Minimax Theorems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Boston, MA
Birkhäuser Boston
1996
|
| Series: | Progress in Nonlinear Differential Equations and Their Applications
24 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | Many boundary value problems are equivalent to Au=O (1) where A : X ---+ Y is a mapping between two Banach spaces. When the problem is variational, there exists a differentiable functional <p : X ---+ lR such that A = <p', i. e. -1' <p(u + tv) - O t The space Y corresponds then to the topological dual X' of X and equation (1) is equivalent to <p'(u) = 0, i. e. (2) (<p'(u), v) = 0, Vv E X. A critical point of <p is a solution u of (2) and the value of <p at u is a critical value of <po How to find critical values? When <p is bounded from below, the infimum c := inf <p x is a natural candidate. Ekeland's variational principle implies the existence of a sequence (un) such that Such a sequence is called a Palais-Smale sequence at level C. The functional <p satisfies the (PS)c condition if any Palais-Smale sequence at level c has a convergent subsequence. If <p is bounded from below and satisfies the (PS)c condition at level c := infx <p, then c is a critical value of <po Following Ambrosetti and Rabinowitz, we consider now the case when 0 and e E X such that lIell > rand inf |
| Physical Description: | 1 Online-Ressource (X, 165 p) |
| ISBN: | 9781461241461 9781461286738 |
| DOI: | 10.1007/978-1-4612-4146-1 |
Staff View
MARC
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| 500 | |a Many boundary value problems are equivalent to Au=O (1) where A : X ---+ Y is a mapping between two Banach spaces. When the problem is variational, there exists a differentiable functional <p : X ---+ lR such that A = <p', i. e. -1' <p(u + tv) - O t The space Y corresponds then to the topological dual X' of X and equation (1) is equivalent to <p'(u) = 0, i. e. (2) (<p'(u), v) = 0, Vv E X. A critical point of <p is a solution u of (2) and the value of <p at u is a critical value of <po How to find critical values? When <p is bounded from below, the infimum c := inf <p x is a natural candidate. Ekeland's variational principle implies the existence of a sequence (un) such that Such a sequence is called a Palais-Smale sequence at level C. The functional <p satisfies the (PS)c condition if any Palais-Smale sequence at level c has a convergent subsequence. If <p is bounded from below and satisfies the (PS)c condition at level c := infx <p, then c is a critical value of <po Following Ambrosetti and Rabinowitz, we consider now the case when 0 and e E X such that lIell > rand inf | ||
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| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:06Z |
| institution | BVB |
| isbn | 9781461241461 9781461286738 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027855647 |
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| physical | 1 Online-Ressource (X, 165 p) |
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| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Birkhäuser Boston |
| record_format | marc |
| series | Progress in Nonlinear Differential Equations and Their Applications |
| series2 | Progress in Nonlinear Differential Equations and Their Applications |
| spelling | Willem, Michel Verfasser aut Minimax Theorems by Michel Willem Boston, MA Birkhäuser Boston 1996 1 Online-Ressource (X, 165 p) txt rdacontent c rdamedia cr rdacarrier Progress in Nonlinear Differential Equations and Their Applications 24 Many boundary value problems are equivalent to Au=O (1) where A : X ---+ Y is a mapping between two Banach spaces. When the problem is variational, there exists a differentiable functional <p : X ---+ lR such that A = <p', i. e. -1' <p(u + tv) - O t The space Y corresponds then to the topological dual X' of X and equation (1) is equivalent to <p'(u) = 0, i. e. (2) (<p'(u), v) = 0, Vv E X. A critical point of <p is a solution u of (2) and the value of <p at u is a critical value of <po How to find critical values? When <p is bounded from below, the infimum c := inf <p x is a natural candidate. Ekeland's variational principle implies the existence of a sequence (un) such that Such a sequence is called a Palais-Smale sequence at level C. The functional <p satisfies the (PS)c condition if any Palais-Smale sequence at level c has a convergent subsequence. If <p is bounded from below and satisfies the (PS)c condition at level c := infx <p, then c is a critical value of <po Following Ambrosetti and Rabinowitz, we consider now the case when 0 and e E X such that lIell > rand inf Mathematics Applications of Mathematics Game Theory, Economics, Social and Behav. Sciences Mathematik Minimax-Theorem (DE-588)4135131-9 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Minimax-Theorem (DE-588)4135131-9 s Partielle Differentialgleichung (DE-588)4044779-0 s 1\p DE-604 Progress in Nonlinear Differential Equations and Their Applications 24 (DE-604)BV036582883 24 https://doi.org/10.1007/978-1-4612-4146-1 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Willem, Michel Minimax Theorems Progress in Nonlinear Differential Equations and Their Applications Mathematics Applications of Mathematics Game Theory, Economics, Social and Behav. Sciences Mathematik Minimax-Theorem (DE-588)4135131-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4135131-9 (DE-588)4044779-0 |
| title | Minimax Theorems |
| title_auth | Minimax Theorems |
| title_exact_search | Minimax Theorems |
| title_full | Minimax Theorems by Michel Willem |
| title_fullStr | Minimax Theorems by Michel Willem |
| title_full_unstemmed | Minimax Theorems by Michel Willem |
| title_short | Minimax Theorems |
| title_sort | minimax theorems |
| topic | Mathematics Applications of Mathematics Game Theory, Economics, Social and Behav. Sciences Mathematik Minimax-Theorem (DE-588)4135131-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Mathematics Applications of Mathematics Game Theory, Economics, Social and Behav. Sciences Mathematik Minimax-Theorem Partielle Differentialgleichung |
| url | https://doi.org/10.1007/978-1-4612-4146-1 |
| volume_link | (DE-604)BV036582883 |
| work_keys_str_mv | AT willemmichel minimaxtheorems |