Fichera's existence principle
{{Short description|Theorem in functional analysis}}
In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954.{{harv|Faedo|1957|p=1}}, {{harv|Valent|1999|p=84}}, {{harv|Leonardi|Passarelli di Napoli|Sbordone|2000|p=221}}. More precisely, given a general vector space {{mvar|V}} and two linear maps from it onto two Banach spaces, the principle states necessary and sufficient conditions for a linear transformation between the two dual Banach spaces to be invertible for every vector in {{mvar|V}}.See {{harvs|last=Fichera|year=1955|year2=1958|loc=pp. 175–177|loc2=pp. 30–35}}, {{harv|Faedo|1957|pp=1–2}}, {{harv|Miranda|1970|pp=123–124}}, {{harv|Valent|1999|p=84}}.
See also
- {{annotated link|Banach fixed-point theorem}}
- {{annotated link|Babuška–Lax–Milgram theorem}}
- {{annotated link|Lax–Milgram theorem}}
- {{annotated link|Lions–Lax–Milgram theorem}}
- {{annotated link|Surjection of Fréchet spaces}}
Notes
{{reflist}}
References
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|last2 = Lanzara
|first2 = Flavia
|author2-link = Flavia Lanzara
|editor-last = Cialdea
|editor-first = Alberto
|editor-link = Alberto Cialdea
|contribution = Some contributions of G. Fichera to the theory of Partial Differential Equations
|contribution-url =
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|publisher = Aracne Editrice
|volume = 7
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|year = 2000
|isbn = 978-88-7999-321-0
|mr = 1913527
|zbl = 1005.35003
|url = https://archive.org/details/homagetogaetanof0000unse/page/79
}}. A survey of Gaetano Fichera's contributions to the theory of partial differential equations, written by two of his pupils.
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|year =1957
|language =Italian
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}}.
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|year =1958
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}}.
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|last =Fichera
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|place =Roma
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|orig-year =1954
|year =1962
|edition =3rd reprint
|pages =XIX+502
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|doi =
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|title =G. Fichera, Lezioni sulle trasformazioni lineari, Vol. I: Introduzione all'Analisi lineare, Istituto Matematico dell'Università di Trieste, 1954 – pag. XVII + 502.
|journal =Bollettino dell'Unione Matematica Italiana
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}}.
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|editor-last = Fichera
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|year = 1955
|pages = 174–227
|place = Roma
|publisher = Edizioni Cremonese
|mr = 0074665
|zbl = 0068.31101
}}. The paper Some recent developments of the theory of boundary value problems for linear partial differential equations describes Fichera's approach to a general theory of boundary value problems for linear partial differential equations through a theorem similar in spirit to the Lax–Milgram theorem.
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|series =Corsi dell'Istituto Nazionale di Alta Matematica
|volume =
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|pages =III+292
|language =Italian
|url =
|doi =
|isbn =
|mr =
|zbl =
}}. A monograph based on lecture notes, taken by Lucilla Bassotti and Luciano De Vito of a course held by Gaetano Fichera at the INdAM: for a review of the book, see {{Citation
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|first =Carlo
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|journal =Bollettino dell'Unione Matematica Italiana
|series =Serie 3
|volume =14
|issue =4
|pages =568–570
|year =1959
|language =Italian
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}}.
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|last1 =Leonardi
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|last3 =Sbordone
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|title = Problemi attuali dell'analisi e della fisica matematica. Atti del 2° simposio internazionale. Dedicato alla memoria del Prof. Gaetano Fichera. Taormina, 15–17 ottobre 1998
|trans-title=Current problems in analysis and mathematical physics. Papers of the 2nd international symposium dedicated to the memory of Prof. Gaetano Fichera. Taormina, 15–17 ottobre 1998
|editor-last = Ricci
|editor-first = Paolo Emilio
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|contribution =On Fichera’s existence principle in functional analysis
|year = 2000
|pages = 221–234
|language= English
|place = Roma
|publisher =Aracne Editrice
|doi=10.4399/978887999264017
|isbn=978-88-7999-264-0
|mr = 1809690
|zbl = 0956.00046
|doi-broken-date =1 November 2024
}}.
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|volume =93
|year =1972
|pages =XIV+283
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|doi =
|id =
|isbn =0-12-449550-8
|mr =0355280
|zbl =0245.35001
}}, reviewed also by {{Citation
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|author-link = Martin Schechter (mathematician)
|title =Theory of Partial Differential Equations, by H. Melvin Lieberstein
|journal =SIAM Review
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|date =October 1973
|jstor =2028753
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}}, and by {{Citation
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|translator-last=Motteler
|translator-first=Zane C.
|title = Partial Differential Equations of Elliptic Type
|place = Berlin – Heidelberg – New York
|publisher = Springer Verlag
|series = Ergebnisse der Mathematik und ihrer Grenzgebiete – 2 Folge
|volume = Band 2
|orig-year = 1955
|year = 1970
|edition = 2nd Revised
|pages = XII+370
|url =
|doi =
|id =
|isbn = 978-3-540-04804-6
|mr = 0284700
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}}.
- {{Citation
|first = Tullio
|last = Valent
|author-link = Tullio Valent
|title = Su un principio generale di esistenza in analisi lineare
|trans-title=On a general existence principle in linear analysis
|language = Italian
|journal = Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali
|series = Series VIII
|url=http://www.bdim.eu/item?id=RLINA_1979_8_66_5_331_0
|volume = 66
|issue = 5
|pages = 331–337
|year = 1979
|zbl =0476.46002
}}.
- {{Citation
|first = Tullio
|last = Valent
|author-link = Tullio Valent
|editor-last = Capriz
|editor-first = Gianfranco
|editor2-last = Grioli
|editor2-first = Giuseppe
|editor2-link = Giuseppe Grioli
|editor3-last = Manacorda
|editor3-first = Tristano
|editor3-link = Tristano Manacorda
|contribution = Existence problems
|title = Interactions between Analysis and Mechanics. The Legacy of Gaetano Fichera. Convegno internazionale (Roma, 22--23 aprile 1998)
|year = 1999
|pages = 83–98
|place = Roma
|series = Atti dei Convegni Lincei
|volume = 148
|publisher = Accademia Nazionale dei Lincei
|url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=36500
|issn=0391-805X
}}. An expository paper detailing the contributions of Gaetano Fichera and his school on the problem of numerical calculation of eigenvalues for general differential operators.
{{Banach spaces}}
{{Functional analysis}}