Approximation of almost quadratic mappings via modular functional

Volume 29, Issue 2, pp 106--117 http://dx.doi.org/10.22436/jmcs.029.02.01

Publication Date: August 24, 2022 Submission Date: February 09, 2022 Revision Date: July 01, 2022 Accteptance Date: July 07, 2022

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Authors

I.-S. Chang - Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Korea. H.-M. Kim - Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Korea. H.-W. Lee - Department of Mathematics, Chungnam National University, 99 Daehangno, Yuseong-gu, Daejeon 34134, Korea.

Abstract

In this paper, we present generalized stability results of refined quadratic functional equation \[ f(ax-by)=abf(x-y)+a(a-b)f(x)+b(b-a)f(y), \] for any fixed nonzero integer numbers \(a,b\in\mathbb{Z}\) with \(a\neq b\) in modular spaces. As results, we generalize stability results of a quadratic functional equation in [{K.-W. Jun, H.-M. Kim, J. Son, Functional Equations in Mathematical Analysis, \(\bf{2012}\) (2012), 153--164}].

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Keywords

• Generalized Hyers-Ulam stability
• modular functional
• modular spaces

MSC

•  39B82
•  39B72
•  16W25

References

• [1] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge (1989)
  • View Article
  • MathSciNet


• [2] C. Borelli, G. L. Forti, On a general Hyers--Ulam stability result, Internat. J. Math. Math. Sci., 18 (1995), 229--236
  • View Article
  • MathSciNet


• [3] N. Brillouët-Belluot, J. Brzdȩk, K. Ciepliński, On some recent developments in Ulam’s type stability, Abstr. Appl. Anal., 2012 (2012), 41 pages
  • View Article
  • MathSciNet
  • MATH


• [4] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59--64
  • View Article
  • MathSciNet


• [5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., River Edge (2002)
  • View Article
  • MathSciNet


• [6] G.-L. Forti, Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations, J. Math. Anal. Appl., 295 (2004), 127--133
  • View Article
  • MathSciNet


• [7] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431--436
  • View Article
  • MathSciNet
  • MATH


• [8] M. B. Ghaemi, M. Choubin, G. Sadeghi, M. E. Gordji, A fixed point approach to stability of quintic functional equations in modular spaces, Kyungpook Math. J., 55 (2015), 313--326
  • View Article
  • MathSciNet


• [9] A. Grabiec, The generalized Hyers--Ulam stability of a class of functional equations, Publ. Math. Debrecen, 48 (1996), 217--235
  • View Article
  • MathSciNet


• [10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222--224
  • View Article
  • MathSciNet


• [11] K.-W. Jun, H.-M. Kim, J. Son, Generalized Hyers–Ulam stability of a quadratic functional equation, Functional Equations in Mathematical Analysis, 2012 (2012), 153--164
  • View Article
  • MathSciNet


• [12] S. Karthikeyan, C. Park, P. Palani, T. R. K. Kumar, Stability of an additive-quartic functional equation in modular spaces, J. Math. Comput. Sci., 26 (2022), 22--40
  • View Article
  • Google Scholar


• [13] H.-M. Kim, Y. S. Hong, Approximate quadratic mappings in modular spaces, Int. J. Pure Appl. Math., 116 (2017), 31--43
  • View Article
  • Google Scholar


• [14] C. Park, A. Bodaghi, S. O. Kim, A fixed point approach to stability of additive mappings in modular spaces without $\Delta_2$-conditions, J. Comput. Anal. Appl., 24 (2018), 1038--1048
  • View Article
  • MathSciNet


• [15] C. Park, J. M. Rassias, A. Bodaghi, S. O. Kim, Approximate homomorphisms from ternary semigroups to modular spaces, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM,, 113 (2019), 2175--2188
  • View Article
  • MathSciNet


• [16] M. Ramdoss, D. Pachaiyappan, I. Hwang, C. Park, Stability of an n-variable mixed type functional equation in probabilistic modular space, AIMS Math., 5 (2020), 5903--5915
  • View Article
  • MathSciNet


• [17] T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297--300
  • View Article
  • MathSciNet


• [18] T. M. Rassias, Functional equations, inequalities and applications, Kluwer Acadmic Publishers, Dordrecht (2003)
  • View Article
  • MathSciNet


• [19] G. Sadeghi, A fixed point approach to stability of functional equations in modular spaces, Bull. Malays. Math. Sci. Soc., 37 (2014), 333--344
  • View Article
  • MathSciNet


• [20] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, New York (1964)
  • MathSciNet
  • Google Scholar