SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

doi:10.1016/S0252-9602(16)30040-6

数学物理学报(英文版) ›› 2016, Vol. 36 ›› Issue (3): 791-801.doi: 10.1016/S0252-9602(16)30040-6

SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

Bouikhalene BELAID¹, Elqorachi ELHOUCIEN²

1. Bouikhalene Belaid, Polydisciplinary Faculty, University Sultan Moulay Slimane, Beni-Mellal, Morocco;
2. Elqorachi Elhoucien, Functional Equations and Applications Team, Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Agadir, Morocco

收稿日期:2015-03-26 修回日期:2015-07-15 出版日期:2016-06-25 发布日期:2016-06-25
作者简介:Bouikhalene BELAID,E-mail:bbouikhalene@yahoo.fr;Elqorachi ELHOUCIEN,E-mail:elqorachi@hotamail.com

SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

Bouikhalene BELAID¹, Elqorachi ELHOUCIEN²

1. Bouikhalene Belaid, Polydisciplinary Faculty, University Sultan Moulay Slimane, Beni-Mellal, Morocco;
2. Elqorachi Elhoucien, Functional Equations and Applications Team, Department of Mathematics, Faculty of Sciences, University Ibn Zohr, Agadir, Morocco

Received:2015-03-26 Revised:2015-07-15 Online:2016-06-25 Published:2016-06-25

摘要/Abstract

摘要：

In this paper we study the solutions and stability of the generalized Wilson's functional equation∫_Gf(xty)dμ(t) +∫_Gf(xtσ(y))dμ(t)=2f(x)g(y), x, y∈G, where G is a locally compact group, σ is a continuous involution of G and μ is an idempotent complex measure with compact support and which is σ-invariant. We show that∫_Gg(xty)dμ(t)+∫_Gg(xtσ(y))dμ(t)=2g(x)g(y) if f≠0 and∫_Gf(t.)dμ(t)≠0, where[∫_Gf(t.)dμ(t)](x)=∫_Gf(tx)dμ(t). We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy)+χ(y)f(xσ(y))=2f(x)g(y)x, y∈G, where χ is a unitary character of G.

关键词: d'Alembert's functional equation, locally compact group, involution, character, complex measure, Wilson's functional equation, Hyers-Ulam stability

Abstract:

Key words: d'Alembert's functional equation, locally compact group, involution, character, complex measure, Wilson's functional equation, Hyers-Ulam stability

中图分类号:

39B82

Bouikhalene BELAID, Elqorachi ELHOUCIEN. SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION[J]. 数学物理学报(英文版), 2016, 36(3): 791-801.

Bouikhalene BELAID, Elqorachi ELHOUCIEN. SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION[J]. Acta mathematica scientia,Series B, 2016, 36(3): 791-801.

参考文献

[1] Badora R. On the stability of cosine functional equation. Wyzszkola Red Krakow Rocz Nauk-Dydakt Pr Mat, 1998, 15: 1-14
[2] Badora R. On the stability of a functional equation for generalized trigonometric functions//Rassias Th M, ed. Functional Equations and Inequalities. Kluwer Academic Publishers, 2000: 1-5
[3] Badora R. Note on the superstability of the Cauchy functional equation. Pub Math Debrecen, 2000, 57(3/4): 421-424
[4] Baker J A. The stability of the cosine equation. Proc Amer Math Soc, 1980, 80: 411-416
[5] 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
[6] 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
[7] Bouikhalene B, Elqorachi E. Stability of a generalization Wilson's equation. Aequat Math, 2015, DOI: 10-1007/s00010-015-0356-0
[8] Bouikhalene B, Elqorachi E. Stability of the spherical functions. Georgian Math J, (to appears)
[9] Chung J, Sahoo P K. Stability of Wilson's functional equations with involutions. Aequat Math, 2015, 89(3): 749-763
[10] D'Alembert, Jean Le Round. Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration. Hist Acad Berlin, 1750: 355-360
[11] Davison T M K. D'Alembert's functional equation on topological groups. Aequat Math, 2008, 76: 33-53
[12] Davison T M K. D'Alembert's functional equation on topological monoids. Publ Math Debrecen, 2009, 75(1/2): 41-66
[13] Ebanks Bruce R, Stetkær H. On Wilson's functional equations. Aequat Math, 2015, 89(2): 339-354
[14] Elqorachi E, Akkouchi M. On generalized d'Alembert and Wilson functional equations. Aequat Math, 2003, 66(3): 241-256
[15] Elqorachi E, Akkouchi M. The superstability of the generalized d'Alembert functional equation. Georgian Math J, 2003, 10: 503-508
[16] Ger R. Superstability is not natural. Rocznik Nauk-Dydakt Prace Mat, 1993, 159(13): 109-123
[17] Hyers, D. H. On the stability of the linear functional equation. Proc Nat Acad Sci, 1941, 27: 222-224
[18] Kannappan Pl. The functional equation f(xy)+f(xy^-1)=2f(x)f(y) for groups. Proc Amer Math Soc, 1968, 19: 69-74
[19] Kannappan Pl, Kim G H. On the stability of the generalized cosine functional equations. Stud Math Ann Acad Paedagogicae Cracowiensis, 2001, 4: 49-57
[20] Kim G H. The stability of d'Alembert and Jensen type functional equations. J Math Anal Appl, 2007, 325: 237-248
[21] Kim G H. On the stability of trigonometric functional equations. Adv Differ Equ, 2007, 2007: Article ID 90405
[22] Kim G H. On the stability of the Pexiderized trigonometric functional equation. Appl Math Comput, 2008, 203(1): 99-105
[23] Lukasik R. Some generalization of the quadratic and Wilson's functional equation. Aequat Math, 2014, 87(1/2): 105-123
[24] Rassias J M. On approximation of approximately linear mappings by linear mappings. J Funct Anal, 1982, 46: 126-130
[25] Rassias Th M. On the stability of linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72: 297-300
[26] Redouani A, Elqorachi E, Rassias Th M. The superstability of d'Alembert's functional equation on step 2 nilpotent groups. Aequat Math, 2007, 74(3): 226-241
[27] Stetkær H. Functional Equations on Groups. Singapore: World Scientific, 2013
[28] Stetkær H. D'Alembert's functional equation on groups. Banach Center Publ, 2013, 99: 173-191
[29] Stetkær H. A link between Wilson's and d'Alembert's functional equations. Aequat Math, 2015, DOI: 10.1007/s00010-015-0336-4
[30] Székelyhidi L. The stability of d'Alembert-type functional equations. Acta Sci Math, 1982, 44: 313-320
[31] Székelyhidi L. On a theorem of Baker, Lawrence and Zorzitto. Proc Amer Math Soc, 1982, 84: 95-96
[32] Wilson W H. On certain related functional equations. Bull Amer Math Soc, 1920, 26: 300-312
[33] Ulam S M. A Collection of Mathematical Problems. New York: Interscience Publ, 1961; Problems in Modern Mathematics. New York: Wiley, 1964

SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	袁益让, 程爱杰, 羊丹平, 李长峰, 杨青. CONVERGENCE ANALYSIS OF MIXED VOLUME ELEMENT-CHARACTERISTIC MIXED VOLUME ELEMENT FOR THREE-DIMENSIONAL CHEMICAL OIL-RECOVERY SEEPAGE COUPLED PROBLEM[J]. 数学物理学报(英文版), 2018, 38(2): 519-545.
[2]	Iz-iddine EL-FASSI, Janusz BRZDȨK, Abdellatif CHAHBI, Samir KABBAJ. ON HYPERSTABILITY OF THE BIADDITIVE FUNCTIONAL EQUATION[J]. 数学物理学报(英文版), 2017, 37(6): 1727-1739.
[3]	王定怀, 周疆, 陈文艺. ANOTHER CHARACTERIZATIONS OF MUCKENHOUPT A_p CLASS[J]. 数学物理学报(英文版), 2017, 37(6): 1761-1774.
[4]	陈中川. RECURRENCE FOR WEIGHTED TRANSLATIONS ON GROUPS[J]. 数学物理学报(英文版), 2016, 36(2): 443-452.
[5]	A. Vinodkumar, K. Malar, M. Gowrisankar, P. Mohankumar. EXISTENCE, UNIQUENESS AND STABILITY OF RANDOM IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2016, 36(2): 428-442.
[6]	M. MURSALEEN, Khursheed J. ANSARI, Asif KHAN. STABILITY OF SOME POSITIVE LINEAR OPERATORS ON COMPACT DISK[J]. 数学物理学报(英文版), 2015, 35(6): 1492-1500.
[7]	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.
[8]	Choonkil PARK. QUADRATIC ρ-FUNCTIONAL INEQUALITIES IN BANACH SPACES[J]. 数学物理学报(英文版), 2015, 35(6): 1501-1510.
[9]	袁益让, 程爱杰, 羊丹平, 李长峰. THEORY AND APPLICATION OF FRACTIONAL STEP CHARACTERISTIC FINITE DIFFERENCE METHOD IN NUMERICAL SIMULATION OF SECOND ORDER ENHANCED OIL PRODUCTION[J]. 数学物理学报(英文版), 2015, 35(6): 1547-1565.
[10]	陈宇, 周忆. SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS[J]. 数学物理学报(英文版), 2015, 35(4): 855-875.
[11]	Krzysztof CIEPLINSKI, Angelika SUROWCZYK. ON THE HYERS-ULAM STABILITY OF AN EQUATION CHARACTERIZING MULTI-QUADRATIC MAPPINGS[J]. 数学物理学报(英文版), 2015, 35(3): 690-702.
[12]	孟媛媛, 王彦英. INVOLUTIONS WITH FIXED POINT SET RP(2m) ? P(2m, 2n + 1)[J]. 数学物理学报(英文版), 2014, 34(2): 331-342.
[13]	吴昭君. AN INEQUALITY ON THE ZEROS AND POLES OF VECTOR VALUED MEROMORPHIC FUNCTIONS[J]. 数学物理学报(英文版), 2014, 34(2): 380-386.
[14]	段萍, 杜金元. RIEMANN-HILBERT CHARACTERIZATION FOR MAIN BESSEL POLYNOMIALS WITH VARYING LARGE NEGATIVE PARAMETERS[J]. 数学物理学报(英文版), 2014, 34(2): 557-567.
[15]	周芳, 米芳, 刘合国. THE MAXIMAL NUMBER OF I_π-CHARACTERS OVER NORMAL SUBGROUPS[J]. 数学物理学报(英文版), 2014, 34(1): 209-214.