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 .
- To do this, be an operator from the Banach space into its dual space .
- The operator has the following properties:
- Then:
- (1) is surjective.
- (2) For each , the fiber is a closed , bounded, and convex subset of .
- (3) If it is also strictly monotonic, it is even a bijection .
Explanation of terminology
With regard to the above-mentioned properties of the operator , the following terms are essential:
- is monotonic if and only if for always holds:
- The operator is strictly monotonic if and only if with always holds:
- The operator is coercive if and only if:
- .
- The operator is hemistent if and only if it holds for always:
- The real-valued function defined on the interval is continuous .
See also
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
- ↑ a b Michael Růžička: Nonlinear Functional Analysis: An Introduction. 2004, p. 63 ff
- ^ A b Philippe G. Ciarlet: Linear and Nonlinear Functional Analysis with Applications. 2013, p. 742 ff
- ^ Philippe Blanchard, Erwin Bruning: Direct methods of the calculus of variations: A textbook. 1982, p. 154 ff
- ↑ The scalar product notation usually used here serves to avoid multiple brackets . The rule here is for the establishment, .
- ↑ Here is the standard mapping of the Banach space .