International Journal of Nonlinear Analysis and Applications

Asymptotic behavior of a radical quadratic functional equation in quasi-β-Banach spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco

2 Department of Mathematics, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco

Abstract

Let $\mathbb{R}$ be the set of real numbers and $\big(Y,\|\cdot\|\big)$  be a real quasi-$\beta$-Banach space. In this paper, we prove the Hyers-Ulam stability on a  restricted domain in quasi-$\beta$-spaces for the following two radical functional equations
$$
f\big(\sqrt{x^{2}+y^{2}}\big)=f(x)+f(y)
$$
and
$$
 f\big(\sqrt{x^{2}+y^{2}}\big)=g(x)+f(y)
$$
where $f,g:\mathbb{R}\to Y$. Also, we discuss an asymptotic behavior for these equations.

Keywords

[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.
[2] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.
[3] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86.
[4] J. Chung, D. Kim and J.M. Rassias, Stability of Jensen-type functional equations on restricted domains in a group and their asymptotic behaviors, J. App. Math. 2012 (2012), Article ID 691981.
[5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.
[6] M.E. Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009), 5629–5643.
[7] M.E. Gordji and M. Parviz, On the Hyers–Ulam stability of the functional equation f p2 x 2 + y2  = f(x) +f(y), Nonlinear Funct. Anal. Appl. 14 (2009), 413–420.
[8] M.E. Gordji, H. Khodaei, A. Ebadian and G.H. Kim, Nearly radical quadratic functional equations in p-2-normed spaces, Abstr. Appl. Anal. 2012 (2012), Article ID 896032.
[9] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.
[10] K. Jun and H. Kim, On the stability of an n-dimensional quadratic and additive functional equation, Math. Inequal. Appl. 9 (2006), 153–165.
[11] S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126–137.
[12] S.S. Kim, Y.J. Cho and M.E. Gordji, On the generalized Hyers-Ulam-Rassias stability problem of radical functional equations, J. Inequal. Appl. 2012 (2012), Article ID 186.
[13] J.M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 276 (2002), 747–762.
[14] J.M. Rassias and M.J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281 (2003), 516–524.
[15] J.M. Rassias and M.J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. 129 (2005), 545–558.
[16] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
[17] Th.M. Rassias, Problem 16; 2. Report of the 27th international symposium on functional equations, Aequationes Math. 39 (1990), 292–293.
[18] Th.M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106–113.
[19] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. BabesBolyai Math. XLIII (1998), 89–124.
[20] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta. Appl. Math. 62 (2000), no. 1, 23–130.
[21] F. Skof, Propriet`a locali e approssimazione di operatori Rend, Semin. Mat. Fis. Milano 53 (1983), 113–129.
[22] J. Tabor, Stability of Cauchy functional equation in quasi-Banach spaces, Ann. Pol. Math. 83 (2004), 243–255.
[23] S.M. Ulam, Problems in Modern Mathematics, Science Editions, John-Wiley & Sons Inc., New York, 1964.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 3
March 2023
Pages 113-120
  • Receive Date: 06 May 2021
  • Revise Date: 03 July 2021
  • Accept Date: 14 July 2021