Hyperstability of a quadratic functional equation on abelian group and inner product spaces

EL-Fassi, Iz-iddine¹ ; Kim, Gwang Hui²

¹ Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco
² Department of Mathematics, Kangnam University, Yongin, Gyoenggi 446-702, Republic of Korea

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 9, p. 5353-5361.

Résumé

Using the fixed point approach, we prove some results on hyperstability of the following quadratic functional equation

$f(x + y + z) + f(x - y) + f(x - z) + f(y - z) = 3[f(x) + f(y) + f(z)],$

in the class of functions from an abelian group into a Banach space.

Détail

DOI : 10.22436/jnsa.009.09.04

Classification : 39B82, 39B52
Keywords: Hyperstability, quadratic functional equation, fixed point theorem.

Bibliographie

[1] Aoki, T. On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan., Volume 2 (1950), pp. 64-66

[2] Bae, J.-H. On the stability of 3-dimensional quadratic functional equation, Bull. Korean Math. Soc., Volume 37 (2000), pp. 477-486

[3] Bae, J.-H.; Jun, K.-W. On the generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, Bull. Korean Math. Soc., Volume 38 (2011), pp. 325-336

[4] Bae, J.-H.; Jung, Y.-S. The Hyers-Ulam stability of the quadratic functional equations on abelian groups, Bull. Korean Math. Soc., Volume 39 (2002), pp. 199-209

[5] Bahyrycz, A.; Piszczek, M. Hyperstability of the Jensen functional equation, Acta Math. Hungar., Volume 142 (2014), pp. 353-365

[6] Bourgin, D. G. Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., Volume 16 (1949), pp. 385-397

[7] Brzdęk, J. Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar., Volume 141 (2013), pp. 58-67

[8] Brzdęk, J. Remarks on hyperstability of the Cauchy functional equation, Aequationes Math., Volume 86 (2013), pp. 255-267

[9] Brzdęk, J. A hyperstability result for the Cauchy equation, Bull. Aust. Math. Soc., Volume 89 (2014), pp. 33-40

[10] Brzdęk, J.; Chudziak, J.; Páles, Z. A fixed point approach to stability of functional equations, Nonlinear Anal., Volume 74 (2011), pp. 6728-6732

[11] Brzdęk, J.; Ciepliński, K. Hyperstability and superstability, Abstr. Appl. Anal., Volume 2013 (2013), pp. 1-13

[12] Czerwik, St. On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, Volume 62 (1992), pp. 59-64

[13] EL-Fassi, Iz.; Kabbaj, S. On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces, Proyecciones, Volume 34 (2015), pp. 359-375

[14] Gajda, Z. On stability of additive mappings, Internat. J. Math. Math. Sci., Volume 14 (1991), pp. 431-434

[15] Găvruţa, P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., Volume 184 (1994), pp. 431-436

[16] Gselmann, E. Hyperstability of a functional equation, Acta Math. Hungar., Volume 124 (2009), pp. 179-188

[17] Hyers, D. H. On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A, Volume 27 (1941), pp. 222-224

[18] Jung, S.-M. On the Hyers-Ulam-Rassias stability of a quadratic functional equation, J. Math. Anal. Appl., Volume 232 (1999), pp. 384-393

[19] Kannappan, P. Quadratic functional equation and inner product spaces, Results Math., Volume 27 (1995), pp. 368-372

[20] Maksa, G.; Páles, Z. Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyhzi. (N.S.), Volume 17 (2001), pp. 107-112

[21] Mirzavaziri, M.; Moslehian, M. S. fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. (N.S.), Volume 37 (2006), pp. 361-376

[22] Piszczek, M. Remark on hyperstability of the general linear equation, Aequationes Math., Volume 88 (2013), pp. 163-168

[23] Rassias, Th. M. On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., Volume 72 (1978), pp. 297-300

[24] Rassias, Th. M. On a modified Hyers-Ulam sequence, J. Math. Anal. Appl., Volume 158 (1991), pp. 106-113

[25] Rassias, J. M. On the stability of the Euler-Lagrange functional equation, Chinese J. Math., Volume 20 (1992), pp. 185-190

[26] Rassias, Th. M. On the stability of the quadratic functional equation and its applications, Studia Univ. Babe-Bolyai Math., Volume 43 (1998), pp. 89-124

[27] Rassis, Th. M. On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., Volume 251 (2000), pp. 264-284

[28] Rassias, Th. M.; Šemrl, P. On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., Volume 114 (1992), pp. 989-993

[29] Ulam, S. M. Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1960