On approximation of approximately quadratic mappings by quadratic mappings
Abstract
In this paper we establish an approximation of approximately quadratic mappings by quadratic mappings, which solves the pertinent Ulam stability problem.
Keywords
Ulam problem; Ulam type problem; general Ulam problem; quadratic mapping; approximately quadratic mapping; approximation; Ulam stability problem; normed linear space; complete normed linear space
References
2. S. Banach, Sur l'equation fonctionelle f(x + y) = f(x) + f(y), Fund. Math. 1 (1930), 123-124.
3. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 77-86.
4. I. Fenyö, Über eine Lösungsmethode gewisser Functionalgleichungen, Acta Math. Acad. Sci. Hung. 7 (1956), 383-396.
5. G.L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143-190.
6. K.G. Grosse-Erdmann, Regularity properties of functional equations and inequalities, Aequationes Math. 37 (1989), 233-251.
7. P.M. Gruber, Stability of Isometrics, Trans. Amer. Math. Soc., U. S. A. 245 (1978), 263-277.
8. D.H. Hyers, The stability of homomorphisms and related topics, Global Analysis-Analysis on Manifolds, Teubner - Texte zur Mathematik 57 (1983), 140-153.
9. A. Járai, L. Székelyhidi, Regularization and general methods in the theory of functional equations, Aequationes Math. 52 (1996), 10-29.
10. A.M. Kagan, Yu.V. Linnik, C.R.C. Rao, Characterization problems in mathematical statistics, John Wiley and Sons, New York 1973.
11. C.G. Khatri, C.R. Rao, Functional equations and characterization of probability laws through linear functions of random variables, J. Multiv. Anal. 2 (1972), 162-173.
12. M. Kuczma, An introduction to the theory of functional equations and inequalities, PWN. UŚ, Warszawa, Kraków, Katowice 1985.
13. S. Kurepa, A cosine functional equation in Hilbert space, Canad. J. Math. 12 (1957), 45-50.
14. L. Paganoni, On a functional equation concerning affine transformations, J. Math. Anal. Appl. 127 (1987), 475-491.
15. J.M. Rassias, On Approximation of Approximately Linear Mappings by Linear Mappings, J. Funct. Anal. 46 (1982), 126-130.
16. J.M. Rassias, On Approximation of Approximately Linear Mappings by Linear Mappings, Bull. Sc. Math. 108 (1984), 445-446.
17. J.M. Rassias, Solution of a Problem of Ulam, J. Approx. Th. 57 (1989), 268-273.
18. J.M. Rassias, Complete Solution of the Multi-dimensional Problem of Ulam, Discuss. Math. 14 (1994), 101-107.
19. J.M. Rassias, Solution of a Stability Problem of Ulam, Discuss. Math. 12 (1992), 95-103.
20. J.M. Rassias, On the Stability of the Euler-Lagrange Functional Equation, Chin. J. Math. 20 (1992), 185-190.
21. J.M. Rassias, On the Stability of the Non-linear Euler-Lagrange Functional Equation in Real Normed Linear Spaces, J. Math. Phys. Sci. 28 (1994), 231-235.
22. J.M. Rassias, On the Stability of the Multi-dimensional Non-linear Euler-Lagrange Functional Equation, Geometry, Analysis and Mechanics, World Sci. Publ. (1994), 275-285.
23. J.M. Rassias, On the Stability of the General Euler-Lagrange Functional Equation, Demonstr. Math. 29 (1996), 755-766.
24. J.M. Rassias, Solution of the Ulam Stability Problem for Euler-Lagrange quadratic mappings, J. Math. Anal. & Applications 220 (1998), 613-639.
25. J.M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matem. 34(54) (1999), 243-252.
26. J.M. Rassias, Solution of the Ulam stability problem for 3-dimensional Euler-Lagrange quadratic mappings, Mathem. Balkanica, 2000.
27. J.M. Rassias, Solution of the Ulam stability problem for Cubic Mappings, Glasnik Matem., 2000.
28. J.M. Rassias, M.J. Rassias, On the Hyers-Ulam Stability of Quadratic Mappings, J. Ind. Math. Soc., 2000.
29. A.L. Rutkin, The solution of the functional equation of d'Alembert's type for commutative groups, Internat. J. Math. Sci. 5 (1982), No 2.
30. F. Skof, Local properties ans approximations of operators (Italian), Rend-Sem. Mat. Fis., Milano 53 (1983), 113-129.
31. L. Székelyhidi, Functional equations on Abelian groups, Acta Math. Acad. Sci. Hungar. 37 (1981), 235-243.
32. L. Székelyhidi, Note on Hyers' theorem, C. R. Math. Rep. Sci. Canada 8 (1986), 127-129.
33. A. Tsutsumi, S. Haruki, On hypoelliptic functional equations, Math. Japonica 36(3) (1991), 581-590.
34. S.M. Ulam, A collection of mathematical problems, Interscience Publishers, Inc., New York 1968, p. 63.
Pedagogical Department E. E., Section of Mathematics and Informatics, National and Capodistrian University of Athens. Grece Greece
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
