On the hyperstability of a quartic functional equation in Banach spaces

Nordine Bounader


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.

Palabras clave

Hyperstability; Quartic functional equation; fixed point theorem

Texto completo:



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

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).

A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2), pp. 353-365, (2014).

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

D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, pp. 223-237, (1951).

J. Brzdek, K. Cienplinski , Hyperstability and superstability. Abstract and Applied Analysis. Article ID 101756 13pp., (2013).

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).

J. Brzdek, Remarks on hyperstability of the the Cauchy equation, Aequations Mathematicae, 86, pp. 255-267, (2013).

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).

J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungarica, 141 (1-2), pp. 58-67, (2013).

J. Brzdek, A hyperstability result for the Cauchy equation. Bulletin of the Australian Mathematical Society 89, pp. 33-40, (2014).

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).

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

E. Gselmann, Hyperstability of a functional equation, Acta Mathematica Hungarica, vol. 124, No. 1-2, pp. 179-188, (2009).

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

S. H. Lee, Im and I. S. Shwang , Quartic functional equations, J. Math. Anal. Appl 307, pp. 387-394, (2005).

G. Maksa, Z. P´ ales, Hyperstability of a class of linear functional equations, Acta Math., vol. 17, no. 2, pp. 107-112, (2001).

M. Piszczek, Remark on hyperstability of the general linear equation, Aequations Mathematicae, (2013).

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

Th. M. Rassias, On a modified HyersUlam sequence, J. Math. Anal. Appl. 158, pp. 106-113, (1991).

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).

J. M. Rassias, solution of the Ulam stability problem for quartic mapping, Glasmik, Matematicki 34 (54), pp. 243-252, (1999).

S. M. Ulam, Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, (1960).

DOI: http://dx.doi.org/10.4067/S0716-09172017000100003

Enlaces refback

  • No hay ningún enlace refback.