On the hyperstability of a quartic functional equation in Banach spaces
DOI:
https://doi.org/10.4067/S0716-09172017000100003Keywords:
Hyperstability, Quartic functional equation, fixed point theoremAbstract
In this paper, we establish some hyperstability results of the following functional equation
f (2x + y) + f (2x - y) = 4(f (x + y) + f (x - y)) + 24f (x) - 6f (y)
in Banach spaces.
References
[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, pp. 64-66, (1950).
[2] C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Mathematica Sinica, English Series, Vol.22, No.6, pp. 1789-1796, (2006).
[3] A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2), pp. 353-365, (2014).
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, pp. 385-397, (1949).
[5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, pp. 223-237, (1951).
[6] J. Brzdek, K. Cienplinski , Hyperstability and superstability. Abstract and Applied Analysis. Article ID 101756 13pp., (2013).
[7] J. Brzdek, J. Chudziak, Z. P´ ales, A fixed point approach to stability of functional equations, Nonlinear Anal., Vol. 74, No. 17, pp. 6728-6732, (2011).
[8] J. Brzdek, Remarks on hyperstability of the the Cauchy equation, Aequations Mathematicae, 86, pp. 255-267, (2013).
[9] J. Brzdek, W. Fechner, M. S. Moslehian, J. Sikorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal. 9, No. 3, pp. 278-326, (2015).
[10] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungarica, 141 (1-2), pp. 58-67, (2013).
[11] J. Brzdek, A hyperstability result for the Cauchy equation. Bulletin of the Australian Mathematical Society 89, pp. 33-40, (2014).
[12] I. I. EL-Fassi, , S. Kabbaj, On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces. Proyecciones (Antofagasta), 34 (4), pp. 359-375, (2015).
[13] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14, pp. 431-434, (1991).
[14] E. Gselmann, Hyperstability of a functional equation, Acta Mathematica Hungarica, vol. 124, No. 1-2, pp. 179-188, (2009).
[15] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27, pp. 222-224, (1941).
[16] S. H. Lee, Im and I. S. Shwang , Quartic functional equations, J. Math. Anal. Appl 307, pp. 387-394, (2005).
[17] G. Maksa, Z. P´ ales, Hyperstability of a class of linear functional equations, Acta Math., vol. 17, no. 2, pp. 107-112, (2001).
[18] M. Piszczek, Remark on hyperstability of the general linear equation, Aequations Mathematicae, (2013).
[19] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, pp. 297-300, (1978).
[20] Th. M. Rassias, On a modified HyersUlam sequence, J. Math. Anal. Appl. 158, pp. 106-113, (1991).
[21] Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114, pp. 989-993, (1992).
[22] J. M. Rassias, solution of the Ulam stability problem for quartic mapping, Glasmik, Matematicki 34 (54), pp. 243-252, (1999).
[23] S. M. Ulam, Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, (1960).
[2] C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Mathematica Sinica, English Series, Vol.22, No.6, pp. 1789-1796, (2006).
[3] A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2), pp. 353-365, (2014).
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, pp. 385-397, (1949).
[5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, pp. 223-237, (1951).
[6] J. Brzdek, K. Cienplinski , Hyperstability and superstability. Abstract and Applied Analysis. Article ID 101756 13pp., (2013).
[7] J. Brzdek, J. Chudziak, Z. P´ ales, A fixed point approach to stability of functional equations, Nonlinear Anal., Vol. 74, No. 17, pp. 6728-6732, (2011).
[8] J. Brzdek, Remarks on hyperstability of the the Cauchy equation, Aequations Mathematicae, 86, pp. 255-267, (2013).
[9] J. Brzdek, W. Fechner, M. S. Moslehian, J. Sikorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal. 9, No. 3, pp. 278-326, (2015).
[10] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungarica, 141 (1-2), pp. 58-67, (2013).
[11] J. Brzdek, A hyperstability result for the Cauchy equation. Bulletin of the Australian Mathematical Society 89, pp. 33-40, (2014).
[12] I. I. EL-Fassi, , S. Kabbaj, On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces. Proyecciones (Antofagasta), 34 (4), pp. 359-375, (2015).
[13] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14, pp. 431-434, (1991).
[14] E. Gselmann, Hyperstability of a functional equation, Acta Mathematica Hungarica, vol. 124, No. 1-2, pp. 179-188, (2009).
[15] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27, pp. 222-224, (1941).
[16] S. H. Lee, Im and I. S. Shwang , Quartic functional equations, J. Math. Anal. Appl 307, pp. 387-394, (2005).
[17] G. Maksa, Z. P´ ales, Hyperstability of a class of linear functional equations, Acta Math., vol. 17, no. 2, pp. 107-112, (2001).
[18] M. Piszczek, Remark on hyperstability of the general linear equation, Aequations Mathematicae, (2013).
[19] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, pp. 297-300, (1978).
[20] Th. M. Rassias, On a modified HyersUlam sequence, J. Math. Anal. Appl. 158, pp. 106-113, (1991).
[21] Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114, pp. 989-993, (1992).
[22] J. M. Rassias, solution of the Ulam stability problem for quartic mapping, Glasmik, Matematicki 34 (54), pp. 243-252, (1999).
[23] S. M. Ulam, Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, (1960).
Published
2017-04-06
How to Cite
[1]
N. Bounader, “On the hyperstability of a quartic functional equation in Banach spaces”, Proyecciones (Antofagasta, On line), vol. 36, no. 1, pp. 29-44, Apr. 2017.
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