ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS

doi:10.1016/S0252-9602(14)60120-X

数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (6): 1749-1760.doi: 10.1016/S0252-9602(14)60120-X

ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS

Driss ZEGLAMI|Ahmed CHARIFI*|Samir KABBAJ

Department of Mathematics, E.N.S.A.M, Moulay Ismail University, Meknes, Morocco； Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco

收稿日期:2013-09-10 修回日期:2014-02-25 出版日期:2014-11-20 发布日期:2014-11-20
通讯作者: Ahmed CHARIFI，charifi2000@yahoo.fr E-mail:zeglamidriss@yahoo.fr；charifi2000@yahoo.fr；samkabbaj@yahoo.fr

ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS

Driss ZEGLAMI|Ahmed CHARIFI*|Samir KABBAJ

Department of Mathematics, E.N.S.A.M, Moulay Ismail University, Meknes, Morocco； Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco

Received:2013-09-10 Revised:2014-02-25 Online:2014-11-20 Published:2014-11-20
Contact: Ahmed CHARIFI，charifi2000@yahoo.fr E-mail:zeglamidriss@yahoo.fr；charifi2000@yahoo.fr；samkabbaj@yahoo.fr

摘要/Abstract

摘要：

The aim of this paper is to investigate the superstability problem for the pex-iderized trigonometric functional equation
∑_φ∈Φ∫_Kf(xkφ(y)k⁻¹)dwK(k) =|Φ|g(x)h(y), x, y ∈ G,
where G is any topological group, K is a compact subgroup of G, !K is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of G and f, g, h are continuous complex-valued functions. Consequently, we have generalized the results of stability for d’Alembert´s and Wilson´s equa-tions by R. Badora, J. Baker, B. Bouikhalene, P. Gavruta, S. Kabbaj, Pl. Kannappan, G. H. Kim, J.M. Rassias, A. Roukbi, L. Sz´ekelyhidi, D. Zeglami, etc.

关键词: superstability, generalized Pexider d´Alembert equation, Wilson´s functional equation, group of morphisms

Abstract:

Key words: superstability, generalized Pexider d´Alembert equation, Wilson´s functional equation, group of morphisms

中图分类号:

39B72

Driss ZEGLAMI, Ahmed CHARIFI, Samir KABBAJ. ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(6): 1749-1760.

Driss ZEGLAMI, Ahmed CHARIFI, Samir KABBAJ. ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS[J]. Acta mathematica scientia,Series B, 2014, 34(6): 1749-1760.

参考文献

[1] Badora R. On Hyers-Ulam stability of Wilson´s functional equation. Aequationes Math, 2000, 60: 211–218

[2] Badora R. On a generalized Wilson functional equation. Georgian Math J, 2005, 12(4): 595–606

[3] Baker J A. The stability of the cosine equation. Proc Amer Math Soc, 1980, 80: 411–416

[4] Baker J A, Lawrence J, Zorzitto F. The stability of the equation f(x + y) = f(x)f(y). Proc Amer Math Soc, 1979, 74: 242–246

[5] Benson C, Jenkins J, Ratkliff G. Bounded K-spherical functions on Heisenberg groups. J Funct Anal, 1992, 105: 409–443

[6] Bouikhalene B. On Hyers-Ulam stability of generalised Wilson´s equations. J Inequal Pure Appl Math, 2004, 5(4): Article 100

[7] Bouikhalene B, Kabbaj S. Gelfand pairs and generalised d´Alembert´s and Cauchy´s functional equations. Georgian Math J, 2005, 12(2): 207–216

[8] Bouikhalene B, Elqorachi E, Rassias J M. The superstability of d´Alembert´s functional equation on the Heisenberg group. Appl Math Lett, 2010, 23(1): 105–109

[9] Brzd¸ek J, Popa D, Xu B. The Hyers-Ulam stability of linear equations of higher orders. Acta Math Hungar, 2008, 120: 1–8

[10] Cao H X, Lv J R, Rassias J M. Superstability for generalized module left derivations and generalized module derivations on a Banach module (II). J Pure & Appl Math, 2009, 10(2): 1–8

[11] Cao P, Xu B. Note on the superstability of d´Alembert type functional equations. Bull Korean Math Soc, 2009, 46(2): 235–243

[12] Cho Y J, Rassias Th M, Saadati R. Stability of Functional Equations in Random Normed Spaces. Springer
Optimization and Its Applications, Vol 86. Springer, 2013

[13] Corovei I. The functional equation f(xy) + f(yx) + f(xy−1) + f(y−1x) = 4f(x)f(y) for nilpotent groups. Buletinul S¸tiint¸ific al Institutului Politehnic Cluj-Napoca, 1977, 20: 25–28

[14] Davison T M K. D´Alembert´s functional equation on topological monoids. Publ Math Debrecen, 2009, 75(1/2): 41–66

[15] Elqorachi E, Akkouchi M. On Hyers-Ulam stability of Cauchy and Wilson equations. Georgian Math J, 2004, 11(1): 69–82

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

[17] Gajda Z. A Generalization of D´Alembert´s Functional Equation. Funkcial Ekvac, 1990, 33: 69–77

[18] Hyers D H, Isac G I, Rassias Th M. Stability of Functional Equations in Several Variables. Boston, Basel, Berlin: Birkh¨auser Verlag, 1998

[19] Kannappan Pl, Kim G H. On the stability of the generalized cosine functional equations. Ann Acad Pedagog Crac Stud Math, 2001, 1: 49–58

[20] Kim G H. The Stability of pexiderized cosine functional equations. Korean J Math, 2008, 16(1): 103–114

[21] Rassias J M. On approximately of approximately linear mappings by linear mappings. J Funct Anal, 1982, 46: 126–130

[22] Rassias J M, Ravi K, Arunkumar M, Senthil Kumar B V. Ulam stability of mixed type cubic and additive functional equation. Functional Ulam Notions (F.U.N) Nova Science Publishers, 2010, Chapter 13, 149–175

[23] Rassias J M, Son E, Kim H M. On the Hyers-Ulam stability of 3D and 4D mixed type mappings. Far East J Math Sci, 2011, 48(1): 83–102

[24] Rassias J M, Rassias M J. Refined Ulam stability for Euler-Lagrange type mappings in Hilbert spaces. Intern J Appl Math Stat, 2007, 7: 126–132

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

[26] Ravi K, Rassias J M, Arunkumar M, Kodandan R. Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation. J Ineq Pure Appl Math, 2009, 10(4): 1–29

[27] Redouani A, Elqorachi E, Rassias Th M. The superstability of d’Alembert’s functional equation on step 2 nilpotent groups. Aequationes Math, 2007, 74(3): 237–248

[28] Roukbi A, Zeglami D, Kabbaj S. Hyers-Ulam stability of Wilson’s functional equation. Math Sciences: Adv Appl, 2013, 22: 19–26

[29] Stetkær H. d’Alembert’s equation and spherical functions. Aeq Math, 1994, 48: 220–227

[30] Stetkaer H. Wilson’s functional equations on groups. Aeq Maths, 1995, 49: 252–275

[31] Stetkær H. Functional equations on abelian groups with involution. Aeq Math, 1997, 54: 144–172

[32] Stetkaer H. Functional equations and matrix-valued spherial function. Aeq Maths, 2005, 69(3): 271–292

[33] Sz´ekelyhidi L. On a theorem of Baker, Lawrence and Zorzitto. Proc Amer Math Soc, 1982, 84: 95–96

[34] Sz´ekelyhidi L. On a stability theorem. C R Math Rep Acad Sci Canada, 1981, 3: 253–255

[35] Tyrala I. The stability of the second generalization of d’Alembert’s functional equation. Scientic Issues,
Mathematics XIII, Cz¸e stochowa, 2008

[36] Ulam S M. A Collection of Mathematical Problems. New York: Interscience Publ, 1961; Problems in Modern Mathematics. New York: Wiley, 1964

[37] Xu T Z, Rassias J M, Rassias M J, Xu W X. A fixed point approach to the stability of quintic and sextic functional equations in quasi- -normed spaces. J Inequal Appl 2010, Art ID 423231

[38] Zeglami D, Roukbi A, Kabbaj S. Hyers-Ulam Stability of Generalized Wilson´s and d´Alembert´s Functional
equations. Afr Mat, 2013, DOI: 10.1007/s13370-013-0199-6

[39] Zeglami D, Kabbaj S, Roukbi A. Superstability of a generalization of the cosine equation. Br J Math Comput Sci, 2014, 4(5): 719–734

ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS

ON THE SUPERSTABILITY OF THE PEXIDER TYPE GENERALIZED TRIGONOMETRIC FUNCTIONAL EQUATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

Metrics

本文评价

推荐阅读 10

[1]	Anna BAHYRYCZ, Krzysztof CIEPLINSKI, Jolanta OLKO. ON AN EQUATION CHARACTERIZING MULTI-CAUCHY-JENSEN MAPPINGS AND ITS HYERS-ULAM STABILITY[J]. 数学物理学报(英文版), 2015, 35(6): 1349-1358.
[2]	许天周, Rassias John Michael, 许婉欣. A FIXED POINT APPROACH TO THE STABILITY OF A GENERAL MIXED ADDITIVE-CUBIC EQUATION ON BANACH MODULES[J]. 数学物理学报(英文版), 2012, 32(3): 866-892.