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Topic: Banach fixed point theorem

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	PlanetMath: Banach fixed point theorem
		Theorem 1 (Banach Fixed Point Theorem) Every contraction has a unique fixed point.
		There is an estimate to this fixed point that can be useful in applications.
		This is version 17 of Banach fixed point theorem, born on 2002-03-07, modified 2004-02-09.
	planetmath.org /encyclopedia/BanachFixedPointTheorem.html (287 words)

	Banach fixed point theorem - Wikipedia, the free encyclopedia
		The Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points.
		The theorem is named after Stefan Banach (1892-1945), and was first stated by Banach in 1922.
		The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions.
	en.wikipedia.org /wiki/Banach_fixed_point_theorem (586 words)

	Recursive Functions (Stanford Encyclopedia of Philosophy)
		Banach [1922] has proved that a contraction on a complete metric space has a unique fixed point, and the proof is a typical iteration.
		Banach's result was obtained as an abstraction of the technique of successive substitution developed in the 19th century by Liouville, Neumann and Volterra to find solutions to integral equations, in which an unknown function appears under an integral sign.
		Either the vector vanishes on a point of the border on one of the squares, thus determining a fixed point of the given function, or there is at least one square on which the vector makes a complete turn while the point moves around the border, and the process can be started again.
	plato.stanford.edu /entries/recursive-functions (6913 words)

	Fixed-point theorem - Wikipedia, the free encyclopedia
		The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
		Every lambda expression has a fixed point, and a fixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression.
		Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.
	en.wikipedia.org /wiki/Fixed-point_theorem (472 words)

	Banach biography
		Banach's father had never given his son much support, but now once he left school he quite openly told Banach that he was now on his own.
		Banach had been on good terms with the Soviet mathematicians before the war started, visiting Moscow several times, and he was treated well by the new Soviet administration.
		There is the Hahn-Banach theorem on the extension of continuous linear functionals, the Banach-Steinhaus theorem on bounded families of mappings, the Banach-Alaoglu theorem, the Banach fixed point theorem and the Banach-Tarski paradoxical decomposition of a ball.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Banach.html (2503 words)

	20. The Initial Theory Library: An Overview
		A theory library is a collection of theories, theory interpretations, and theory constituents (e.g., definitions and theorems) which serves as a database of mathematics.
		The main interest of this section is the statement and proof of the Knaster fixed point theorem which states that on a complete partial order with a least element and a greatest element, any monotone mapping has a fixed point.
		The fundamental theorem of arithmetic and the infinity of primes are proved.
	imps.mcmaster.ca /manual/node26.html (1261 words)

	PlanetMath: proof of Banach fixed point theorem
		"proof of Banach fixed point theorem" is owned by pbruin.
		This is version 1 of proof of Banach fixed point theorem, born on 2002-11-10.
		Ok, so it seems you like to denote the fixed point by x^* as well.
	planetmath.org /encyclopedia/ProofOfBanachFixedPointTheorem.html (154 words)

	EE290n Lecture 7 Notes (Site not responding. Last check: 2007-10-10)
		(Banach fixed point theorem) If a functional process F is D-causal and the metric space is complete, then F has exactly 1 fixed point.
		Then an example of the use of the Banach theorem is presented to show how to obtain the fixed point as the limit of a sequence.
		A similar theorem applies in the case that the metric space be compact.
	ptolemy.eecs.berkeley.edu /~eal/ee290n/lec7.scribe.html (611 words)

	Colloquium, 11/16/2001 (Site not responding. Last check: 2007-10-10)
		Banach fixed point theorem, one of the most import results in the nonlinear analysis, has a broad application to both pure and applied mathematics.
		This talk will be focused on the application of Banach point theorem to differential equations, such as the existence and uniqueness of solutions, local bifurcation, etc.
		In particular, we will use Banach fixed point theorem to prove the existence of an important class of solutions, the traveling wave, for the reaction-diffusion equations that have been served as the models for many problems from sciences.
	www.math.uah.edu /colloquia/11-16-2001.html (98 words)

	5. Exercises
		Theorems include a variant of the definition in which iota does not occur, as well as the familiar condition for the existence of the limit.
		The first file proves the Bell-LaPadula ``Fundamental Theorem of Security'' in the context of the theory of an arbitrary deterministic state machine with a start state, augmented by an unspecified notion of ``secure state.'' (A similar theorem also holds for nondeterministic state machines.) The proof is by induction on the accessible states of the machine.
		For instance, the theorem that the -property is preserved under get-write is the dual of the theorem* that the simple security property is preserved under get-read; hence, it suffices to prove the latter, and the former follows automatically.
	imps.mcmaster.ca /manual/node9.html (2551 words)

	Fixed point theorem (Site not responding. Last check: 2007-10-10)
		fixed point theorem is the easiest of the lot, the one which applies to
		"fixed point" here is a point that is contained in its image.
		homology of a point (as contrasted to being convex).
	www.groupsrv.com /science/about9734.html (1935 words)

	Stefan Banach (Site not responding. Last check: 2007-10-10)
		After a few days Banach had the main idea for the required counterexample and Steinhaus and Banach wrote a joint paper, which they presented for publication.
		Banach founded modern functional analysis and made major contributions to the theory of topological vector spaces.
		Banach had been on good terms with the Soviet mathematicians before the war started, and he was treated well by the new Soviet administration.
	www.stetson.edu /~efriedma/periodictable/html/Bh.html (806 words)

	BanachFPT.html
		The Banach Fixed Point Theorem is a very good example of the sort of theorem that the author of this quote would approve.
		Tell us that under a certain condition there is a unique fixed point.
		Tell us that the fixed point is the limit of a certain computable sequence.
	www.umsl.edu /~siegel/SetTheoryandTopology/BanachFPT.html (208 words)

	The Catholic University of America - Mathematics Department
		Riemann-Stieltjes integral; equicontinuous families of functions and Arzela-Ascoli theorem; Tietze's extension theorem; Baire category theorem; differentiation and integration of a function of several variables; fixed point theorem; implicit function theorem; inverse function theorem; existence and uniqueness theorems for ordinary differential equations.
		Periodic points, fixed points, bifurcation, 1-dimensional chaos, Cantor sets, 2-dimensional chaos, dynamics of linear functions, nonlinear maps, fractals, capacity dimension, Lyapunov dimension, Julia sets and the Mandelbrot set, iterated function systems, systems of differential equations, the Lorenz system.
		Mean value theorem for Banach space-valued piecewise differentiable functions, differentiable and Lipschitzian operators, construction of epsilon-approximate solution to a differential equation, existence and uniqueness of local solutions, extensions to a maximal solution.
	math.cua.edu /courses.cfm (2191 words)

	Table of contents for Library of Congress control number 2005925884
		The Theorem of Arzela-Ascoli Convergence of sequences of functions, power series, convergence theorems, uniformly convergent sequences, norms on function spaces, theorem of Arzela-Ascoli on the uniform convergence of sequences of uniformly bounded and equicontinuous functions.
		The Convergence Theorems of Lebesgue Integration Theory Monotone convergence theorem of B. Levi, Fatou's lemma, dominated convergence theorem of H. Lebesgue, parameter dependent integrals, differentiation under the integral sign.
		The Theorem of Egorov Measurable functions and their properties, measurable sets, measurable functions as limits of simple functions, the composition of a measurable function with a continuous function is measurable, Jensen's inequality for convex functions, theorem of Egorov on almost uniform convergence of measurable functions, the abstract concept of a measure.
	www.loc.gov /catdir/toc/fy0606/2005925884.html (742 words)

	A Constructive Fixed-Point Theorem and the Feedback Semantics of Timed Systems
		In particular, any connected network of timed systems can be modeled as a single system with feedback, and the system behavior is the fixed point of the corresponding system equation, when it exists.
		Moreover, the Banach fixed-point theorem is constructive: it provides a method to construct the unique fixed point through iteration.
		The existence and uniqueness of behaviors for such systems comes from the fixed-point theorem of Priess-Crampe, but this theorem gives no constructive method to compute the fixed point.
	chess.eecs.berkeley.edu /pubs/52.html (478 words)

	Orðasafn: B
		Banach's fixed point theorem fastapunktssetning Banachs, herpingarsetning Banachs.
		bisecting point miðpunktur, = middle point, = midpoint.
		branch point kvíslipunktur, kvíslunarpunktur, = point of ramification, = ramification point.
	www.hi.is /~mmh/ord/safn/safnB.html (1233 words)

	Guests of the Algebra and Logic Group at the University of Saskatchewan (Site not responding. Last check: 2007-10-10)
		One has furthermore a generalization of this singlevalued fixed point theorem to multivalued mappings (again as it is the case in the metric situation).
		I will point out, as well, a consequence of (1) concerning the representation of forms of a given degree by linear combinations of Pfister forms of a given degree.
		While for fixed N this upper bound is primitive recursive, as a function of N it involves the notorious Ackermann function (and thus is not primitive recursive).
	math.usask.ca /fvk/alggtalk.htm (4901 words)

	Schauder's Fixed Point Theorem
		This is a theorem for all continuous functions of a certain kind - no linearity.
		The rest of the proof follows as in theorem 3.3 in 933.
		The content of this theorem is that to show a set is weakly compact, we need only look at sequences
	www.math.unl.edu /~s-bbockel1/929/node18.html (175 words)

	Schauder Fixed Point Theorem
		which does not have the fixed point property, and therefore is not compact.
		is continuous but does not have a fixed point.
		As an application of the Schauder and/or the Banach Fixed Point Theorem, consider the nonlinear integral operator
	www.math.unl.edu /~s-bbockel1/933-notes/node5.html (322 words)

	Postmodern Analysis (Site not responding. Last check: 2007-10-10)
		Theorem of Dini, upper and lower semicontinuous functions, the characteristic function of a set
		Monotone convergence theorem of B. Levi, Fatou's lemma, dominated convergence theorem of H. Lebesgue, parameter dependent integrals, differentiation under the integral sign
		Measurable functions and their properties, measurable sets, measurable functions as limits of simple functions, the composition of a measurable function with a continuous function is measurable, Jensen's inequality for convex functions, theorem of Egorov on almost uniform convergence of measurable functions
	www.mis.mpg.de /jjost/publications/postmod.html (575 words)

	Iterative Function System (Site not responding. Last check: 2007-10-10)
		Mathematically speaking (and very loosely) the theory of IFS is a corollary of the Banach Contraction Mapping Theorem (also known as the Banach-Caccioppoli fixed point theorem).
		Thus by the CMT, this map has a unique fixed point, and iterating the map on any input from the space will get you to the fixed point (it is even guaranteed to converge fairly quickly).
		The translation is, roughly, systems in general have a solution iff it has a "fixed point": a solution is a point that maps to itself under the transformation in question.
	c2.com /cgi/wiki?IterativeFunctionSystem (373 words)

	fixedpoint
		Periodic points and fixed points of semiflows of holomorphic maps
		Fixed points and nonlinear evolution equations with constraints
		Fixed points of morphisms of spaces admissible in the sense of Klee
	www.math.technion.ac.il /institute/fixedpoint.htm (586 words)

	Sample-Path Stability of Non-Stationary Dynamic Economic Systems
		We place our study within the mathematical theory of random dynamical systems and apply the concept of a random fixed point which is tailor-made for the study of the long-term behavior of sample-paths in stochastic systems.
		The main tool for the application of this approach is a Banach-type fixed point theorem for non-stationary random dynamical systems which is proved here.
		The concept and the theorem are thoroughly explained and illustrated by two examples from stochastic growth theory.
	ideas.repec.org /p/zur/iewwpx/046.html (399 words)

	MATH292_Exam1
		General theory for differential equations: nth order ODE in IR^m are equivalent to 1st order ODE in IR^{nm}, Banach fixed point theorem, existence and uniqueness results for differential equations.
		Sample Problems for Exam 1 - Please turn solutions to this list of problems in a neat and very organized way, by Wednesday, Feb. 23rd.
		Depending upon your performance, you may earn up to 10 extra points to Exam 1.
	www.math.rutgers.edu /~eteixeir/MATH292SPRING06/MATH292_Exam1.html (108 words)

	AMCA: On Banach fixed point theorems for partial metric spaces by Oscar Valero (Site not responding. Last check: 2007-10-10)
		On Banach fixed point theorems for partial metric spaces
		S. Oltra, O. Valero, Banach's fixed point theorem for partial metric spaces, preprint.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/n/c/57.htm (132 words)

	Dynamic Programming (Site not responding. Last check: 2007-10-10)
		Contraction operators and the Banach fixed point theorem.
		Optimality of bold play in subfair casinos with fixed goal.
		One-optimal solutions and their relation to the average cost criterion.
	www.math.tau.ac.il /~isaco/DynProgr.html (185 words)

	MSCS.html
		is continuous if it is continuous at every point.
		Note: It is not necessarily the case that the set of limit points of
		has no limit points but is its own closure.
	www.umsl.edu /~siegel/SetTheoryandTopology/MSCS.html (331 words)

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