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Solution and stability of an \(n\)-variable additive functional equation. (English) Zbl 1458.39018
Summary: In this paper, we investigate the general solution and the Hyers-Ulam stability of \(n\)-variable additive functional equation of the form \[ \operatorname{Im}\left(\sum_{i=1}^n(-1)^{i+1}x_i\right)=\sum_{i=1}^n(-1)^{i+1}\operatorname{Im} (x_i), \] where \(n\) is a positive integer with \(n \ge 2\), in Banach spaces by using the direct method.
39B52 Functional equations for functions with more general domains and/or ranges
39B72 Systems of functional equations and inequalities
39B82 Stability, separation, extension, and related topics for functional equations
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. · Zbl 0685.39006
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. · Zbl 0040.35501
[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. · Zbl 1165.39022
[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. · Zbl 1024.39008
[5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86. · Zbl 0549.39006
[6] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. · Zbl 0779.39003
[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.
[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. · Zbl 1179.39034
[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.
[10] C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach alge- bras, Bull. Sci. Math. 132 (2008), 87-96. · Zbl 1140.39016
[11] C. Park and J. Cui, Generalized stability of C∗-ternary quadratic mappings, Abs. Appl. Anal. 2007 (2007), Article ID 23282. · Zbl 1158.39020
[12] C. Park and A. Najati, Homomorphisms and derivations in C∗-algebras, Abs. Appl. Anal. 2007 (2007), Article ID 80630. · Zbl 1157.39017
[13] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297-300. · Zbl 0398.47040
[14] Th. M. Rassias, On the stability of the functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284. · Zbl 0964.39026
[15] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan, London, 2003.
[16] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964. · Zbl 0137.24201
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