Poincaré–Miranda theorem

(Redirected from Poincaré-Miranda theorem) Generalisation of the intermediate value theorem

In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. It says as follows:

Consider

n

continuous, real-valued functions of

n

variables,

f_{1},\ldots ,f_{n}\colon ^{n}\to \mathbb {R}

. Assume that for each variable

x_{i}

, the function

f_{i}

is nonpositive when

x_{i}=-1

and nonnegative when

x_{i}=1

. Then there is a point in the

n

-dimensional cube

^{n}

in which all functions are simultaneously equal to

0

The theorem is named after Henri Poincaré — who conjectured it in 1883 — and Carlo Miranda — who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem.

Intuitive description

A graphical representation of Poincaré–Miranda theorem for n = 2

The picture on the right shows an illustration of the Poincaré–Miranda theorem for n = 2 functions. Consider a couple of functions (f,g) whose domain of definition is (i.e., the unit square). The function f is negative on the left boundary and positive on the right boundary (green sides of the square), while the function g is negative on the lower boundary and positive on the upper boundary (red sides of the square). When we go from left to right along any path, we must go through a point in which f is 0. Therefore, there must be a "wall" separating the left from the right, along which f is 0 (green curve inside the square). Similarly, there must be a "wall" separating the top from the bottom, along which g is 0 (red curve inside the square). These walls must intersect in a point in which both functions are 0 (blue point inside the square).

Generalizations

The simplest generalization, as a matter of fact a corollary, of this theorem is the following one. For every variable x_i, let a_i be any value in the range . Then there is a point in the unit cube in which for all i:

f_{i}=a_{i}

This statement can be reduced to the original one by a simple translation of axes,

x_{i}^{\prime }=x_{i}\qquad y_{i}^{\prime }=y_{i}-a_{i}\qquad \forall i\in \{1,\dots ,n\}

where

x_i are the coordinates in the domain of the function
y_i are the coordinates in the codomain of the function.

By using topological degree theory it is possible to prove yet another generalization. Poincare-Miranda was also generalized to infinite-dimensional spaces.

References

Miranda, Carlo (1940), "Un'osservazione su un teorema di Brouwer", Bollettino dell'Unione Matematica Italiana, Serie 2 (in Italian), 3: 5–7, JFM 66.0217.01, MR 0004775, Zbl 0024.02203
Kulpa, Wladyslaw (June 1997), "The Poincaré-Miranda Theorem", The American Mathematical Monthly, 104 (6): 545–550, doi:10.2307/2975081, JSTOR 2975081, MR 1453657, Zbl 0891.47040
Dugundji, James; Granas, Andrzej (2003), Fixed Point Theory, Springer Monographs in Mathematics, New York: Springer-Verlag, pp. xv+690, ISBN 0-387-00173-5, MR 1987179, Zbl 1025.47002
Vrahatis, Michael N. (2016-04-01). "Generalization of the Bolzano theorem for simplices". Topology and its Applications. 202: 40–46. doi:10.1016/j.topol.2015.12.066. ISSN 0166-8641.
Vrahatis, Michael N. (1989). "A short proof and a generalization of Miranda's existence theorem". Proceedings of the American Mathematical Society. 107 (3): 701–703. doi:10.1090/S0002-9939-1989-0993760-8. ISSN 0002-9939.
Schäfer, Uwe (2007-12-05). "A Fixed Point Theorem Based on Miranda". Fixed Point Theory and Applications. 2007 (1): 078706. doi:10.1155/2007/78706. ISSN 1687-1812.
Ahlbach, Connor (2013-05-12). "A Discrete Approach to the Poincaré–Miranda Theorem". HMC Senior Theses.

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