Theorem from Minty-Browder

The set of Minty-Browder or set of Browder and Minty , English Minty-Browder theorem is a mathematical theorem of nonlinear functional analysis . It goes back to the work of the two mathematicians George Minty and Felix Browder in 1962 and 1963.

The theorem deals with the question of the conditions under which a monotonic operator is surjective on a separable reflexive Banach space over the field of real numbers . It is also known as the main theorem of the theory of monotonic operators and is considered a nonlinear analogue of the Lax-Milgram theorem . The theorem is widely used in the solution of non-linear boundary value problems in the calculus of variations . The proof of the theorem is based on Brouwer's fixed point theorem and the Galerkin method .

Formulation of the sentence

Following the representation by Růžička or Ciarlet , Minty-Browder's theorem can be stated as follows:

Given a separable reflexive Banach space over . ${\displaystyle X}$ ${\displaystyle \mathbb {R} }$

To do this, be an operator from the Banach space into its dual space . ${\displaystyle A\colon X\to X^{'}=L(X,\mathbb {R} )}$

The operator has the following properties: ${\displaystyle A}$

(a) is monotonic . ${\displaystyle A}$

(b) is coercive . ${\displaystyle A}$

(c) is hemistent. ${\displaystyle A}$

Then:

(1) is surjective. ${\displaystyle A}$

(2) For each , the fiber is a closed , bounded, and convex subset of . ${\displaystyle x^{'}\in X^{'}}$ ${\displaystyle A^{-1}(x^{'})}$ ${\displaystyle X}$

(3) If it is also strictly monotonic, it is even a bijection . ${\displaystyle A}$ ${\displaystyle A}$

Explanation of terminology

With regard to the above-mentioned properties of the operator , the following terms are essential: ${\displaystyle A\colon X\to X^{'}}$

• ${\displaystyle A}$is monotonic if and only if for always holds: ${\displaystyle u,v\in X}$

${\displaystyle \langle A(u)-A(v)\;,\;u-v\rangle \geq 0}$

• The operator is strictly monotonic if and only if with always holds: ${\displaystyle A}$ ${\displaystyle u,v\in X}$ ${\displaystyle u\neq v}$

${\displaystyle \langle A(u)-A(v)\;,\;u-v\rangle >0}$

• The operator is coercive if and only if: ${\displaystyle A}$

${\displaystyle \lim _{\|x\|\to {\infty }}{\frac {\langle A(x)\;,\;x\rangle }{\|x\|}}={\infty }}$.

• The operator is hemistent if and only if it holds for always: ${\displaystyle A}$ ${\displaystyle u,v,w\in X}$

The real-valued function defined on the interval is continuous . ${\displaystyle [0\;,\;1]}$ ${\displaystyle t\mapsto \langle A(u+tv)\;,\;w\rangle }$

See also

• Lax-Milgram's theorem

Sources and background literature

• Philippe Blanchard , Erwin Brüning : Direct methods of the calculus of variations . A textbook. Springer Verlag , Vienna, New York 1982, ISBN 3-211-81692-5 . MR0687073
• Felix E. Browder: Nonlinear elliptic boundary value problems . In: Bulletin of the American Mathematical Society . tape 69 , 1963, pp. 862-874 ( [1] ). MR0156116
• Philippe G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications . Society for Industrial and Applied Mathematics , Philadelphia, PA 2013, ISBN 978-1-61197-258-0 . MR3136903
• George J. Minty: On a "monotonicity" method for the solution of non-linear equations in Banach spaces . In: National Academy of Sciences . tape 50 , 1963, pp. 1038-1041 , JSTOR : 71840 . MR0162159
• George J. Minty: Monotonic (nonlinear) operators in Hilbert space . In: Duke Mathematical Journal . tape 29 , 1962, pp. 341-346 ( [2] ). MR0169064
• Michael Růžička: Nonlinear Functional Analysis . An introduction. Springer Verlag, Berlin, Heidelberg (among others) 2004, ISBN 978-3-540-20066-6 .

Individual evidence

1. ↑ ^{a b} Michael Růžička: Nonlinear Functional Analysis: An Introduction. 2004, p. 63 ff
2. ^ ^{A b} Philippe G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications. 2013, p. 742 ff
3. ^ Philippe Blanchard, Erwin Bruning: Direct methods of the calculus of variations: A textbook. 1982, p. 154 ff
4. ↑ The scalar product notation usually used here serves to avoid multiple brackets . The rule here is for the establishment, . ${\displaystyle x\in X\;,\;f\in X^{'}}$ ${\displaystyle \langle f\;,\;x\rangle =f(x)\in \mathbb {R} }$
5. ↑ Here is the standard mapping of the Banach space . ${\displaystyle x\mapsto \|x\|\in \mathbb {R} \;(x\in X)}$ ${\displaystyle X}$