# American Institute of Mathematical Sciences

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 & 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. 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.), (1995), 11. [4] V. Benci, An algebraic approach to nonstandard analysis,, in Calculus of Variations and Partial differential equations, (1999), 285. [5] V. Benci, Ultrafunctions and generalized solutions,, Advanced Nonlinear Studies, 13 (2013), 461. [6] V. Benci and M. Di Nasso, Alpha-theory: An elementary axiomatic for nonstandard analysis,, Expositiones Mathematicae, 21 (2003), 355. 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 (2010), 1. doi: 10.1090/conm/530/10438. [8] V. Benci, L. Horsten and S. Wenmackers, Non-Archimedean probability,, Milan J. Math., 81 (2012), 121. 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, (2012). [11] V. Benci and L. Luperi Baglini, Basic Properties of ultrafunctions,, to appear on the proceedings of WNLDE 2012, (2012). [12] D. J. Brown and A. Robinson, Nonstandard exchange economies,, Econometrica, 43 (1974), 41. doi: 10.2307/1913412. [13] J. F. Colombeau, Elementary introduction to new generalized functions,, North-Holland, (1985). [14] P. Du Bois-Reymond, Über die Paradoxen des Infinit är-Calcüls},, Math. Annalen, 11 (1877), 150. [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. 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, (2002). [18] O. Heaviside, Electromagnetic Theory. Including An Account of Heaviside's Unpublished Notes for a Fourth Volume,, Chelsea Publishing Company, (1971). [19] D. Hilbert, Grundlagen der Geometrie,, in English, (1980). [20] H. J. Keisler, Foundations of Infinitesimal Calculus,, Prindle, (1976). [21] T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici,, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, 4 (): 1892. [22] E. Nelson, Internal Set Theory: A new approach to nonstandard analysis,, Bull. Amer. Math. Soc., 83 (1977), 1165. doi: 10.1090/S0002-9904-1977-14398-X. [23] A. Robinson, Non-standard analysis,, Proceedings of the Royal Academy of Sciences, 64 (1961), 432. [24] M. Sato, Theory of Hyperfunctions, I,, Journal of the Faculty of Science, 8 (1959), 139. [25] M. Sato, Theory of Hyperfunctions, II,, Journal of the Faculty of Science, 8 (1960), 387. [26] L. Schwartz, Théorie des Distributions,, Hermann, (1966). doi: 10.5802/aif.68. [27] L. Schwartz, Mathematics for the Physical Sciences,, Hermann, (1966). [28] G. Veronese, Il continuo rettilineo e l'assioma V di Archimede,, Memorie della Reale Accademia dei Lincei, 6 (1889), 603.

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. 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.), (1995), 11. [4] V. Benci, An algebraic approach to nonstandard analysis,, in Calculus of Variations and Partial differential equations, (1999), 285. [5] V. Benci, Ultrafunctions and generalized solutions,, Advanced Nonlinear Studies, 13 (2013), 461. [6] V. Benci and M. Di Nasso, Alpha-theory: An elementary axiomatic for nonstandard analysis,, Expositiones Mathematicae, 21 (2003), 355. 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 (2010), 1. doi: 10.1090/conm/530/10438. [8] V. Benci, L. Horsten and S. Wenmackers, Non-Archimedean probability,, Milan J. Math., 81 (2012), 121. 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, (2012). [11] V. Benci and L. Luperi Baglini, Basic Properties of ultrafunctions,, to appear on the proceedings of WNLDE 2012, (2012). [12] D. J. Brown and A. Robinson, Nonstandard exchange economies,, Econometrica, 43 (1974), 41. doi: 10.2307/1913412. [13] J. F. Colombeau, Elementary introduction to new generalized functions,, North-Holland, (1985). [14] P. Du Bois-Reymond, Über die Paradoxen des Infinit är-Calcüls},, Math. Annalen, 11 (1877), 150. [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. 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, (2002). [18] O. Heaviside, Electromagnetic Theory. Including An Account of Heaviside's Unpublished Notes for a Fourth Volume,, Chelsea Publishing Company, (1971). [19] D. Hilbert, Grundlagen der Geometrie,, in English, (1980). [20] H. J. Keisler, Foundations of Infinitesimal Calculus,, Prindle, (1976). [21] T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici,, Atti del R. Istituto Veneto di Scienze Lettere ed Arti, 4 (): 1892. [22] E. Nelson, Internal Set Theory: A new approach to nonstandard analysis,, Bull. Amer. Math. Soc., 83 (1977), 1165. doi: 10.1090/S0002-9904-1977-14398-X. [23] A. Robinson, Non-standard analysis,, Proceedings of the Royal Academy of Sciences, 64 (1961), 432. [24] M. Sato, Theory of Hyperfunctions, I,, Journal of the Faculty of Science, 8 (1959), 139. [25] M. Sato, Theory of Hyperfunctions, II,, Journal of the Faculty of Science, 8 (1960), 387. [26] L. Schwartz, Théorie des Distributions,, Hermann, (1966). doi: 10.5802/aif.68. [27] L. Schwartz, Mathematics for the Physical Sciences,, Hermann, (1966). [28] G. Veronese, Il continuo rettilineo e l'assioma V di Archimede,, Memorie della Reale Accademia dei Lincei, 6 (1889), 603.
 [1] Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313 [2] Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967 [3] Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 [4] John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337 [5] Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485 [6] John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269 [7] Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759 [8] Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835 [9] Corentin Audiard. On the non-homogeneous boundary value problem for Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3861-3884. doi: 10.3934/dcds.2013.33.3861 [10] Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065 [11] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [12] Nassif Ghoussoub. Superposition of selfdual functionals in non-homogeneous boundary value problems and differential systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 187-220. doi: 10.3934/dcds.2008.21.187 [13] Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624 [14] Jong-Shenq Guo, Masahiko Shimojo. Blowing up at zero points of potential for an initial boundary value problem. Communications on Pure & Applied Analysis, 2011, 10 (1) : 161-177. doi: 10.3934/cpaa.2011.10.161 [15] Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104 [16] Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099 [17] Toyohiko Aiki, Joost Hulshof, Nobuyuki Kenmochi, Adrian Muntean. Analysis of non-equilibrium evolution problems: Selected topics in material and life sciences. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : i-iii. doi: 10.3934/dcdss.2014.7.1i [18] Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263 [19] Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187 [20] Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834

2017 Impact Factor: 0.561