Brouwer Fixed Point Theorem in the General Case

In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of εn with a non empty interior. This article is based on [15].Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Agata Darmochwał. Compact spaces. Formalized Mathematics, 1(2):383-386, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in En/T. Formalized Mathematics, 12(3):301-306, 2004.Artur Korniłowicz and Yasunari Shidama. Brouwer fixed point theorem for disks on the plane. Formalized Mathematics, 13(2):333-336, 2005.Yatsuka Nakamura, Andrzej Trybulec, and Czesław Byliński. Bounded domains and unbounded domains. Formalized Mathematics, 8(1):1-13, 1999.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Karol Sieklucki. Geometria i topologia. PWN, 1979.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Mirosław Wysocki and Agata Darmochwał. Subsets of topological spaces. Formalized Mathematics, 1(1):231-237, 1990

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

We show the direct applicability of the Brouwer fixed point theorem for the existence of equilibrium points and periodic solutions for differential systems on general domains satisfying geometric conditions at the boundary. We develop a general approach for arbitrary bound sets and present applications to the case of convex and star-shaped domains. We also provide an answer to a question raised in a recent paper of Cid and Mawhin.Comment: 22 pages, 3 figure

A Brouwer fixed point theorem for graph endomorphisms

We prove a Lefschetz formula for general simple graphs which equates the Lefschetz number L(T) of an endomorphism T with the sum of the degrees i(x) of simplices in G which are fixed by T. The degree i(x) of x with respect to T is defined as a graded sign of the permutation T induces on the simplex x multiplied by -1 if the dimension of x is odd. The Lefschetz number is defined as in the continuum as the super trace of T induced on cohomology. In the special case where T is the identity, the formula becomes the Euler-Poincare formula equating combinatorial and cohomological Euler characteristic. The theorem assures in general that if L(T) is nonzero, then T has a fixed clique. A special case is a discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. Unlike in the continuum, the fixed point theorem proven here looks for fixed cliques, complete subgraphs which play now the role of "points" in the graph. Fixed points can so be vertices, edges, fixed triangles etc. If A denotes the automorphism group of a graph, we also look at the average Lefschetz number L(G) which is the average of L(T) over A. We prove that this is the Euler characteristic of the graph G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function zeta(T,z) is a product of two dynamical zeta functions and therefore has an analytic continuation as a rational function which is explicitly given by a product formula involving only the dimension and the signature of prime orbits of simplices in G.Comment: 24 pages, 6 figure

Sperner Lemma, Fixed Point Theorems, and the Existence of Equilibrium

In characterizing the existence of general equilibrium, existing studies mainly draw on Brouwer and Kakutani fixed point theorems and, to some extent, Gale-Nikaido-Debreu lemma. In this paper, we show that Sperner lemma can play a role as an alternative powerful tool for the same purpose. Specifically, Sperner lemma can be used to prove those theorems as well as the lemma. Additionally, Kakutani theorem is shown as a corollary of Gale-Nikaido-Debreu lemma. For a demonstration of the use of Sperner lemma to prove general equilibrium existence, we consider two competitive economies marked either by production goods or financial assets. In each case, we successfully provide another proof on the existence of a general equilibrium using only Sperner lemma and without a need to call on the fixed point theorems or the lemma

Existence and Uniqueness of Non-linear, Possibly Degenerate Parabolic PDEs, with Applications to Flow in Porous Media

In the branch of mathematical analysis known as functional analysis, one mainly studies functions defined on vector spaces. For partial differential equations (PDEs), this analysis has proven to be a mighty resource of understanding and modelling the behavior of the equations. Throughout this thesis, the work will focus of theory of function spaces and existence and uniqueness theorems for variational formulations in normed vector spaces. We will recast PDEs as variational problems with operators acting on normed spaces, and further seek to prove the existence and uniqueness of a solution by assigning certain properties to the operator. The outline of this thesis is as follows: In Chapter 1, we summarize the Basic Notions of Functional Analysis relevant for the later work in the thesis. We define operators, discuss monotonicity, present the theory of Sobolev spaces, and illustrate the finite element method, giving short hints to the future relevancy of the described properties. Linear Problems have been extensively studied in the past. In Chapter 2, we present three important theorems illustrating the conditions for existence and uniqueness of solutions for variational formulations of the type: (i) Galerkin formulations in Hilbert spaces: The Lax-Milgram Theorem, (ii) Petrov-Galerkin formulations in Hilbert spaces: The Babuška-Lax-Milgram Theorem, (iii) Petrov-Galerkin formulations in Banach spaces: The Banach-Nečas-Babuška Theorem, and give their proofs. Chapter 3 is dedicated to the study of Non-linear Problems. We seek to extend the ideas of the previous chapter to variational formulations containing a non-linearity b(·) depending on the solution we seek. This has a major application in the analysis of non-linear PDEs, which in general may not possess analytical solutions. To attack these types of problems, we define a weak formulation of the main problem, and discretize the domain of where a solution is sought. Next, existence and uniqueness is established through fixed point theorems, which will be given with proof. We will focus our study on two central problems: The Richards equation (a non-linear, possibly degenerate parabolic PDE) and a transport equation modelling reactive flow in porous media (two coupled PDEs). For the fully discrete (non-linear) formulation of Richards equation we show results for (i) a Lipschitz continuous non-linearity. Here we consider three cases: 3(a) First, a linearization scheme is proposed. We prove existence and uniqueness by using the Lax-Milgram Theorem in combination with the Banach Fixed Point Theorem. (b) Second, we make the assumption that the non-linearity is strongly monotone. Here, existence is proven by the Brouwer Fixed Point Theorem (c) Third, we let the non-linearity be monotone and add a regularization term to the fully discrete formulation. Here, we prove existence as in the previous step, and lastly show convergence of the regularized scheme to the fully discrete scheme. (ii) a Hölder continuous non-linearity. We give two results: (a) First, we prove existence for a monotone and bounded non-linearity. (b) Second, we state the result of existence for a strongly monotone non-linearity by the Brouwer Fixed Point Theorem. In the applications of Brouwer Fixed Point Theorem, the uniqueness of the problem is proved by assuming there exists two solutions and obtaining a contradiction through inequalities by showing estimates that can not be true. Lastly, in Chapter 4, a mathematical model of Two-phase Flow in porous media is studied. We discuss the case of a Lipschitz continuous saturation, and show for the first time a proof of existence and uniqueness of a solution for the fully discrete (non-linear) scheme, assuming the saturation to be Hölder continuous and strongly monotonically increasing. This is done by creating a regularization of the fully discrete scheme, further proving existence with the Brouwer Fixed Point Theorem, and finally showing convergence with the help of an a priori estimate.Masteroppgave i anvendt og beregningsorientert matematikkMAMN-MABMAB39

Connected Choice and the Brouwer Fixed Point Theorem

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\H{o}nig's Lemma. While we can present two independent proofs for dimension three and upwards that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upwards. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater or equal to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.Comment: 36 page

Pseudo-rotations of the open annulus

In this paper, we study pseudo-rotations of the open annulus, \emph{i.e.} conservative homeomorphisms of the open annulus whose rotation set is reduced to a single irrational number (the angle of the pseudo-rotation). We prove in particular that, for every pseudo-rotation $h$ of angle $\rho$, the rigid rotation of angle $\rho$ is in the closure of the conjugacy class of $h$. We also prove that pseudo-rotations are not persistent in $C^r$ topology for any $r\geq 0$.Comment: 25 page

An index for Brouwer homeomorphisms and homotopy Brouwer theory

We use the homotopy Brouwer theory of Handel to define a Poincar{\'e} index between two orbits for an orientation preserving fixed point free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.Comment: To appear in Ergodic Theory and Dynamical System

Constant Rank Bimatrix Games are PPAD-hard

The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash equilibrium (NE) of a rank-$0$, i.e., zero-sum game is equivalent to linear programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an FPTAS for constant rank games, and asked if there exists a polynomial time algorithm to compute an exact NE. Adsul et al. (2011) answered this question affirmatively for rank-$1$ games, leaving rank-2 and beyond unresolved. In this paper we show that NE computation in games with rank $\ge 3$, is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets, and provides a simpler proof of PPAD-hardness for NE computation in bimatrix games. In addition, we get: * An equivalence between 2D-Linear-FIXP and PPAD, improving a result by Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD. * NE computation in a bimatrix game with convex set of Nash equilibria is as hard as solving a simple stochastic game. * Computing a symmetric NE of a symmetric bimatrix game with rank $\ge 6$ is PPAD-hard. * Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP) piecewise-linear function is PPAD-hard. The status of rank-$2$ games remains unresolved