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Is There A Generalization Of Brouwer's Fixed Point Theorem?

Banach fixed point theorem?

  • What are some applications of this theorem? I've already seen it in the proof of the Picard–Lindelöf theorem and the Inverse function theorem. So any applications besides those two would be good. Thanks!

  • Answer:

    The same principle underlying Picard-Lindelof can be used to prove local existence and uniqueness for the Cauchy problem for nonlinear partial differential equations (the technique goes something like the following: for a linear partial differential evolution equation, we can re-write it usually as F_t = LF where L is a linear partial differential operator. Using the Hille-Yoshida theorm, if L satisfies certain equations, L is the generator of a one parameter quasi-contration semigroup, so we can write F = e^{tL} F(0) The Picard (or Nash-Moser) iteration coupled to a priori estimates allows one to obtain a local solution. -------------------------- Through the Banach fixed point theorem (or just the spirit of the proof), I think it is also possible to generalize the concept of topological degree to maps on the Banach space. I vaguely remember someone giving a presentation about it in some seminar three years ago, but I cannot find the notes at the moment. --------------------------- The Perron-Frobenius theorm can also be generalized to operators (think of them as inifinite dimensional matrices); a theorem called Krein-Rutman. ------------------------------- Edit: can't believe I forgot about the method of continuity. See theorem 5.2 in Gilbarg and Trudinger, "Elliptic Partial Differential Equations of Second Order". I am too sleepy to copy down the theorem and its proof here. Believe me when I say that the proof uses essentially the Banach fixed point theorem. BTW, the Banach fixed point theorem is also often called contraction mapping principle, this may allow you to broaden your search.

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I thought you were asking for very specific applications of Banach's Fixed Point theorem, which unfortunately I am not particularly knowledgeable about. If you are speaking of Fixed Point Theorems in a more general context (though I am not sure how "interchangeable" they really are as each has required a different insightful proof for the different domain), however... JCS is correct that Fixed Point Theory is quite relevant to theoretical computer science. I would actually say that fixed point theory is one of the most fundamental and unifying aspects of theoretical computer science. It arises crucially in game theory (Nash's result and Kakutani's FPT), formal verification and model checking (where fixed point logics are defined), dynamical systems (obviously, as this is primarily a continuous domain in which stability notions are important), distributed computing (where Brouwer's FPT and Sperner's Lemma have an almost "automated" use in proving results about possibility or impossibility for classes of consensus problems), and with great philosophical depth in recursion theory. I imagine that the relevance of FPT to recursion theory is perhaps the *most fundamental* form, but I do not have a proof of this. :D

Gwen

Another application of Banach (and other related fixed-point theorems: you can't really separate them today, there are scores of them) is in Numerical Analysis: most classical and modern iterative methods for finding roots and solving linear equations (bisection, false position, secant, Newton, Gauss-Seidel, steepest descent methods, etc.) can be recast in a fixed-point setting and these theorems (Banach included) may then be used to prove convergence. For the classical methods, this is not the usually done, because the contraction condition is usually too strong, leading to an underestimation of convergence rates and/or too restrictive convergence conditions (for example, in the Newton method it predicts, if I remember correctly, a convergence rate that is at least liner, instead of the quadratic one), but in specialist applications, where you have to develop a method taylored to your problem, the fixed point/contracting principle approach is, in many occasions, the first line of attack to prove convergence. An enterely different application is in Theoretical Computer Science (I only know this because I work in Logic and talk regularly to people involved in TCS), on a topic called Domain Theory, that was initially developed to solve problems in programming language semantics', but today is expanding to some parts of pure mathematics (topology and geometry mainly, I think). Domain Theory started by studying continuous functions in complete posets endowed with the order topology (the said continuity is relative to this topology) and the application of a fixed-point theorem due to Tarski; but recently, this topology was strenghtened to something close to a metric and this opened the door to more powerful fixed-point theorems, Banach included. See the links below, if this piqued your curiosity.

JCS

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