Degree of reproduction

The degree of mapping is an aid in nonlinear analysis to prove the existence of solutions to nonlinear equations . With its help, one can, for example , prove Brouwer's Fixed Point Theorem, Borsuk-Ulam's theorem, or the Jordanian curve theorem. In the finite dimensional (for continuous functions) it is called Brouwer's degree of mapping; its extension to Banach spaces (for compact perturbations of identity) is called the leray-schauderscher degree of representation. ${\displaystyle f(x)=y}$

Brouwer's degree of representation

Brouwer's degree of mapping, named after LEJ Brouwer , assigns an integer to a continuous function for open, limited and given . The decisive factor for the applications is the fact that the equation can already be solved when the degree of mapping is different from zero. If the degree of mapping disappears , no statement can be made about solvability. ${\displaystyle f\colon {\overline {\Omega }}\subset \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle y\in \mathbb {R} ^{n}\setminus f(\partial \Omega )}$ ${\displaystyle d(f,\Omega ,y)}$ ${\displaystyle f(x)=y}$ ${\displaystyle d(f,\Omega ,y)}$ ${\displaystyle d(f,\Omega ,y)}$

Axiomatic definition

Brouwer's degree of mapping is a function

${\displaystyle d\colon \{(f,\Omega ,y)\ |\ \Omega \subset \mathbb {R} ^{n}\ \mathrm {offen,beschr{\ddot {a}}nkt} \ ,\ f\colon {\overline {\Omega }}\rightarrow \mathbb {R} ^{n}\ {\textrm {stetig}}\ ,\ y\in \mathbb {R} ^{n}\setminus f(\partial \Omega )\}\rightarrow \mathbb {Z} }$

with the following characteristics:

• ${\displaystyle d(\mathrm {id} _{\overline {\Omega }},\Omega ,y)=1}$for everyone . ${\displaystyle y\in \Omega }$
• Disassembly property:

${\displaystyle d(f,\Omega ,y)=d(f,\Omega _{1},y)+d(f,\Omega _{2},y)}$if there are disjoint open subsets of such that . ${\displaystyle \Omega _{1},\Omega _{2}}$ ${\displaystyle \Omega }$ ${\displaystyle y\not \in f({\overline {\Omega }}\setminus (\Omega _{1}\cup \Omega _{2}))}$

• Homotopy invariance:

${\displaystyle t\mapsto d(F(t,\cdot ),\Omega ,y(t))}$is constant with respect to if and are continuous with for all and . ${\displaystyle t\in [0,1]}$ ${\displaystyle F\colon [0,1]\times {\overline {\Omega }}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle y\colon [0,1]\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle y(t)\not =F(t,x)}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle x\in \partial \Omega }$

One can show that such a function exists and that it is unique.

Important properties of Brouwer's degree of reproduction

• Is , then the equation is solvable. ${\displaystyle d(f,\Omega ,y)\neq 0}$ ${\displaystyle f(x)=y}$ ${\displaystyle \Omega }$

• If with then applies In particular, the degree of mapping is clearly defined by the values . ${\displaystyle g\in C({\bar {\Omega }})}$
  ${\displaystyle \max\{|f(x)-g(x)|\,\colon x\in \partial \Omega \}<\mathrm {dist} (y,f(\partial \Omega )),}$
  ${\displaystyle d(f,\Omega ,y)=d(g,\Omega ,y).}$
  ${\displaystyle \partial \Omega }$

• If and are in the same connected component of , then we write briefly for to indicate that the degree of mapping does not depend on the point but on the component. ${\displaystyle y_{1}}$ ${\displaystyle y_{2}}$ ${\displaystyle Z}$ ${\displaystyle \mathbb {R} ^{n}\setminus f(\partial \Omega )}$ ${\displaystyle d(f,\Omega ,y_{1})=d(f,\Omega ,y_{2}).}$
  ${\displaystyle d(f,\Omega ,Z)}$ ${\displaystyle d(f,\Omega ,y)}$

• Let and be continuous and the bounded connected components of as well as , then the Leray product formula holds in which only finitely many summands are different from zero. ${\displaystyle f\colon {\overline {\Omega }}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle g\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle K_{i}}$ ${\displaystyle \mathbb {R} ^{n}\setminus f(\partial \Omega )}$ ${\displaystyle y\in \mathbb {R} ^{n}\setminus (g\circ f)(\partial \Omega )}$
  ${\displaystyle d(g\circ f,\Omega ,y)=\sum _{i}d(f,\Omega ,K_{i})\cdot d(g,K_{i},y),}$
  

Representations of the degree of mapping

• If, in addition, is continuously differentiable and all points in are regular , i.e. the determinant of the Jacobian matrix is not zero at these points , then the following applies: Is not continuously differentiable, then one can choose a function that has the same degree of mapping as Has. ${\displaystyle f}$ ${\displaystyle \Omega }$ ${\displaystyle f^{-1}(y)}$ ${\displaystyle J(f)(x)}$ ${\displaystyle x\in f^{-1}(y)}$
  ${\displaystyle d(f,\Omega ,y)=\sum _{x\in f^{-1}(y)}\mathrm {sgn} \left(\det(J(f)(x))\right)\,.}$
  ${\displaystyle f}$ ${\displaystyle g\in C^{1}(\Omega )\cap C({\bar {\Omega }})}$ ${\displaystyle f}$
• Be steadily up again and continuously differentiable up , not a critical point. Also choose a family of continuous functions from to with and for all , here denotes the closed ball with a radius around zero. Then there exists such that the integral formula holds for all . ${\displaystyle f\colon {\overline {\Omega }}\to \mathbb {R} ^{n}}$ ${\displaystyle {\overline {\Omega }}}$ ${\displaystyle \Omega }$ ${\displaystyle y\notin f(\partial \Omega )}$ ${\displaystyle (\phi _{\epsilon })_{\epsilon >0}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \operatorname {supp} (\phi _{\epsilon })\subset {\overline {K_{\epsilon }(0)}}}$ ${\displaystyle \textstyle \int _{\mathbb {R} ^{n}}\phi _{\epsilon }(x)\mathrm {d} x=1}$ ${\displaystyle \epsilon >0}$ ${\displaystyle {\overline {K_{\epsilon }(0)}}\subset \mathbb {R} ^{n}}$ ${\displaystyle \epsilon }$ ${\displaystyle \epsilon _{0}(f,y)}$
  ${\displaystyle d(f,\Omega ,y)=\int _{\Omega }\phi _{\epsilon }(f(x)-y)J(f)(x)\mathrm {d} x}$
  ${\displaystyle \epsilon \geq \epsilon _{0}(f,y)}$

Circulation number

As a special case, Brouwer's degree of mapping includes the rotation number, which is important in function theory . If you identify with , Brouwer's degree of mapping is also defined for the complex plane. A closed curve can be understood as a continuous image of . The unit circle around point zero is referred to with. That is, there is a continuous and surjective mapping . If now , because of the continuity of the degree of mapping, the expression for all continuous continuations of the same number is. It applies now ${\displaystyle \operatorname {ind} }$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \gamma \colon [0,1]\to \mathbb {C} }$ ${\displaystyle \mathbb {S} (0)}$ ${\displaystyle \mathbb {S} (0)\subset \mathbb {C} }$ ${\displaystyle f\colon \mathbb {S} (0)\to \operatorname {Bild} (\gamma )}$ ${\displaystyle a\notin \gamma =f(\mathbb {S} (0))}$ ${\displaystyle d(f,K_{1}(0),a)}$ ${\displaystyle f}$

${\displaystyle d(f,K_{1}(0),a)=\sum _{x\in f^{-1}(a)}\mathrm {sgn} \left(\det(J(f)(x))\right)=\sum _{x\in f^{-1}(a)}{\frac {1}{2\pi i}}\int _{f(S_{x}^{+})}{\frac {\mathrm {d} z}{z-a}}={\frac {1}{2\pi i}}\int _{f(\mathbb {S} (0))}{\frac {\mathrm {d} z}{z-a}}=\operatorname {ind} (f(S),a),}$

here denotes a sufficiently small circular ring around . In particular, to justify the last equal sign, a few facts from the topology are necessary. ${\displaystyle S_{x}^{+}}$ ${\displaystyle x}$

The leray-Schauder degree of representation

The Leray-Schauder degree of mapping is an analogue of Brouwer's degree of mapping for (infinitely dimensional) Banach spaces. This degree of mapping was defined in 1934 by J. Leray and J. Schauder. However, it is not possible to define the degree of mapping for any continuous functions, but only allow compact perturbations of the identity.

Compact disruptions of identity

Let be Banach spaces and a subset of the Banach space . A function is called a compact operator if ${\displaystyle X,Y}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle K\colon M\rightarrow Y}$

• ${\displaystyle K}$ is continuous and, if
• ${\displaystyle K}$ maps restricted sets to relatively compact sets . In other words, is a compact subset of . ${\displaystyle B\subset M}$ ${\displaystyle {\overline {T(B)}}}$ ${\displaystyle Y}$

An operator that can be represented as with a compact operator is called compact perturbation of identity. ${\displaystyle F\colon M\subset X\rightarrow X}$ ${\displaystyle F=\operatorname {Id} -K}$ ${\displaystyle K}$

Compact homotopy

A compact homotopy is a homotopy between compact operators. Let it be open and restricted and for an operator-valued function with compact operators . This operator valued function is compact homotopy on , if at all one does so ${\displaystyle M\subset X}$ ${\displaystyle K\colon t\mapsto K(t)}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle K(t)\colon M\subset X\rightarrow X}$ ${\displaystyle K}$ ${\displaystyle M}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$

${\displaystyle \|K(t_{1})(x)-K(t_{2})(x)\|_{X}\leq \varepsilon }$

applies to everyone and with . ${\displaystyle x\in M}$ ${\displaystyle t_{1},t_{2}\in [0,1]}$ ${\displaystyle |t_{1}-t_{2}|<\delta }$

definition

Be a compact disorder of identity, open and limited and . Then the leray-schauder degree of mapping is an integer , so that the following properties apply: ${\displaystyle F=\operatorname {Id} -K\colon {\overline {M}}\subset X\rightarrow X}$ ${\displaystyle M\subset X}$ ${\displaystyle y\not \in F(\partial M)}$ ${\displaystyle d(F,M,y)\in \mathbb {Z} }$

• Is , then the equation is solvable. ${\displaystyle d(F,M,y)\neq 0}$ ${\displaystyle F(x)=y}$

• Homotopy invariance: If a compact homotopy is open with for all and , the degree of mapping is independent of . ${\displaystyle K}$ ${\displaystyle {\overline {M}}}$ ${\displaystyle K(t)(x)\neq x}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle x\in \partial M}$ ${\displaystyle d((\operatorname {Id} -K)(t),M,y)}$ ${\displaystyle t\in [0,1]}$

example

The most important method for calculating the Leray-Schauder degree of mapping leads, just as with Brouwer's degree of mapping, via the homotopy invariance.

For example, if you are interested in whether the equation has a solution in , you first look for a suitable space so that is a compact operator. In order to prove solvability, one now indirectly assumes that on holds because there is nothing else to show. ${\displaystyle x-F_{0}(x)x=y}$ ${\displaystyle {\overline {\Omega }}}$ ${\displaystyle F_{0}}$ ${\displaystyle x-F_{0}(x)\neq y}$ ${\displaystyle \partial \Omega }$

Then you look for a compact homotopy with and for everyone and . This homotopy should be chosen in such a way that one can prove for the leray-schauder degree of representation . From this follows namely for all and thus the existence of one with . ${\displaystyle H}$ ${\displaystyle H(1)=F_{0}}$ ${\displaystyle x-H(t)(x)\neq y}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle x\in \partial \Omega }$ ${\displaystyle d(I-H(0),\Omega ,y)\neq 0}$ ${\displaystyle d(I-H(t),\Omega ,y)\neq 0}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle x\in \Omega }$ ${\displaystyle x-F_{0}(x)x=y}$

For a concrete example, consider the initial value problem

${\displaystyle x'=f(t,x)}$

for and given. It can be shown that it has a solution of at least if is continuous and optionally on a suitable applies. To see this, one writes the system of differential equations into the system ${\displaystyle t\in [0,a]}$ ${\displaystyle x(0)=x_{0}}$ ${\displaystyle f\colon [0,a]\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle |f(t,x)|\leq B(1+|x|)}$ ${\displaystyle [0,a]\times \mathbb {R} ^{n}}$ ${\displaystyle B\geq 0}$

${\displaystyle x(t)=x_{0}+\int _{0}^{t}f(\tau ,x(\tau ))\mathrm {d} \tau }$

of integral equations . Since both equations are equivalent, it suffices to show that the integral equation has a continuous solution. This is then also differentiable . Therefore we choose the space of the continuous function on the interval with the maximum norm . You also bet ${\displaystyle X=C([0,a])}$ ${\displaystyle [0,a]}$ ${\displaystyle \textstyle \|x\|=\max _{t\in [0,a]}|x(t)|}$

${\displaystyle F_{0}(x)(t):=x_{0}+\int _{0}^{t}f(\tau ,x(\tau ))\mathrm {d} \tau \,.}$

Due to Arzelà-Ascoli's theorem, is a compact operator and a compact homotopy. Since the existence of a solution is examined by, is set. Since it was presupposed, one can show that it is enough to vote with one and, due to the homotopy invariance, receives ${\displaystyle F_{0}}$ ${\displaystyle H(t)(x):=t\cdot F_{0}(x)}$ ${\displaystyle x-F_{0}(x)=0}$ ${\displaystyle y=0}$ ${\displaystyle |f(t,x)|\leq B(1+|x|)}$ ${\displaystyle \Omega :=B_{r}(0)}$ ${\displaystyle r>(|x_{0}|+B\cdot a)e^{-Ba}}$

${\displaystyle d(I-F_{0},B_{r}(0),y)=d(I,B_{r}(0),y)=1\,.}$

This shows that the integral equation has at least one continuous solution.

Mappings between manifolds

Be

${\displaystyle f\colon M\rightarrow N}$

a continuous mapping between n-dimensional, compact, oriented manifolds. (n is a natural number.)

The orientation of the manifolds induces isomorphisms

${\displaystyle H_{n}(M,\mathbb {Z} )\cong \mathbb {Z} ,H_{n}(N,\mathbb {Z} )\cong \mathbb {Z} }$.

The homomorphism induced by f

${\displaystyle f_{*}\colon H_{n}(M,\mathbb {Z} )\rightarrow H_{n}(N,\mathbb {Z} )}$

is the multiplication by an integer d, this is the degree of mapping of f.

literature

• Klaus Deimling: Nonlinear Functional Analysis . 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 .
• Michael Růžička: Nonlinear Functional Analysis . 1st edition. Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-20066-5 .
• Andrzej Granas, James Dugundji: Fixed point theory . 1st edition. Springer-Verlag, Berlin / Heidelberg 2003, ISBN 978-0-387-00173-9 .

Individual evidence

1. ^ Klaus Deimling: Nonlinear Functional Analysis. 1st edition. Springer-Verlag, Berlin / Heidelberg 1985, ISBN 3-540-13928-1 , page 37.