Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.

Abstract

In this paper,  we have improved some of the results in [C. Choi and   B. Lee, Stability of Mixed Additive-Quadratic and Additive--Drygas Functional Equations. Results Math.  75  no. 1 (2020), Paper No. 38]. Indeed, we investigate the Hyers-Ulam stability problem of the following   functional equations
\begin{align*}
2\varphi(x + y) + \varphi(x - y) &= 3\varphi(x)+ 3\varphi(y)  \\
2\psi(x + y) + \psi(x - y) &= 3\psi(x) + 2\psi(y) + \psi(-y).
\end{align*}
We also consider the Pexider type functional equation $2\psi(x + y) + \psi(x - y) = f(x) + g(y),$ and the additive functional equation
$2\psi(x + y) + \psi(x - y) = 3\psi(x) + \psi(y).$

Keywords

###### ##### References
[1] B. Batko, Stability of an alternative functional equation, J. Math. Anal. Appl., 339~ (2008), pp. 303-311.

[2] C. Choi and B. Lee, Stability of Mixed Additive-Quadratic and Additive-Drygas Functional Equations, Results Math., 75 (2020), Paper No. 38.

[3] J. Chung, Stability of conditional Cauchy functional equations, Aequat. Math., 83 (2012), pp. 313-320.

[4] J. Chung and J.M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl., 419 (2014), pp. 1065-1075.

[5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publ. Co., 2002.

[6] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), pp. 222-224.

[7] D.H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, 1998.

[8] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, 2011.

[9] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009.

[10] P. Kaskasem, A. Janchada and C. Klin-eam, On approximate solutions of the generalized radical cubic functional equation in quasi $beta$-Banach spaces, Sahand Commun. Math. Anal., 17 (2020), pp. 69-90.

[11] B. Khosravi, M.B. Moghimi and A. Najati, Asymptotic aspect of Drygas, quadratic and Jensen functional equations in metric abelian groups, Acta Math. Hungar., 155 (2018), pp. 248-265.

[12] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publ. and Silesian Univ. Press, Warsaw, 1985.

[13] Y.-H. Lee, S.-M. Jung and M.Th. Rassias, On an $n$-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), pp. 13-16.

[14] Y.-H. Lee, S.-M. Jung and M.Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 12 (2018), pp. 43-61.

[15] M. Maysami Sadr, Stability of additive functional equation on discrete quantum semigroups, Sahand Commun. Math. Anal., 8 (2017), pp. 73-81.

[16] D. Molaei and A. Najati, Hyperstability of the general linear equation on restricted domains, Acta Math. Hungar., 149 (2016), pp. 238-253.

[17] A. Najati and Soon-Mo Jung, Approximately quadratic mappings on restricted domains, J. Inequal. Appl., (2010), Art. ID 503458, 10 pages.

[18] A. Najati and Th.M. Rassias, Stability of the Pexiderized Cauchy and Jensen's equations on restricted domains, Sahand Commun. Math. Anal., 8 (2010), pp. 125-135.

[19] J.C. Oxtoby, Measure and Category, Springer, New York, 1980.

[20] J. Senasukh and S. Saejung, On the hyperstability of the Drygas functional equation on a restricted domain, Bull. Aust. Math. Soc., 102 (2020), pp. 126-137.