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In this paper, we investigate the stability problem in the spirit of Hyers-Ulam, Rassias and G·avruta for the quadratic functional equation:
f(2x + y) + f(2x ¡ y) = 2f(x + y) + 2f(x ¡ y) + 4f(x) ¡ 2f(y) in 2-Banach spaces. These results extend the generalized Hyers-Ulam stability results by the
quadratic functional equation in normed spaces to 2-Banach spaces.
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 Y.J. Cho, P.C.S. Lin, S.S. Kim, A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publishers, New York, 2001.
 S.C. Chong, W.-G. Park, Hyers-Ulam stability of functional equations in 2-Banach spaces, Int. J. Math. Anal. 6(2012), 951-961.
 P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86.
 S. Czerwik, On the stability of the quadratic mappings in normed spaces, Abh. Math. Sem. Univ. Hamburg. 62 (1992), 59-64.
 R.W. Freese, Y.J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, New York, 2001.
 S. GÄahler, Lineare 2-normierte RÄaume, Math. Nachr. 28 (1964), 1-43 (German).
 P. G·avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.
 M. Eshaghi Gordji, H. Khodaei, On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations, Abs. Appl. Anal. Vol. 2009, Article ID 923476, 11 pages.
 D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222-224.
 W.-G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011) 193-202.
 B.M. Patel and A.B. Patel, Stability od quartic functional equations in 2-Banach spaces, Int. J. Math. Anal. 7 (2013), 1097-1107.
 J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130.
 Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.
 K. Ravi, R. Murali and M. Arunkumar, The generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, JIPAM. 9(2008), Issue 1, Article 20, 5pp.
 F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. 53 (1983), 113-129.
 S.M. Ulam, A Collection of Mathematical Problems, Interscience Publ. New York, 1960.
 A. White, 2-Banach spaces, Math. Nachr. 42 (1969) 43-60.