August  2014, 7(4): 593-616. doi: 10.3934/dcdss.2014.7.593

Ultrafunctions and applications

1. 

Dipartimento di Matematica Applicata, Università degli Studi di Pisa, Via F. Buonarroti 1/c, Pisa, Italy

2. 

University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Received  October 2013 Revised  December 2013 Published  February 2014

This paper deals with a new kind of generalized functions, called ``ultrafunctions", which have been introduced recently in [5] and developed in [10] and [11]. Their peculiarity is that they are based on a Non Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultrafunctions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions. Some applications of this kind will be presented in the second part of this paper.
Citation: Vieri Benci, Lorenzo Luperi Baglini. Ultrafunctions and applications. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 593-616. doi: 10.3934/dcdss.2014.7.593
References:
[1]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618260.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[3]

V. Benci, A construction of a nonstandard universe, in Advances of Dynamical Systems and Quantum Physics (eds. S. Albeverio et al.), World Scientific, Singapore, 1995, 11-21.

[4]

V. Benci, An algebraic approach to nonstandard analysis, in Calculus of Variations and Partial differential equations, (eds. G.Buttazzo, et al.), Springer, Berlin, 1999, 285-326.

[5]

V. Benci, Ultrafunctions and generalized solutions, Advanced Nonlinear Studies, 13 (2013), 461-486.

[6]

V. Benci and M. Di Nasso, Alpha-theory: An elementary axiomatic for nonstandard analysis, Expositiones Mathematicae, 21 (2003), 355-386. doi: 10.1016/S0723-0869(03)80038-5.

[7]

V. Benci, S. Galatolo and M. Ghimenti, An elementary approach to stochastic differential equations using the infinitesimals, in Contemporary Mathematics, 530, Ultrafilters across Mathematics, American Mathematical Society, 2010, 1-22. doi: 10.1090/conm/530/10438.

[8]

V. Benci, L. Horsten and S. Wenmackers, Non-Archimedean probability, Milan J. Math., 81 (2012), 121-151. doi: 10.1007/s00032-012-0191-x.

[9]

V. Benci and L. Luperi Baglini, An Algebra of ultrafunctions and distributions, in preparation.

[10]

V. Benci and L. Luperi Baglini, A model problem for ultrafunctions, to appear on the proceedings of Variational and Topological Methods, Flagstaff, (2012), EJDE, arXiv:1212.1370.

[11]

V. Benci and L. Luperi Baglini, Basic Properties of ultrafunctions, to appear on the proceedings of WNLDE 2012, PNLDE, Birkhäuser, arXiv:1302.7156.

[12]

D. J. Brown and A. Robinson, Nonstandard exchange economies, Econometrica, 43 (1974), 41-55. doi: 10.2307/1913412.

[13]

J. F. Colombeau, Elementary introduction to new generalized functions, North-Holland, Amsterdam, 1985.

[14]

P. Du Bois-Reymond, Über die Paradoxen des Infinit är-Calcüls}, Math. Annalen, 11 (1877), 150-167.

[15]

Ph. Ehrlich, The Rise of non-Archimedean mathematics and the roots of a misconception I: The Emergence of non-Archimedean systems of magnitudes, Arch. Hist. Exact Sci., 60 (2006), 1-121. doi: 10.1007/s00407-005-0102-4.

[16]

M. Goze and R. Lutz, Nonstandard analysis, a practical guide with applications, Springer Lect. Notes Math., 881 (1981).

[17]

M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Springer Series Mathematics and Its Applications, 537, 2002.

[18]

O. Heaviside, Electromagnetic Theory. Including An Account of Heaviside's Unpublished Notes for a Fourth Volume, Chelsea Publishing Company, Incorporated, 1971.

[19]

D. Hilbert, Grundlagen der Geometrie, in English, [The Foundations of Geometry], 2nd edition, The Open Court Publishing Company, Chicago, 1980.

[20]

H. J. Keisler, Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston, 1976.

[21]

T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, Venezia, (Serie 7), 4 (1892-93), 1765-1815.

[22]

E. Nelson, Internal Set Theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc., 83 (1977), 1165-1198. doi: 10.1090/S0002-9904-1977-14398-X.

[23]

A. Robinson, Non-standard analysis, Proceedings of the Royal Academy of Sciences, Amsterdam (Series A), 64 (1961), 432-440.

[24]

M. Sato, Theory of Hyperfunctions, I, Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 8 (1959), 139-193.

[25]

M. Sato, Theory of Hyperfunctions, II, Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 8 (1960), 387-437.

[26]

L. Schwartz, Théorie des Distributions, Hermann, Paris, 2 vols., (1950/1951), new edition, 1966. doi: 10.5802/aif.68.

[27]

L. Schwartz, Mathematics for the Physical Sciences, Hermann, Paris, 1966.

[28]

G. Veronese, Il continuo rettilineo e l'assioma V di Archimede, Memorie della Reale Accademia dei Lincei, Atti della Classe di Scienze Naturali, Fisiche e Matematiche, 6 (1889), 603-624.

show all references

References:
[1]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, 2007. doi: 10.1017/CBO9780511618260.

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[3]

V. Benci, A construction of a nonstandard universe, in Advances of Dynamical Systems and Quantum Physics (eds. S. Albeverio et al.), World Scientific, Singapore, 1995, 11-21.

[4]

V. Benci, An algebraic approach to nonstandard analysis, in Calculus of Variations and Partial differential equations, (eds. G.Buttazzo, et al.), Springer, Berlin, 1999, 285-326.

[5]

V. Benci, Ultrafunctions and generalized solutions, Advanced Nonlinear Studies, 13 (2013), 461-486.

[6]

V. Benci and M. Di Nasso, Alpha-theory: An elementary axiomatic for nonstandard analysis, Expositiones Mathematicae, 21 (2003), 355-386. doi: 10.1016/S0723-0869(03)80038-5.

[7]

V. Benci, S. Galatolo and M. Ghimenti, An elementary approach to stochastic differential equations using the infinitesimals, in Contemporary Mathematics, 530, Ultrafilters across Mathematics, American Mathematical Society, 2010, 1-22. doi: 10.1090/conm/530/10438.

[8]

V. Benci, L. Horsten and S. Wenmackers, Non-Archimedean probability, Milan J. Math., 81 (2012), 121-151. doi: 10.1007/s00032-012-0191-x.

[9]

V. Benci and L. Luperi Baglini, An Algebra of ultrafunctions and distributions, in preparation.

[10]

V. Benci and L. Luperi Baglini, A model problem for ultrafunctions, to appear on the proceedings of Variational and Topological Methods, Flagstaff, (2012), EJDE, arXiv:1212.1370.

[11]

V. Benci and L. Luperi Baglini, Basic Properties of ultrafunctions, to appear on the proceedings of WNLDE 2012, PNLDE, Birkhäuser, arXiv:1302.7156.

[12]

D. J. Brown and A. Robinson, Nonstandard exchange economies, Econometrica, 43 (1974), 41-55. doi: 10.2307/1913412.

[13]

J. F. Colombeau, Elementary introduction to new generalized functions, North-Holland, Amsterdam, 1985.

[14]

P. Du Bois-Reymond, Über die Paradoxen des Infinit är-Calcüls}, Math. Annalen, 11 (1877), 150-167.

[15]

Ph. Ehrlich, The Rise of non-Archimedean mathematics and the roots of a misconception I: The Emergence of non-Archimedean systems of magnitudes, Arch. Hist. Exact Sci., 60 (2006), 1-121. doi: 10.1007/s00407-005-0102-4.

[16]

M. Goze and R. Lutz, Nonstandard analysis, a practical guide with applications, Springer Lect. Notes Math., 881 (1981).

[17]

M. Grosser, M. Kunzinger, M. Oberguggenberger and R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Springer Series Mathematics and Its Applications, 537, 2002.

[18]

O. Heaviside, Electromagnetic Theory. Including An Account of Heaviside's Unpublished Notes for a Fourth Volume, Chelsea Publishing Company, Incorporated, 1971.

[19]

D. Hilbert, Grundlagen der Geometrie, in English, [The Foundations of Geometry], 2nd edition, The Open Court Publishing Company, Chicago, 1980.

[20]

H. J. Keisler, Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston, 1976.

[21]

T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, Venezia, (Serie 7), 4 (1892-93), 1765-1815.

[22]

E. Nelson, Internal Set Theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc., 83 (1977), 1165-1198. doi: 10.1090/S0002-9904-1977-14398-X.

[23]

A. Robinson, Non-standard analysis, Proceedings of the Royal Academy of Sciences, Amsterdam (Series A), 64 (1961), 432-440.

[24]

M. Sato, Theory of Hyperfunctions, I, Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 8 (1959), 139-193.

[25]

M. Sato, Theory of Hyperfunctions, II, Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 8 (1960), 387-437.

[26]

L. Schwartz, Théorie des Distributions, Hermann, Paris, 2 vols., (1950/1951), new edition, 1966. doi: 10.5802/aif.68.

[27]

L. Schwartz, Mathematics for the Physical Sciences, Hermann, Paris, 1966.

[28]

G. Veronese, Il continuo rettilineo e l'assioma V di Archimede, Memorie della Reale Accademia dei Lincei, Atti della Classe di Scienze Naturali, Fisiche e Matematiche, 6 (1889), 603-624.

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