Fixed point

A fixed point of a mapping $ F $ on a set $ X $ is a point $ x \in X $ for which $ F ( x) = x $. Proofs of the existence of fixed points and methods for finding them are important mathematical problems, since the solution of every equation $ f ( x) = 0 $ reduces, by transforming it to $ x \pm f ( x) = x $, to finding a fixed point of the mapping $ F = I \pm f $, where $ I $ is the identity mapping. Depending on the structure on $ X $, or the properties of $ F $, there arise various fixed-point principles. Of greatest interest is the case when $ X $ is a topological space and $ F $ is a continuous operator in some sense.

The simplest among them is the contraction-mapping principle (cf. also Contracting-mapping principle). Let $ X $ be a complete metric space and $ F: X \rightarrow X $ an operator such that

$$ \tag{1 } \rho ( F ( x),\ F ( y)) \leq \ q \rho ( x, y),\ \ 0 < q < 1. $$

Then $ F $ has precisely one fixed point $ \overline{x}\; $, which can be obtained as the limit of successive approximations $ x _ {n} = F ( x _ {n - 1 } ) $, $ n = 1, 2 \dots $ where $ x _ {0} \in X $ is arbitrary. This principle not only establishes the existence of a fixed point, but also indicates a method for finding it, and it is fairly easy to estimate the rate of convergence of the sequence $ \{ x _ {n} \} $ to $ \overline{x}\; $. The condition (1) cannot, in general, be replaced by

$$ \tag{2 } \rho ( F ( x),\ F ( y)) < \ \rho ( x, y); $$

however, if $ X $ is compact, then (2), as before, ensures that $ F $ has a unique fixed point.

More general is the generalized contraction principle. Suppose, as above, that $ X $ is a complete metric space, $ F: X \rightarrow X $ and

$$ \tag{3 } \rho ( F ( x),\ F ( y)) \leq \ q ( \alpha , \beta ) \rho ( x, y) $$

for $ \alpha \leq \rho ( x, y) \leq \beta $, where $ q ( \alpha , \beta ) < 1 $ for $ 0 < \alpha \leq \beta < \infty $. Then $ F $ has a unique fixed point. If $ X $ is a Banach space, then (1) is nothing but a Lipschitz condition for $ F $ with a constant less than 1. The contraction principle is extensively used to prove the existence and uniqueness of solutions of algebraic, differential, integral, and other equations and to find approximate solutions of them.

There are other conditions of a topological nature that guarantee the existence of a fixed point for an operator $ F $. The best known of them is Schauder's principle. Let $ X $ be a Banach space and $ F $ a completely-continuous operator mapping a bounded convex closed set $ C \subset X $ into itself. Then $ F $ has in $ C $ at least one fixed point. However, in this case the question of the number of fixed points remains open and there is no indication of a method for finding them.

Example (Peano's theorem). Let $ f ( t, x) $ be continuous in both variables in a domain $ | t - t _ {0} | \leq a $, $ | x - x _ {0} | \leq b $, and let $ \beta = \max | f ( t, x) | $ in this domain. If $ h = \min \{ a, b/ \beta \} $, then on the interval $ [ t _ {0} - h, t _ {0} + h] $ there is at least one solution of the equation

$$ \tag{4 } x ^ \prime ( t) = \ f ( t, x) $$

such that

$$ \tag{5 } x ( t _ {0} ) = x _ {0} . $$

Equation (4) together with (5) is equivalent to the integral equation

$$ x ( t) = \ x _ {0} + \int\limits _ {t _ {0} } ^ { t } f ( \tau , x ( \tau )) d \tau . $$

The operator

$$ F ( x) = \ x _ {0} + \int\limits _ {t _ {0} } ^ { t } f ( \tau , x ( \tau )) d \tau $$

maps, under the conditions of the theorem, the ball $ \| x - x _ {0} \| \leq b $ of the space $ C [ t _ {0} - h, t _ {0} + h] $ into itself and it is completely continuous on this ball. Therefore, by Schauder's principle, $ F $ has a fixed point, which is also a solution of the Cauchy problem (see [4], [5]). A generalization of Schauder's principle is Tikhonov's principle. Let $ X $ be a separable locally convex space and $ F $ a continuous operator mapping a convex compact set $ C \subset X $ into itself. Then $ F $ has in $ C $ at least one fixed point. There are also other generalizations of Schauder's principle, among them to many-valued mappings, but in all cases one has to assume that $ C $ is convex, for without this Schauder's theorem and its generalizations become false. One can combine Schauder's principle and the contraction principle. Let $ F $ be an operator that transforms a bounded closed convex set $ C $ of a Banach space $ X $ into itself and that can be represented in the form $ F = F _ {1} + F _ {2} $, where $ F _ {1} $ is completely continuous and $ F _ {2} $ contracting. Then $ F $ has at least one fixed point in $ C $.

Principles of Schauder type can be extended in the following way to non-compact operators. Let $ M $ be a bounded set in a complete metric space $ X $. The measure of non-compactness $ \chi ( M) $ of this set is defined as the greatest lower bound of those $ \epsilon > 0 $ for which there is a finite $ \epsilon $- net for $ M $( cf. Net (of sets in a topological space)). For compact sets $ \chi ( M) = 0 $. An operator $ F: X \rightarrow X $ is said to be compressing if $ \chi ( F ( M)) < \chi ( M) $ for any non-compact bounded set $ M \subset X $. Suppose that a compressing operator $ F $ transforms a bounded convex closed set $ C \subset X $ into itself. Then $ F $ has at least one fixed point in $ C $. In Banach spaces one can introduce other measures of non-compactness, and by varying them one can obtain various versions of the theorem, which make it possible to prove the existence of solutions of various differential, integral and other equations with operators that need not be completely continuous.

An appeal to more subtle topological concepts leads to stronger criteria for the existence of fixed points. Suppose that on the boundary $ \partial M $ of a bounded domain $ M $ in a Banach space $ X $ a non-degenerate vector field $ \Phi $ is given, that is, every point $ x \in \partial M $ is put in correspondence with a non-zero vector $ \Phi ( x) $. To such a field one can assign under certain conditions an integer, the so-called index (rotation) $ \gamma ( \Phi , \partial M) $ of $ \Phi $ on $ \partial M $. Suppose, to begin with, that $ X $ is finite dimensional and that $ \Phi $ is continuous on $ \partial M $. Then $ \gamma ( \Phi , \partial M) $ is defined as the topological degree of the mapping $ \Psi ( x) = \Phi ( x)/ \| \Phi ( x) \| $ of $ \partial M $ onto the unit sphere $ \| x \| = 1 $( cf. Degree of a mapping). Now let $ X $ be an infinite-dimensional Banach space and let $ \Phi ( x) = x - F ( x) $, where $ F $ is a completely-continuous operator on $ \overline{M}\; $. Such fields are called completely continuous.

Suppose that a finite-dimensional subspace $ X _ {n} $ gives a fairly good approximation to $ F ( \overline{M}\; ) $ and that $ P _ {n} $ is the projection operator of $ F ( \overline{M}\; ) $ onto $ X _ {n} $. If $ \| P _ {n} F ( x) - F ( x) \| $ is sufficiently small for $ x \in \partial M $, then the field $ \Phi _ {n} = I - P _ {n} F $ is also continuous on $ \partial M \cap X _ {n} $ and its index $ \gamma _ {n} $ does not depend on the choice of the approximating subspace $ X _ {n} $ nor on $ P _ {n} $. This number $ \gamma _ {n} $ is called the index of the completely-continuous vector field $ \Phi $ on $ \partial M $ and is denoted by $ \gamma ( \Phi , \partial M) $. An important property of the rotation is the fact that it does not change under homotopy transformations of $ \Phi $.

The Leray–Schauder principle. Suppose that on the closure $ \overline{M}\; $ of a bounded domain $ M $ in a Banach space $ X $ one is given a completely-continuous vector field $ \Phi $ that is non-degenerate on $ \partial M $ and suppose that $ \gamma ( \Phi , \partial M) \neq 0 $. Then $ \Phi $ vanishes at at least one point $ x \in M $, that is, the operator $ F $ has in $ M $ at least one fixed point. The invariance of the index under homotopy transformations makes it possible to compute the index in the following way. From the given field $ \Phi ( x) $ one constructs a family of fields $ \Phi ( x, \lambda ) $, $ \alpha \leq \lambda \leq \beta $, such that they are all homotopic to each other and $ \Phi ( x, \lambda _ {0} ) = \Phi ( x) $ for some $ \lambda _ {0} \in [ \alpha , \beta ] $. If for another $ \lambda \in [ \alpha , \beta ] $ the index of $ \Phi ( x, \lambda ) $ is easy to compute, and is $ k $, then $ \gamma ( \Phi , \partial M) = k $ too. By this device, using the degree of a mapping to establish that completely-continuous operators have a fixed point, one can prove that some fairly complicated partial differential equations of high order have solutions.

By strengthening the conditions on the space one can weaken the restrictions on the operator. For example, an operator $ F $ is called non-expansive if $ \rho ( F ( x), F ( y)) \leq \rho ( x, y) $. Suppose that the Banach space is uniformly convex (for example, a Hilbert space, cf. Banach space) and that $ F $ is a non-expansive operator taking a bounded closed convex set $ C \subset X $ into itself. Then $ F $ has in $ C $ at least one fixed point.

All preceding fixed-point principles assume the continuity of $ F $. If $ X $ is endowed with the structure of a partially ordered set, then in some cases the requirement of continuity can be dropped.

The Birkhoff–Tarski principle. Let $ X $ be a complete lattice and $ F $ an isotone operator (cf. Isotone mapping) from $ X $ to $ X $. Then $ F $ has at least one fixed point. There is another version of this principle. Let $ X $ be a conditionally-complete lattice, that is, every bounded subset in $ X $ has in $ X $ a least upper and a greatest lower bound. If $ F $ is isotone and maps the ordered interval $ [ a, b] = \{ {x } : {a \leq x \leq b } \} \subset X $ into itself, then $ F $ has in $ [ a, b] $ at least one fixed point.

A combination of topological and order conditions leads to new fixed-point principles. For example, let $ X $ be a partially ordered Banach space and $ F $ a continuous isotone operator mapping the ordered interval $ [ a, b] $ into itself. If the semi-order on $ X $ is regular, that is, if every monotone increasing order-bounded sequence $ \{ x _ {n} \} \subset X $ converges in the norm of $ X $, then $ F $ has in $ [ a, b] $ at least one fixed point. Here the conditions of the theorem do not require a lattice order on $ X $, that is, not for every pair of elements $ x, y \in X $ their sup and inf need exist in $ X $.

Finally, a fixed point of a linear operator $ F $ is an eigen element of it corresponding to the eigen value 1.

References

[1]	L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1968) (Translated from Russian) MR0539144 Zbl 0044.32501
[2]	M.A. Krasnosel'skii, "Topological methods in the theory of nonlinear integral equations" , Pergamon (1964) (Translated from Russian) Zbl 0111.30303
[3]	M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian)
[4]	V.V. Nemytskii, Uspekhi Mat. Nauk , 1 : 1 (1946) pp. 141–174
[5]	J. Leray, J. Schauder, "Topology and functional equations" Uspekhi Mat. Nauk , 1 : 3–4 (1946) pp. 71–95 (In Russian) Zbl 0060.27704
[6]	B.N. Sadovskii, "Limit-compact and condensing operators" Russian Math. Surveys , 27 : 1 (1972) pp. 85–155 Uspekhi Mat. Nauk , 27 : 1 (1972) pp. 81–146 MR0428132

Comments

See also Brouwer theorem; Lefschetz theorem.

References

[a1]	J. Dugundji, A. Granas, "Fixed point theory" , PWN (1982) MR0660439 Zbl 0483.47038

A fixed point of a fractional-linear transformation $ A $ of the closed complex plane $ \overline{\mathbf C}\; $ is a point $ \rho \in \overline{\mathbf C}\; $ for which

$$ \rho = \ \frac{a \rho + b }{c \rho + d } , $$

where

$$ A: \overline{\mathbf C}\; \rightarrow \overline{\mathbf C}\; ,\ \ z \rightarrow w = A ( z) = \ \frac{az + b }{cz + d } , $$

is the fractional-linear transformation, $ a, b, c, d \in \mathbf C $ and $ ad - bc \neq 0 $, $ \overline{\mathbf C}\; = \mathbf C \cup \{ \infty \} $. If $ A \neq I $( where $ I $ is the identity transformation $ w = z $), then $ A $ has one or two fixed points. By means of the fixed points one can classify the fractional-linear mappings (cf. Fractional-linear mapping) ( $ I $ is excluded from the discussion).

O.M. Fomenko

Comments

Fractional-linear transformations are also called Möbius transformations. For their classification according to fixed points see also [a1].

References

[a1]	H. Schwerdtfeger, "Geometry of complex numbers" , Dover, reprint (1979) MR0620163 Zbl 0484.51007

For a fixed point of a system of ordinary differential equations or of a dynamical system see Equilibrium position.