgenstern O. 1944, Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey Proof {Aσκ 2}

13 FPT in Infinite-dimensional VS generalise Brower FPT Brower FPT straightforward generalization to infinite dim VS, is not possible. Because using the unit ball of an arbitrary Hilbert space instead of Euclidean space, Because the unit spheres of infinite-dimensional Hilbert spaces are not compact. For example, in the Hilbert space l 2 of square-integrable real (or complex) sequences, consider the map S : l 2 l 2 which sends a sequence (x n ) from the closed unit ball of l 2 to the sequence (y n ) defined by The map S : l 2 l 2 is continuous, has its image in the unit sphere of l 2, but does not have a fixed point. {Ασκ 1} The generalization of the Brouwer fixed point theorem to infinite dimensional spaces should include a compactness assumption

14 FPT4 Schauder Every Convex Compact subspace of a Banach Space Y, is a Fixed Point TS Proof Comments Schauder FPT was foreshadowed by Birkhoff and Kellog 1922, Trans. Am. Math. Soc. 23,

15 FPT in Topological Spaces FPT Banach Contraction Map (Y, d) be a non-empty complete metric space. S : Y Y be a contraction map on Y, i.e: there is a real number 0< α < 1 such that d(sx,sy) α d(x,y), for all x, y in Y. α is called the contraction constant of S Then 1) the map S admits a unique fixed point ξ in Y. 2) lim n S n ψ = ξ ξ is Attractive FP with basin the whole space Y 3)The speed of convergence is estimated by: d(s n ψ, ξ) αn d(sψ, ξ) 1 α d(s n+1 ψ, ξ) α 1 α d(sn ψ, ξ) d(s n+1 ψ, ξ) α d(s n ψ, ξ) The smallest contraction constant α is called the Lipschitz constant.

16 Proof uniqueness of the PF Let ξ, η be distinct FP of S: S[ξ] = ξ, S[η] = η. Then: d(ξ,η) = d(s[ξ],s[η]) α d(ξ,η) therefore: ξ = η. Aτοπον The sequence ψ n =S n ψ, is Cauchy sequence, therefore ψ n =S n ψ converges in Y d(s m ψ, S n ψ) d(s m ψ, S m-1 ψ) + d(s m-1 ψ, S n ψ) d(s m ψ, S m-1 ψ) + d(s m-1 ψ, S m-2 ψ)+...+ d(s n+1 ψ, S n ψ)= (α m-1 + α m α n ) d(sψ, ψ)= (Lemma 1) d(sψ, ψ) an d(sψ, ψ) (Lemma 2) 1 α Since α<1, d(s m ψ, S n ψ) 0, therefore ψ n = S n ψ, n=0,1,2,... Cauchy sequence = an α m 1 α The limit of the sequence ψ n =S n ψ, is a FP of S Let ξ= lim n S n ψ S[ξ]= S[ lim n S n ψ]= lim n S[S n ψ] = lim n [S n+1 ψ] = ξ, (Lemma 3)

17 Lemma 1: d(s n+1 ψ, S n ψ) α n d(sψ, ψ), n=1,2,3,... Βy induction: n=1: d(s 2 ψ, Sψ) α d(sψ, ψ), true from the definition of the contraction Suppose: d(s k+1 ψ, S k ψ) α k d(sψ, ψ) We shall show that: d(s k+2 ψ, S k+1 ψ) α k+1 d(sψ, ψ) Indeed: d(s k+2 ψ, S k+1 ψ) α d(s k+1 ψ, S k ψ) α α k d(sψ, ψ) = α k+1 d(sψ, ψ) Lemma 2: (α m-1 + α m α n )(1-α) = α n - α m Lemma 3 Every Contraction S is Uniformly Continuous

18 Comments 1.Τhe requirement d(sx, Sy) < d(x, y) for all unequal x and y is in general not enough to ensure the existence of a fixed point. Example the map S : [1, ) [1, ) with S(x) = x + 1/x, which lacks a fixed point. 2.Search for a distance d for which X complete, S contraction Converse of the Banach contraction Map [Czesław Bessaga 1959] Let S : X X be a map of an abstract set X, such that each iterate S n has a unique fixed point. Let α be a real number, 0 < α < 1. Then there exists a complete metric d on X such that S is contractive, and α is the contraction constant. 3.Construction of the FP 4.Metric Spaces are suitable structures for the study of Approximation Problems Παραδειγμα συναρτησης Συστολης ως προς την Ευκλειδια Αποσταση που δεν είναι συστολη ως προς άλλη αποσταση 0.5

19 Παραδειγμα συναρτησης που δεν είναι συστολη ως προς την Ευκλειδια Αποσταση και είναι Συστολη ως προς άλλη αποσταση Banach FPT is the key to the proof of the Picard- Lindelof Theorem about the existence and uniqueness of solutions to certain Ordinary Differential Equations. The sought solution of the ODE is expressed as a fixed point of a suitable integral operator S which transforms continuous functions into continuous functions. The Banach fixed point theorem is then used to show that this integral operator has a unique fixed point. The ODE dy dt = Φ(y(t), t), y(t 0) = y 0 has exactly one solution if Φ is Lipschitz continuous in y, continuous in t as long as y(t) stays bounded. ψ 0 (t)=y 0 t ψ n+1 (t) = ψ 0 + dx Φ( t 0 ψ n (x), x) = S[ψ n ](t) The Picard sequence ψ n, converges to the unique solution of the ODE

20 6. S n may be a Contraction, even if S is not a Contraction ie, S may have Periodic Points but not Fixed Points Example: Y= the class of Continuous maps on [a,b] x Sψ(x) = (Sψ)(x)= dt ψ a S n ψ(x) = S = b-a S n = (b a)n n! 1 x (n 1)! dt a ψ(t) (x t) n 1 (t), x in [a,b], a contraction for large enough n. where S = sup { Sy, y =1}, the Norm of the Map S Find the FP of (the selected) S n {Ασκ 0.3}

21 Convergent Sequence: Knowledge of the Limit required, Increasing Accuracy Cauchy Sequence: Knowledge of the Limit not required, Increasing Precision Brown R. F., Furi M., Górniewicz L., Schauder J., Jiang B., Editors 2005, Handbook of Topological Fixed Point Theory, Springer, Dordrecht, The Netherlands

22 FPE and Differential Equations Once a differential equation has been formulated as a FPE, numerical methods that search for fixed points in a function space can be used. [D. R. Smart 1974, Fixed Point Theorems, Cambridge University Press, New York, Ch. 6]. Interval techniques may also be used to bound the solution of a FPE. [R. E. Moore 1966, Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, NJ] Krasnoselskii's FPT is useful in differential equations [J. Franklin 1980, Methods of Mathematical Economics, Springer-Verlag, New York, p. 277]. Consider the fixed-point equation x = f(x) + g(x) for x in a Banach space X. Let X be a non-empty closed convex set in B. f: X X continuous map, with compact range f[x]. g: X X contraction map (the range of g need not be compact). If y + g(x) is in X for y in f[x] and x in X, then there is a fixed point of the map S(x) = f(x) + g(x).

23 5. FPT in ODE Picard-Lindeloff Method Picard Iteration as Contraction Map T. A. Burton 1985, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, New York, Chapter 3 (pages ). J. K. Hale 1963, Oscillations in Nonlinear Systems, McGraw-Hill Book Company, New York, Appendix (pages ). P. Hartman 1964, Ordinary Differential Equations, John Wiley & Sons, New York, Ch. 12 ]

24 6. FPT in PDE E. L. Allgower 1977, "Application of a Fixed Point Search Algorithm to Nonlinear Boundary Value Problems Having Several Solutions," in S. Karamardian (ed.), Fixed Points: Algorithms and Applications, Academic Press, New York. I. Stakgold 1979, Green's Functions and Boundary Value Problems, John Wiley &; Sons, New York, pages