Solution and stability of an $n$-variable additive functional equation
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Abstract
In this paper, we investigate the general solution and the Hyers-Ulam stability of $n$-variable additive functional equation of the form$$\Im\left(\sum_{i=1}^{n}(-1)^{i+1}x_i\right)=\sum_{i=1}^{n}(-1)^{i+1}\Im (x_i),$$where $n$ is a positive integer with $n \ge 2$, in Banach spaces by using the direct method.
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References
[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. Google Scholar
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. Google Scholar
[3] E. Baktash, Y. Cho, M. Jalili, R. Saadati and S. M. Vaezpour, On the stability of cubic mappingsand quadratic mappings in random normed spaces, J. Inequal. Appl. 2008 (2008), Article ID 902187. Google Scholar
[4] I. Chang, E. Lee and H. Kim, On the Hyers-Ulam-Rassias stability of a quadratic functional equations, Math. Inequal. Appl. 6 (2003), 87–95. Google Scholar
[5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. Google Scholar
[6] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. Google Scholar
[7] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34, Progress in Nonlinear Differential Equations and Their Applications, Birkh ̈auser, Boston, 1998. Google Scholar
[8] M. Eshaghi Gordji and H. Khodaie, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009), 5629–5643. Google Scholar
[9] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 267–278. Google Scholar
[10] C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach alge- bras, Bull. Sci. Math. 132 (2008), 87–96. Google Scholar
[11] C. Park and J. Cui, Generalized stability of C∗-ternary quadratic mappings, Abs. Appl. Anal. 2007 (2007), Article ID 23282. Google Scholar
[12] C. Park and A. Najati, Homomorphisms and derivations in C∗-algebras, Abs. Appl. Anal. 2007 (2007), Article ID 80630. Google Scholar
[13] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300. Google Scholar
[14] Th. M. Rassias, On the stability of the functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264–284. Google Scholar
[15] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan, London, 2003. Google Scholar
[16] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964. Google Scholar