On new stability results for generalized Cauchy functional equations on groups by using Brzdȩk's fixed point theorem

详细信息    查看全文

• 作者：Laddawan Aiemsomboon ; Wutiphol Sintunavarat
• 关键词：Brzdȩk’s fixed point theorem ; generalized Cauchy function equations ; Ulam–Hyers’s stability
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：18
• 期：1
• 页码：45-59
• 全文大小：595 KB
• 参考文献：1.J. Brzdȩk, A note on stability of additive mappings. In: Stability of Mappings of Hyers-Ulam Type, Hadronic Press, Palm Harbor, FL, 1994, 19–22.
  2.Brzdȩk J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141, 58–67 (2013)CrossRef MathSciNet
  3.J. Brzdȩk, Stability of additivity and fixed point methods. Fixed Point Theory Appl. 2013 (2013), doi:10.​1186/​1687-1812-2013-285 , 9 pages.
  4.Brzdȩk J., Ciepliński K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. 74, 6861–6867 (2011)CrossRef MathSciNet
  5.Hayes W., Jackson K.R.: A survey of shadowing methods for numerical solutions of ordinary differential equations. Appl. Numer. Math. 53, 299–321 (2005)CrossRef MathSciNet MATH
  6.Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)CrossRef MathSciNet
  7.K. Palmer, Shadowing in Dynamical Systems. Theory and Applications. Mathematics and Its Applications 501, Kluwer Academic Publishers, Dordrecht, 2000.
  8.S. Yu. Pilyugin, Shadowing in Dynamical Systems. Lectures Notes in Mathematics 1706, Springer-Verlag, Berlin, 1999.
  9.Rassias T.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)CrossRef MathSciNet MATH
  10.T. M. Rassias, Problem 16; 2. Report of the 27th International Symposium on Functional Equations, Aequ. Math. 39, 1990, 292–293.
  11.Rassias T.M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)CrossRef MathSciNet MATH
  12.S., S. : Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos. Chaos Solitons Fractals 35, 238–245 (2008)CrossRef MathSciNet
  13.S. M. Ulam, A Collection of Mathematical Problems. Interscience, New York, 1960.
  
• 作者单位：Laddawan Aiemsomboon (1)
  Wutiphol Sintunavarat (1)
  
  1. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

In this work we present a new type of stability results for generalized Cauchy functional equation of the form $$f(ax \ast by) = af(x) \diamond bf(y),$$where \({a, b \in \mathbb{N}}\) and \({f}\) is a mapping from a commutative semigroup (\({G_1, \ast}\)) to a commutative group (\({G_2, \diamond}\)). Using this form we generalize, extend and complement some earlier classical results concerning the stability of additive Cauchy functional equations. Our results are improvement and generalization of main results of Brzdȩk [Fixed Point Theory Appl. 2013 (2013), doi:10.​1186/​1687-1812-2013-285:285] and many results in literature. Some of the stability results for many types of functional equations are given here to illustrate the usability of the obtained results. Keywords Brzdȩk’s fixed point theorem generalized Cauchy function equations Ulam–Hyers’s stability Mathematics Subject Classification 47H10 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 (13) References1.J. Brzdȩk, A note on stability of additive mappings. In: Stability of Mappings of Hyers-Ulam Type, Hadronic Press, Palm Harbor, FL, 1994, 19–22.2.Brzdȩk J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141, 58–67 (2013)CrossRefMathSciNet3.J. Brzdȩk, Stability of additivity and fixed point methods. Fixed Point Theory Appl. 2013 (2013), doi:10.​1186/​1687-1812-2013-285, 9 pages.4.Brzdȩk J., Ciepliński K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. 74, 6861–6867 (2011)CrossRefMathSciNet5.Hayes W., Jackson K.R.: A survey of shadowing methods for numerical solutions of ordinary differential equations. Appl. Numer. Math. 53, 299–321 (2005)CrossRefMathSciNetMATH6.Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)CrossRefMathSciNet7.K. Palmer, Shadowing in Dynamical Systems. Theory and Applications. Mathematics and Its Applications 501, Kluwer Academic Publishers, Dordrecht, 2000.8.S. Yu. Pilyugin, Shadowing in Dynamical Systems. Lectures Notes in Mathematics 1706, Springer-Verlag, Berlin, 1999.9.Rassias T.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)CrossRefMathSciNetMATH10.T. M. Rassias, Problem 16; 2. Report of the 27th International Symposium on Functional Equations, Aequ. Math. 39, 1990, 292–293.11.Rassias T.M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)CrossRefMathSciNetMATH12.S., S. : Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos. Chaos Solitons Fractals 35, 238–245 (2008)CrossRefMathSciNet13.S. M. Ulam, A Collection of Mathematical Problems. Interscience, New York, 1960. About this Article Title On new stability results for generalized Cauchy functional equations on groups by using Brzdȩk’s fixed point theorem Journal Journal of Fixed Point Theory and Applications Volume 18, Issue 1 , pp 45-59 Cover Date2016-03 DOI 10.1007/s11784-015-0259-7 Print ISSN 1661-7738 Online ISSN 1661-7746 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Analysis Mathematical Methods in Physics Keywords 47H10 39B82 Brzdȩk’s fixed point theorem generalized Cauchy function equations Ulam–Hyers’s stability Authors Laddawan Aiemsomboon (1) Wutiphol Sintunavarat (1) Author Affiliations 1. Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani, 12121, Thailand Continue reading... To view the rest of this content please follow the download PDF link above.