Fichera's existence principle

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.^[1] More precisely, given a general vector space $V$ 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 $V$ .^[2]

Notes

^ (Faedo 1957, p. 1), (Valent 1999, p. 84), (Leonardi, Passarelli di Napoli & Sbordone 2000, p. 221).
^ See (Fichera 1955, pp. 175–177, 1958, pp. 30–35), (Faedo 1957, pp. 1–2), (Miranda 1970, pp. 123–124), (Valent 1999, p. 84).

References

Cialdea, Alberto; Lanzara, Flavia (2000), "Some contributions of G. Fichera to the theory of Partial Differential Equations", in Cialdea, Alberto (ed.), Homage to Gaetano Fichera, Quaderni di Matematica, vol. 7, Aracne Editrice, pp. 79–143, ISBN 978-88-7999-321-0, MR 1913527, Zbl 1005.35003. A survey of Gaetano Fichera's contributions to the theory of partial differential equations, written by two of his pupils.
Faedo, Sandro (1957), "Su un principio di esistenza nell'analisi lineare" [On an existence principle in linear analysis], Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, Serie 3 (in Italian), 11 (1–2): 1–8, MR 0096957, Zbl 0087.32304.
Faedo, Sandro (1958), "Applicazione ai problemi di derivata obliqua di un principio esistenziale e di una legge di dualità fra le formule di maggiorazione" [Application to oblique derivative problems of an existence principle and a duality law between estimates], Rendiconti di Matematica e delle sue Applicazioni, V Serie (in Italian), 16: 515–532, doi:10.1007/978-3-642-10918-8_4, ISBN 978-3-642-10916-4, MR 0096959, Zbl 0105.29902.
Fichera, Gaetano (1962) [1954], Lezioni sulle trasformazioni lineari. Volume I: Introduzione all'analisi lineare [Lectures on linear transformations. Volume I: Introduction to linear analysis] (in Italian) (3rd reprint ed.), Roma: Libreria Eredi Virgilio Veschi, pp. XIX+502, MR 0067346, Zbl 0057.33601: for a review of the book, see Ghizzetti, Aldo (1954), "G. Fichera, Lezioni sulle trasformazioni lineari, Vol. I: Introduzione all'Analisi lineare, Istituto Matematico dell'Università di Trieste, 1954 – pag. XVII + 502.", Bollettino dell'Unione Matematica Italiana, Serie 3 (in Italian), 9 (4): 457–459.
Fichera, Gaetano (1955), "Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parziali lineari", in Fichera, G. (ed.), Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali – Trieste 25–28 Agosto 1954 (in Italian), Roma: Edizioni Cremonese, pp. 174–227, 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.
Fichera, Gaetano (1958), Premesse ad una teoria generale dei problemi al contorno per le equazioni differenziali [Premises to a general theory of boundary value problems for differential equations], Corsi dell'Istituto Nazionale di Alta Matematica (in Italian), Roma: Libreria Eredi Virgilio Veschi, pp. III+292. 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 Miranda, Carlo (1959), "G. Fichera, Premesse ad una teoria generale dei problemi al contorno per le equazioni differenziali, Libreria Eredi V., Roma", Bollettino dell'Unione Matematica Italiana, Serie 3 (in Italian), 14 (4): 568–570.
Leonardi, Salvatore; Passarelli di Napoli, Antonia; Sbordone, Carlo (2000), "On Fichera's existence principle in functional analysis", in Ricci, Paolo Emilio (ed.), 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 [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], Roma: Aracne Editrice, pp. 221–234, doi:10.4399/978887999264017 (inactive 1 November 2024), ISBN 978-88-7999-264-0, MR 1809690, Zbl 0956.00046{{citation}}: CS1 maint: DOI inactive as of November 2024 (link).
Lieberstein, H. Melvin (1972), Theory of partial differential equations, Mathematics in Science and Engineering, vol. 93, New York and London: Academic Press, pp. XIV+283, ISBN 0-12-449550-8, MR 0355280, Zbl 0245.35001, reviewed also by Schechter, Martin (October 1973), "Theory of Partial Differential Equations, by H. Melvin Lieberstein", SIAM Review, 15 (4): 809, doi:10.1137/1015120, JSTOR 2028753, and by Appleyard, David F. (March 1973), "Telegraphic Reviews. Theory of Partial Differential Equations. H. Melvin Lieberstein", The American Mathematical Monthly, 80 (3): 340, JSTOR 2318482
Miranda, Carlo (1970) [1955], Partial Differential Equations of Elliptic Type, Ergebnisse der Mathematik und ihrer Grenzgebiete – 2 Folge, vol. Band 2, translated by Motteler, Zane C. (2nd Revised ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. XII+370, ISBN 978-3-540-04804-6, MR 0284700, Zbl 0198.14101.
Valent, Tullio (1979), "Su un principio generale di esistenza in analisi lineare" [On a general existence principle in linear analysis], Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Series VIII (in Italian), 66 (5): 331–337, Zbl 0476.46002.
Valent, Tullio (1999), "Existence problems", in Capriz, Gianfranco; Grioli, Giuseppe; Manacorda, Tristano (eds.), Interactions between Analysis and Mechanics. The Legacy of Gaetano Fichera. Convegno internazionale (Roma, 22--23 aprile 1998), Atti dei Convegni Lincei, vol. 148, Roma: Accademia Nazionale dei Lincei, pp. 83–98, 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.

[1] (Faedo 1957, p. 1), (Valent 1999, p. 84), (Leonardi, Passarelli di Napoli & Sbordone 2000, p. 221).

[2] See (Fichera 1955, pp. 175–177, 1958, pp. 30–35), (Faedo 1957, pp. 1–2), (Miranda 1970, pp. 123–124), (Valent 1999, p. 84).

[1]

[2]

v t e Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

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References