Hyers-Ulam stability for a class of quadratic functional equations via a typical form. (English) Zbl 1291.39053

Summary: We investigate the following typical form of a certain class of quadratic functional equations: \(f(ax+by)+f(ax-by)+c[f(x+y)+f(x-y)-2f(x)-2f(y)]=2a^2f(x)+2b^2f(y)\). Furthermore, we provide a systematic program to prove the generalized Hyers-Ulam stability for the class of functional equations via the stability for the typical form.


39B82 Stability, separation, extension, and related topics for functional equations
Full Text: DOI


[1] Ulam, S. M., A Collection of Mathematical Problems (1964), New York, NY, USA: Interscience Publisher, New York, NY, USA · Zbl 0086.24101
[2] Hyers, D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224 (1941) · JFM 67.0424.01
[3] Aoki, T., On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2, 1-2, 64-66 (1950) · Zbl 0040.35501
[4] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 2, 297-300 (1978) · Zbl 0398.47040
[5] Găvruţa, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184, 3, 431-436 (1994) · Zbl 0818.46043
[6] Skof, F., Proprieta’ locali e approssimazione di operatori, Rendiconti del Seminario Matematico e Fisico di Milano, 53, 1, 113-129 (1983) · Zbl 0599.39007
[7] Cholewa, P. W., Remarks on the stability of functional equations, Aequationes Mathematicae, 27, 1, 76-86 (1984) · Zbl 0549.39006
[8] Czerwik, S., On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 62, 1, 59-64 (1992) · Zbl 0779.39003
[9] Park, C. G., On the stability of the quadratic mapping in Banach modules, Journal of Mathematical Analysis and Applications, 276, 1, 135-144 (2002) · Zbl 1017.39010
[10] Eskandani, G.; Vaezi, H.; Dehghan, Y. N., Stability of a mixed additive and quadratic functional equation in non-archimedean banach modules, Taiwanese Journal of Mathematics, 14, 4, 1309-1324 (2010) · Zbl 1218.39026
[11] Moradlou, F.; Vaezi, H. V.; Eskandani, G. Z., Hyers-Ulam-Rassias stability of a quadratic and additive functional equation in quasi-Banach spaces, Mediterranean Journal of Mathematics, 6, 2, 233-248 (2009) · Zbl 1183.39021
[12] Rassias, J. M., Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings, Journal of Mathematical Analysis and Applications, 220, 2, 613-639 (1998) · Zbl 0928.39014
[13] Gordji, M. E.; Khodaei, H., On the generalized hyers-ulam-rassias stability of quadratic functional equations, Abstract and Applied Analysis, 2009 (2009) · Zbl 1167.39014
[14] Gordji, M. E.; Savadkouhi, M. B.; Park, C., Quadratic-quartic functional equations in RN-spaces, Journal of Inequalities and Applications, 2009 (2009) · Zbl 1187.39036
[15] Jun, K. W.; Kim, H. M.; Chang, I. S., On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation, Journal of Computational Analysis and Applications, 7, 1, 21-33 (2005) · Zbl 1087.39029
[16] Jun, K. W.; Kim, H. M.; Son, J., Generalized Hyers-Ulam stability of a quadratic functional equation, Functional Equations in Mathematical Analysis, 2012, 153-164 (2012) · Zbl 1248.39037
[17] Ravi, K.; Arunkumar, M.; Rassias, J. M., Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematics and Statistics, 3, 8, 36-46 (2008) · Zbl 1144.39029
[18] Kim, H. M.; Rassias, J. M.; Lee, J., Fuzzy approximation of Euler-Lagrange quadratic mappings, Journal of Inequalities and Applications, 2013 (2013) · Zbl 1286.39014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.