In Mathématiques, for a application F of a Ensemble E in itself, an element X of E is a not fixed of F if F (X) = X .

Examples:

• in the plan, the symmetry compared to a point has admits a single fixed point: has

• the opposite application (definite on the whole of the real nonnull) admits two fixed points: -1 and 1

Graphically, the fixed points of a function F (where the variable is real) are obtained by plotting the straight line of equation there = X : all the points of intersection of the curve representative of F with this line are then the fixed points of F .

All the functions do not have necessarily a fixed point; for example, the function $x\; \backslash \; mapsto\; x+1$ does not have any, because there does not exist any real number X equal to x+1 .

Not fixes and recurring continuations

One considers the function continues $f:\; E\; \backslash \; mapsto\; E$ and (u_n) the recurring continuation defined by its initial value u₀ and by the relation of recurrence u_n+1=f (u_n) . In this case, if (u_n) converges, it necessarily does it towards a fixed point of F .

It should be noted that such a continuation does not converge inevitably , even if F has a fixed point.

Not fixes gravitational

A not fixes gravitational of an application F is a fixed point X ₀ of F such as it exists a Voisinage of $x\_0$ on which the continuation of number realities $x,\; \backslash \; F\; (X),\; \backslash \; F\; (F\; (X)),\; \backslash \; F\; (F\; (F\; (X))),\; \backslash \; \backslash \; dots$ converges towards $x\_0$.

For example, the function cosine admits a single fixed point, which is gravitational.

However, all the fixed points of a function are not necessarily gravitational. Thus, the real function $x\; \backslash \; mapsto\; x^2+x$ has a single point fixes into 0, which is not gravitational.

The gravitational fixed points are a particular case of the mathematical concept of Attracteur.

Theorems of the fixed point

There exists several Théorème S making it possible to determine that an application satisfying to certain criteria has a fixed point. Most known is the following:

That is to say E a metric Space complete provided with a distance $d$ and $f:\; E\; \backslash \; mapsto\; E$ a contracting application (i.e. there exists $k\; \backslash \; in\; such\; as\; for\; all$ (X,\; there)\; \backslash \; in\; E^2$,$ d\; (F\; (X),\; F\; (there))\; \backslash \; the\; k.d\; (X,\; there)$).\; Then$ f$has\; a\; single\; point\; fixes$ l$.$

This result makes it possible to say that any continuation of the form $u\_\; \{n+1\}\; =f\; (u\_n)$ converges towards $l$ and that $d\; (u\_n,\; L)\; \backslash \; the\; k^nd\; (u\_0,\; L)$, which makes it possible to be an estimate the speed of convergence of the continuation.

Use automatically

The Automatique consists in manufacturing systems which converge towards a point fixes (but regulated arbitrarily by the operator) and which names the not instruction .

See too

• Autorégulation
• Théorèmes of point fixes

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