Poincaré–Miranda theorem

{{Short description|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 {{mvar|n}} functions in {{mvar|n}} dimensions. It says as follows:

::Consider $n$ continuous, real-valued functions of $n$ variables, $f_1,\ldots, f_n\colon [-1,1]^n\to \R$ . 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 $[-1,1]^n$ 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.{{Citation |last=Miranda |first=Carlo |title=Un'osservazione su un teorema di Brouwer |url= |journal=Bollettino dell'Unione Matematica Italiana |volume=3 |pages=5–7 |year=1940 |series=Serie 2 |language=Italian |jfm=66.0217.01 |mr=0004775 |zbl=0024.02203 |author-link=Carlo Miranda}}{{citation |last=Kulpa |first=Wladyslaw |title=The Poincaré-Miranda Theorem |date=June 1997 |journal=The American Mathematical Monthly |volume=104 |issue=6 |pages=545–550 |doi=10.2307/2975081 |jstor=2975081 |mr=1453657 |zbl=0891.47040}}{{Rp|page=545}}{{citation |last1=Dugundji |first1=James |title=Fixed Point Theory |pages=xv+690 |year=2003 |series=Springer Monographs in Mathematics |place=New York |publisher=Springer-Verlag |isbn=0-387-00173-5 |mr=1987179 |zbl=1025.47002 |last2=Granas |first2=Andrzej |author-link1=James Dugundji}} It is sometimes called the Miranda theorem or the Bolzano–Poincaré–Miranda theorem.{{Cite journal |last=Vrahatis |first=Michael N. |date=2016-04-01 |title=Generalization of the Bolzano theorem for simplices |url=https://www.sciencedirect.com/science/article/pii/S0166864115005994 |journal=Topology and its Applications |language=en |volume=202 |pages=40–46 |doi=10.1016/j.topol.2015.12.066 |issn=0166-8641}}

Intuitive description

{{Plain image with caption|image=PoincareMiranda.png|caption=A graphical representation of Poincaré–Miranda theorem for {{math|n {{=}} 2}}|width=350px|align=right|caption position=bottom|triangle=triangle}}

The picture on the right shows an illustration of the Poincaré–Miranda theorem for {{math|n {{=}} 2}} functions. Consider a couple of functions {{math|(f,g)}} whose domain of definition is {{math|[-1,1]{{sup|2}}}} (i.e., the unit square). The function {{mvar|f}} is negative on the left boundary and positive on the right boundary (green sides of the square), while the function {{mvar|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 {{mvar|f}} is {{math|0}}. Therefore, there must be a "wall" separating the left from the right, along which {{mvar|f}} is {{math|0}} (green curve inside the square). Similarly, there must be a "wall" separating the top from the bottom, along which {{mvar|g}} is {{math|0}} (red curve inside the square). These walls must intersect in a point in which both functions are {{math|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 {{math|x{{sub|i}}}}, let {{math|a{{sub|i}}}} be any value in the range {{math|[sup{{sub|x{{sub|i}} {{=}} 0}} f{{sub|i}}, inf{{sub|x{{sub|i}} {{=}} 1}} f{{sub|i}}]}}.

Then there is a point in the unit cube in which for all {{mvar|i}}:

: $f_i=a_i$ .

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

: $x^\prime_i=x_i\qquad y^\prime_i=y_i-a_i\qquad \forall i\in\{1,\dots,n\}$

where

{{math|x{{sub|i}}}} are the coordinates in the domain of the function
{{math|y{{sub|i}}}} are the coordinates in the codomain of the function.

By using topological degree theory it is possible to prove yet another generalization.{{Cite journal |last=Vrahatis |first=Michael N. |date=1989 |title=A short proof and a generalization of Miranda's existence theorem |url=https://www.ams.org/proc/1989-107-03/S0002-9939-1989-0993760-8/ |journal=Proceedings of the American Mathematical Society |language=en |volume=107 |issue=3 |pages=701–703 |doi=10.1090/S0002-9939-1989-0993760-8 |issn=0002-9939|doi-access=free }} Poincare-Miranda was also generalized to infinite-dimensional spaces.{{Cite journal |last=Schäfer |first=Uwe |date=2007-12-05 |title=A Fixed Point Theorem Based on Miranda |journal=Fixed Point Theory and Applications |language=en |volume=2007 |issue=1 |pages=078706 |doi=10.1155/2007/78706 |issn=1687-1812|doi-access=free }}

Poincaré–Miranda theorem

Intuitive description

Generalizations

See also

References

Further reading