# On the stability of 2-dimensional Pexider quadratic functional equation in non-Archimedean spaces

Document Type : Research Paper

Author

Department of Mathematics, Urmia University, Urmia, Iran.

Abstract

In this papers we investigate the Hyers-Ulam stability of the following $2$-dimensional Pexider quadratic  functional equation $$f(x+y,z+w)+f(x-y,z-w)=2g(x,z)+2g(y,w)$$ in non-Archimedean normed spaces.

Keywords

#### References

[1] M. R. Abdollahpour and M. Th. Rassias, Hyers-Ulam stability of hypergeometric differential equation, Aequationes Mathematicae, 93 (4) (2019), 691|698.
[2] M. A. Abolfathi, A. Ebadian and R. Aghalary, Stability of mixed additive-quadratic Jensen type functional equation in non-Archimedean `-fuzzy normed spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat., 60 (2) (2014), 307|319.
[3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 245{251.
[4] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Vol.31, 1989.
[5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Soc., 57 (1951), 223{237.
[6] N. Brillouet-Belluot, J. Brzdek, and K. Cieplinski, On some recent developments in Ulam's type stability, Abstract and Applied Analysis, vol. 2012, Article ID 716936, 41 pages, 2012.
[7] J. Brzdek and K. Cieplinski, A  xed point theorem and the Hyers-Ulam stability in non-Archimedean spaces, Journal of Mathematical Analysis and Applications, 400 (1) (2013), 68{75.
[8] P. W. Cholewa, Remarks on the stability of functional equations, Aequ. Math., 27 (1984), 76{86.
[9] K. Cieplinski and T. Z. Xu, Approximate multi-Jensen and multi-quadratic mappings in 2-Banach spaces, Carpathian Journal of Mathematics, 29 (2013), 159{166.
[10] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scienti c, 2002.
[10] Z. Gajda, On the stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431{434.
[11] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of the approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431{436.
[12] F. Q. Gouvea, p-Adic Numbers. An Introduction, Universitext, Springer-Verlag, Berlin, 1997.
[13] K. Hensel,  Uber eine neue Begundung der Theorie der algebraischen Zahlen. Jahres, Deutschen Mathematiker-Vereinigung, 6 (1897), 83{88.
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222{224.
[15] D. H. Hyers, G. Isac and Th. M. Rssias, Stability of Functional Equation in Several Variables, Birkhauser, Basel, 1998.
[16] G. Isac, Th. M. Rassias, Stability of  -additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci., 19 (1996), 219{228.
[17] S. M. Jung, Hyers-Ulam-Rassias Stability of functional Equation in Nonlinear Analysis,Springer, 2011.
[18] PI Kannappan, Functional Equations and Inequalitites With Applications, Springer,2009.
[19] Y. Lee, S. M. Jung and M. Th. Rassias, On an n-dimnsional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), 13{16.
[20] W. G. Park and J. H. Bae , On a bi-quadratic functional equation and its stability, Nonlinear Anal., 62 (2005), 643{654.
[21] C. Park and M. Th. Rassias, Additive functional equations and partial multipliers in C*-algebras, Rev. R. Acad. Cienc. Exactas, Serie A. Matematicas, 113 (3) (2019), 2261{2275.
[22] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297{300.
[23] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (1) (2000), 23{130.
[24] Th. M. Rassias, Functional quations and Inequalities, Kluwer Academic Publishers, 2000.
[25] A. M. Robert, A Course in p-Adic Analysis, Grad. Texts in Math., vol. 198, Springer-Verlag, New York, 2000.
[26] P. K. Sahoo and P. Kannappan, Introduction to Functional Equations, CRC Press, 2011.
[27] N. Shilkret, Non-Archimedian Banach algebras, Ph. D. thesis, Polytechnic University,1968.
[28] T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl., 272 (2) (2002), 604{616.
[29] S. M. Ulam, Problem in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.
[30] T. Z. Xu, On the stability of multi-Jensen mappings in  -normed spaces, Applied Mathematics Letters, 25 (11) (2012), 1866{1870.
[31] T. Z. Xu, J. M. Rassias, and Wan Xin Xu, A  xed point approach to the stability of a general mixed additive-cubic equation on Banach modules, Acta Mathematica Scientia,32B (3) (2012), 866{892.

### History

• Receive Date: 24 December 2021
• Revise Date: 11 February 2022
• Accept Date: 13 February 2022
• First Publish Date: 28 February 2022