RANDOM APPROXIMATION OF AN ADDITIVE FUNCTIONAL EQUATION OF m-APPOLLONIUS TYPE

doi:10.1016/S0252-9602(12)60142-8

摘要/Abstract

摘要：

In this paper, using the fixed-point and direct methods, we prove the Hyers-Ulam stability of the following m-Appolonius type functional equation:
∑^m_i₌₁f(z − x_i) = mf(z −1/m²∑^m_i₌₁x_i)−1/m∑_1≤i<j≤mf(x_i + x_j),
where m is a natural number greater than 1, in random normed spaces.

关键词: Hyers-Ulam stability, additive functional equation, random normed space, fixed point method

Abstract:

Key words: Hyers-Ulam stability, additive functional equation, random normed space, fixed point method

中图分类号:

39B82

Hassan Azadi Kenary. RANDOM APPROXIMATION OF AN ADDITIVE FUNCTIONAL EQUATION OF m-APPOLLONIUS TYPE[J]. 数学物理学报(英文版), 2012, 32(5): 1813-1825.

Hassan Azadi Kenary. RANDOM APPROXIMATION OF AN ADDITIVE FUNCTIONAL EQUATION OF m-APPOLLONIUS TYPE[J]. Acta mathematica scientia,Series B, 2012, 32(5): 1813-1825.

参考文献

[1] Arriola L M, Beyer W A. Stability of the Cauchy functional equation over p-adic fields. Real Anal Exchange, 2005/06, 31: 125–132

[2] Azadi Kenary H. Stability of a cubic functional equation in random normed spaces. J Math Extension, 2009, 4(1): 1–11

[3] Azadi Kenary H. Non-Archimedean stability of Cauchy-Jensen type functional equation. Int J Nonlinear Anal Appl, 2010, 1(2): 1–10

[4] Azadi Kenary H. Hyers-Rassias stability of the Pexiderial functional equation. to appear in Ital J Pure Appl Math

[5] Azadi Kenary H. Stability of a Pexiderial functional equation in random normed spaces. Rend Circ Mat Palermo, 2011, 60: 59–68

[6] Cholewa P W. Remarks on the stability of functional equations. Aequat Math, 1984, 27: 76–86

[7] Czerwik S. Functional Equations and Inequalities in Several Variables. River Edge, NJ: World Scientific, 2002

[8] Eshaghi Gordji M, Bavand Savadkouhi M. Stability of mixed type cubic and quartic functional equations in random normed spaces. J Inequal Appl, 2009, 2009: Article ID 527462

[9] Eshaghi Gordji M, Bavand Savadkouhi M, Park C. Quadratic-quartic functional equations in RN-spaces. J Inequal Appl, 2009, 2009: Article ID 868423

[10] Eshaghi Gordji M, Khodaei H. Stability of Functional Equations. Lap Lambert Academic Publishing, 2010

[11] Eshaghi Gordji M, Zolfaghari S, Rassias J M, Savadkouhi M B. Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces. Abst Appl Anal, 2009, 2009: Article ID 417473

[12] Fechner W. Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequat Math, 2006, 71: 149–161

[13] Gˇavruta P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl, 1994, 184: 431–436

[14] Hyers D H. On the stability of the linear functional equation. Proc Natl Acad Sci USA, 1941, 27: 222–224

[15] Hyers D H, Isac G, Rassias T H. Stability of Functional Equations in Several Variables. Basel: Birkhäuser, 1998

[16] Jun K, Kim H. On the Hyers-Ulam-Rassias stability problem for approximately k-additive mappings and functional inequalities. Math Inequal Appl, 2007, 10: 895–908

[17] Jung S M. Hyers-Ulam-Rassias stability of Jensen´s equation and its application. Proc Amer Math Soc, 1998, 126: 3137–3143

[18] Khodaei H, Rassias T H. Approximately generalized additive functions in several variabels. Int J Nonlinear Anal Appl, 2010, 1(1): 22–41

[19] Li L, Chung J, Kim D. Stability of Jensen equations in the space of generalized functions. J Math Anal Appl, 2004, 299: 578–586

[20] Mihet D, Radu V. On the stability of the additive Cauchy functional equation in random normed spaces. J Math Anal Appl, 2008, 343: 567–572

[21] Moradlou F, Vaezi H, Park C. Fixed Points and Stability of an Additive Functional Equation of n-Apollonius Type in C-Algebras. Abs Appl Anal, 2008, 2008: Article ID 672618

[22] Park C G, Wee H J. Generalized Hyers-Ulam-Rassias stability of a functional equation in three variabels. J Chungcheong Math Soc, 2005, 18: 41–49

[23] Park C. Fuzzy stability of a functional equation associated with inner product spaces. Fuzzy Sets and Systems, 2009, 160: 1632–1642

[24] Park C. Generalized Hyers-Ulam-Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over C*-algebras. J Comput Appl Math, 2005, 180: 279–291

[25] Park C, Hou J, Oh S. Homomorphisms between JC-algebras and Lie C-algebras. Acta Math Sin (Engl Ser), 2005, 21(6): 1391–1398

[26] Parnami J C, Vasudeva H L. On Jensen’s functional equation. Aequat Math, 1992, 43: 211–218

[27] Rassias T H. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72: 297–300

[28] Rassias T H. On the stability of functional equations and a problem of Ulam. Acta Applicandae Math, 2000, 62: 23–130

[29] R¨atz J. On inequalities associated with the Jordan-von Neumann functional equation. Aequat Math, 2003, 66: 191–200

[30] Saadati R, Park C. Non-Archimedean L-fuzzy normed spaces and stability of functional equations (in press).

[31] Saadati R, Vaezpour M, Cho Y J. A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”. Inequal Appl, 2009, 2009: Article ID 214530

[32] Saadati R, Zohdi M M, Vaezpour S M. Nonlinear L-random stability of an ACQ-functional equation. J Inequal Appl, 2011, 2011: Article ID 194394

[33] Schewizer B, Sklar A. Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. New York: North-Holland, 1983

[34] Skof S. Local properties and approximation of operators. Rend Sem Mat Fis Milano, 1983, 53: 113–129

[35] Ulam S M. Problems in Modern Mathematics. Science Editions, New York: John Wiley and Sons, 1964

[36] Xu T Z, Rassias J M, Xu W X. Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. J Math Phys, 2010, 51: 093508

[37] Xu T Z, Rassias J M, Xu W X. Intuitionistic fuzzy stability of a general mixed additive-cubic equation. J Math Phys, 2010, 51: 063519

[38] Xu T Z, Rassias J M, Xu W X. On the stability of a general mixed additive-cubic functional equation in random normed spaces. J Inequ Appl, 2010, 2010: Article ID 328473

[39] Xu T Z, Rassias J M, Xu W X. A fixed point approach to the stability of a general mixed AQCQfunctional equation in non-Archimedean normed spaces. Discrete Dynamics in Nature and Society, 2010, 2010: Article ID 812545