Multiplicative type functional equations in a ring

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• 作者：Jaeyoung Chung ; Jeongwook Chang
• 关键词：Exponential function ; free abelian group ; general solution ; multiplicative function ; number theory ; Ulam–Hyers stability
• 刊名：Aequationes Mathematicae
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：90
• 期：2
• 页码：367-379
• 全文大小：499 KB
• 参考文献：1.Albert M., Baker J.A.: Bounded solutions of a functional inequality. Can. Math. Bull. 25, 491–495 (1982)MathSciNet CrossRef MATH
  2.Baker J.A., Lawrence J., Zorzitto F.: The stability of the equation f(x + y)) = f(x)f(y). Proc. Am. Math. Soc. 74, 242–246 (1979)MathSciNet MATH
  3.Chung, J.: On an exponential functional inequality and its distributional version. Can. Math. Bull. doi:10.​4153/​CMB-2014-012-x
  4.Chung, J., Chang, J.: On two functional equations originating from number theory, to appear in Proc. Indian Acad. Sci.
  5.Chung, J., Riedel, T., Sahoo, P.K.: Stability of functional equations arising from number theory and determinant of matrices (preprint)
  6.Chung J.K., Sahoo P.K.: General solution of some functional equations related to the determinant of some symmetric matrices. Demonstratio Math. 35, 539–544 (2002)MathSciNet MATH
  7.Chudziak J., Tabor J.: On the stability of the Goła̧b-Schinzel functional equation. Jour. Math. Anal. Appl. 302, 196–200 (2005)MathSciNet CrossRef MATH
  8.Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of functional equations in several variables. Birkhauser (1998)
  9.Houston K.B., Sahoo P.K.: On two functional equations and their solutions. Appl. Math. Lett. 21, 974–977 (2008)MathSciNet CrossRef MATH
  10.Jung S.M., Bae J.H.: Some functional equations arising from number theory. Proc. Indian Acad. Sci. Math. Sci. 113(2), 91–98 (2003)MathSciNet CrossRef MATH
  11.Lehmer D.H.: A cotangent analogue of continued fractions. Duke Math. J. 4(2), 323–340 (1938)MathSciNet CrossRef MATH
  12.Riedel T., Sahoo P.K.: On a generalization of a functional equation associated with the distance between the probability distributions. Publ. Math. Debrecen 46(1–2), 125–135 (1995)MathSciNet MATH
  13.Najdecki, J.A.: On stability of functional equation connected with the Reynolds operator. J. Inequal. Appl. 2007, Article ID 79816, 3 pages (2007)
  14.Sahoo, P.K.: Solved and unsolved problems, problem 2, News Letter of the European Math. Soc., 58, 43–44 (2005)
  15.Taussky O.: Sums of squares. Am. Math. Monthly 77, 805–830 (1970)MathSciNet CrossRef MATH
  16.Todd J.: A problem on arc tangent relations. Am. Math. Monthly 56, 517–528 (1949)MathSciNet CrossRef MATH
  
• 作者单位：Jaeyoung Chung (1)
  Jeongwook Chang (2)
  
  1. Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea
  2. Department of Mathematics Education, Dankook University, Yongin, 448-701, Korea
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Combinatorics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8903

文摘

Let R be a ring, \({\mathbb{F}}\) be a field and \({K \subset \mathbb{R}}\) an integral domain. In this paper we investigate general solutions \({f : K^2\to \mathbb{R}^+}\) of the functional equations $$\begin{array}{ll}f(ux-vy, uy+vx)\,=\,f(x, y)f(u, v),\\f(ux+vy, uy-vx)\,=\,f(x, y)f(u, v)\end{array}$$for all \({x, y\in K}\), and general solutions \({f:R^2\to \mathbb{R}^+}\) of the functional equations $$\begin{array}{ll}f(ux+vy, uy+vx)\,=\,f(x, y)f(u, v),\\ f(ux-vy, uy-vx)\,=\,f(x, y)f(u, v)\end{array}$$for all \({x, y\in R}\). The above functional equations arise from number theory and are connected with the characterizations of the determinant and permanent of two-by-two matrices. Keywords Exponential function free abelian group general solution multiplicative function number theory Ulam–Hyers stability Mathematics Subject Classification 39B82 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (16) References1.Albert M., Baker J.A.: Bounded solutions of a functional inequality. Can. Math. Bull. 25, 491–495 (1982)MathSciNetCrossRefMATH2.Baker J.A., Lawrence J., Zorzitto F.: The stability of the equation f(x + y)) = f(x)f(y). Proc. Am. Math. Soc. 74, 242–246 (1979)MathSciNetMATH3.Chung, J.: On an exponential functional inequality and its distributional version. Can. Math. Bull. doi:10.​4153/​CMB-2014-012-x 4.Chung, J., Chang, J.: On two functional equations originating from number theory, to appear in Proc. Indian Acad. Sci.5.Chung, J., Riedel, T., Sahoo, P.K.: Stability of functional equations arising from number theory and determinant of matrices (preprint)6.Chung J.K., Sahoo P.K.: General solution of some functional equations related to the determinant of some symmetric matrices. Demonstratio Math. 35, 539–544 (2002)MathSciNetMATH7.Chudziak J., Tabor J.: On the stability of the Goła̧b-Schinzel functional equation. Jour. Math. Anal. Appl. 302, 196–200 (2005)MathSciNetCrossRefMATH8.Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of functional equations in several variables. Birkhauser (1998)9.Houston K.B., Sahoo P.K.: On two functional equations and their solutions. Appl. Math. Lett. 21, 974–977 (2008)MathSciNetCrossRefMATH10.Jung S.M., Bae J.H.: Some functional equations arising from number theory. Proc. Indian Acad. Sci. Math. Sci. 113(2), 91–98 (2003)MathSciNetCrossRefMATH11.Lehmer D.H.: A cotangent analogue of continued fractions. Duke Math. J. 4(2), 323–340 (1938)MathSciNetCrossRefMATH12.Riedel T., Sahoo P.K.: On a generalization of a functional equation associated with the distance between the probability distributions. Publ. Math. Debrecen 46(1–2), 125–135 (1995)MathSciNetMATH13.Najdecki, J.A.: On stability of functional equation connected with the Reynolds operator. J. Inequal. Appl. 2007, Article ID 79816, 3 pages (2007)14.Sahoo, P.K.: Solved and unsolved problems, problem 2, News Letter of the European Math. Soc., 58, 43–44 (2005)15.Taussky O.: Sums of squares. Am. Math. Monthly 77, 805–830 (1970)MathSciNetCrossRefMATH16.Todd J.: A problem on arc tangent relations. Am. Math. Monthly 56, 517–528 (1949)MathSciNetCrossRefMATH About this Article Title Multiplicative type functional equations in a ring Journal Aequationes mathematicae Volume 90, Issue 2 , pp 367-379 Cover Date2016-04 DOI 10.1007/s00010-015-0340-8 Print ISSN 0001-9054 Online ISSN 1420-8903 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Analysis Combinatorics Keywords 39B82 Exponential function free abelian group general solution multiplicative function number theory Ulam–Hyers stability Industry Sectors Finance, Business & Banking Authors Jaeyoung Chung (1) Jeongwook Chang (2) Author Affiliations 1. Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea 2. Department of Mathematics Education, Dankook University, Yongin, 448-701, Korea Continue reading... To view the rest of this content please follow the download PDF link above.