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Proof of the Brouwer Fixed-Point Theorem: a Simple-Calculus Approach

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Proof of the Brouwer Fixed-Point Theorem: a Simple-Calculus Approach Ekonomi dan Keuangan Indonesia Vol 39, No. 3, 1991 Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach Irsan Azhary Saleh ABSTRAK Kendatipun ieorema titik-tetap Brouwer merupakan landasan penting untuk memahami model-model keseimbangan umum yang relatif telah dikenal luas, namun konsep dan pembuktian dari teorema yang bersangkutan ini kurang mendapat perhatian atau hampir tidak pernah dianalisa sebeiumnya di Indone• sia. Tulisan ini bertujuan untuk membuktikan teorema titik tetap Brouwer dengan menggunakan pendekatan kalkulus, suatu metode pembuktian yang relatif sangat sederhana jika dibandingkan dengan metode-metode lain yang secara konvenstonal telah sering digunakan, yakni antara lain misalnya: teori homologi, bentuk-bentuk diferensial, argumen kombinatorial, serta topologi geometris. 217 1 Saleh 1. INTRODUCTION The Brouwer fixed-point theorem is one of the most important results in modern mathematics. It is easy to state but hard to prove. The statement is easy enough to be understood by anyone, even by someone who cannot add or substract. This is in essence what the Brouwer theorem says in everyday terms: Think of a person is sitting down with a cup of coffee. Gently and continuously he or she swtrls the coffee about in the cup. He or she then puts the cup down, and lets the motion subside. When the coffee is still, Brouwer says there is ai least one point in the coffee that has returned to the exact spot in the cup where it was when the person first sat down. Regardless of its simplicity in this everyday terms, however, the proof of the Brouwer theorem has been usually difficult that it has 'been taught only in graduate courses on topology. The Brouwer theorem has many applications in various mathe• matical disciplines. In mathematical analysis, as an example, it is an essential tool in the theory of ordinary differential equations. In its extension by Schauder and Leray, the fixed-point theorem is used to establish previous• ly unattainable results in the theory of nonlinear partial differential equations and integral equations, Browder (1967), Brezis and Browder (1975). In mathe• matical economics, as another example, in its primitive application the Brouwer theorem is used to prove the existence of an equilibrium in a pure exchange economy. Although this type of an economy is very simplistic and unrealistic, the existence proof contains the essential features of arguments for existence in more general equilibrium models. One of the famous interesting applications of the Brouwer theorem outside of topology is a proof of the minimax theorem of game theory given by von Neumann in 1928 (see Nikaido, 1970, Chapter 7). It should be noted that in terms of the last two examples the Brouwer fixed-point theorem, as it was originally stated in I9I0 and commonly proved, is a pure existence theorem. The theorem, however, was later put in a more useful form. Extensions in this direction were initiated by von Neumann (1938) in connection with general economic equilibrium model, which is often considered as the first sophisticated application of the Brouwer fixed-point theorem to an economic problem. Somewhat later, Kakutani (I94I) generalized the Brouwer theorem to correspondences, which not only allows numerous applications in economic theory and its related fields but also has stimulated further extensions by authors, such as: Eilenberg and Montgomery (1946) working on multivalued transformation which relies on algebraic topological notions; Nash (1950) proving the existence of equilibria for games where the players' preferences are representable by continuous quasi-concave utilities and the strategy sets are simplexes; Debreu (1952) proving the existence of equilibrium for abstract economics; Arrow and Debreu (1954) proving the existence of Walrasian equilibrium of an economy; Moore (1968) replacing the term semi-continuity with hemi-continuity in 218 Proof of the Brouwer Fixed-Point Theorem: A Simple-Calculus Approach referring to correspondences, which helps to avoid confusion with semi-conti• nuity of real-valued functions; and, even further, less than two decades ago. Scarf (1973) who has developed a remarkable algorithm that can easily be implemented on computers to give a method of approximating a fixed-point for a continuous mapping of a price vector into itself. This fixed-point algo• rithm, for practical purposes and policy analysis, has contributed significantly in solving complex and computable general equilibrium models. Comparable to or in point of Scarf algorithm are, among others, works due to Kuhn (1969), MacKinnon (in Karamardian, 1977), and Todd (1980). | In Indonesia it seems to be evident that, for the last few years, there has been a growing interest in the methods of general equilibrium analysis, in particular with regard to the topics of computable general equilibrium model. Such an interest is relevant to the current tendency of globalization at almost any level of the economy, because via general equilibrium approach one can explore how to solve complex models which simulate many of the simul• taneous interdependencies which exist in an economy. Of course, there are many situations in which the theoretically and computationally simpler methods of partial equilibrium analysis are appropriate. However, there are certain problems where the income and other-price effects operate to shift the supply and demand curves used in partial equilibrium analysis and where these feedback effects are the central feature of interest. Examples of this are the formation of a tariff customs union or a change in the tax on one major production input such as capital. In this type of problem, a general equi• librium model is appropriate because we are interested in what effects the policy change will have on many different markets and because what happens in other markets crucially affects any one market of interest. Furthermore, it is a characteristic of such policy changes that they are not marginal so that linear approximation (as in economies with leontief technologies) is a poor way to specify relevant demand and supply functions, Cornwall (1984). We are now already aware that Brouwer fixed point theorem (and understanding its proof) is a basic mathematical tool used in showing the existence of solution concepts in game theory and economics, especially in regard to general equilibrium analysis. Unfortunately, the exposition on this, at least in Indonesia, seems to be quite limited and rather abandoned. Therefore, as the title reflects, the purpose of this paper is to give a relative nonrigorous note for understanding the proof of the Brouwer fixed-point theorem. While there are many excellent readings available on the topic we are discussing about, most of them are inaccessible to a typical well-trained economist. Generally speaking, the reason, why proofs of the Brouwer fixed- point theorem have been difficult because they have used combinatorial argu• ments, differential forms, homology theory, or geometric topology. That is why, in this paper, we are taking a somewhat significantly different direction; all we need to now is just calculus, plus a supporting tool of analysis called contraction-mapping principle which is going to be discussed in Section II. In Section III, we can then proceed with the formal proof of the Brouwer 219 Saleh theorem. John Milnor (1978) deserves the full credit for his initiating work on using calculus for the proof of the theorem. The bulk of Section III is, in essence, a recast of Milnor's article which is only four pages long! So, what the author is trying to do is to rewrite out Milnor's original article in great detail and, hopefully, in much less rigorous fashion. II. CONTRACTIONS MAPPING PRINCIPLE Fixed-point theorem is a one that refers to an equation x = /(x). (1) Usually, the theorem gives conditions for the existence of a solution. The function / may be thought of as a mapping. Then a solution x is a point that the mapping leaves fixed. There are many varieties of fixed point theorems. Some gives conditions for uniqueness or multiplicity of solutions. Some fixed-point theorems are constructive; most are not. The Brouwer theorem is not. It just tells us a solution exists; it is up to us to find it. If it is true that equation (1) is a fixed-point equation, one can then infer that every equation is a fixed point equation. The reason is simple. Suppose this equation is given: ^(x) = 0. Then we can write the fixed-point equation x = x + g(x) (2) or, if we prefer, X = x-83s(x). In general we can write the equation ^(x) = 0 cis the fixed-point equation x = x + (j>[g{x)] / (3) provided that <j)[g] = 0 iff g = 0. So when we talk generally about fixed- point equations, we are talking about all equations. The contraction mapping principle is an exception among fixed-point theorems: it is constructive. We can use the method of proof to construct a solution. The method is called the method of successive approximations. In its elementary form, the contraction mapping theorem says this: Let M be a closed set of real numbers, and suppose the function / maps M into itself. Finally, suppose / is a contraction mapping: \fia)-fm<&\a-b\, (4) 220 Proof of the. Brouwer Fixed-Point Theorem: A Simple-Calculus Approach where 0 <6 <1. Conclusion: The mapping / has a unique fixed point in Af. Uniqueness is easy to prove. If there were two fixed points, a and 6, then the contraction property implies |a-6| = |/(a)-/(6)|<%-6|. (5) Then a - 6 = 0 because 0 < ^ < 1. To prove the existence of a fixed point, we will use an iterative scheme. Start with any XQ in M, and compute the successive approximations ^n + i = /K) (n = 0,l,2,...). (6) We will prove that the sequence converges. That will suffice. If x^-x x, then the limit x must lie in M because all the x„ lie in AI and M is closed {M contains all its limit points).
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