On the stability of quadratic functional equations. (English) Zbl 1146.39045

Summary: Let \(X,Y\) be vector spaces and \(k\) a fixed positive integer. It is shown that a mapping \(f(kx+y)+f(kx-y)=2k^{2}f(x)+2f(y)\) for all \(x,y\in X\) if and only if the mapping \(f:X\rightarrow Y\) satisfies \(f(x+y)+f(x-y)=2f(x)+2f(y)\) for all \(x,y\in X\). Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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[1] S. M. Ulam, “Problems in Modern Mathematics,” p. xvii+150, John Wiley & Sons, New York, NY, USA, 1960. · Zbl 0137.24201
[2] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[3] T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[4] Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 · doi:10.2307/2042795
[5] Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, no. 2-3, pp. 292-293, 309, 1990.
[6] Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[7] Th. M. Rassias and P. \vSemrl, “On the Hyers-Ulam stability of linear mappings,” Journal of Mathematical Analysis and Applications, vol. 173, no. 2, pp. 325-338, 1993. · Zbl 0789.46037 · doi:10.1006/jmaa.1993.1070
[8] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 4, pp. 445-446, 1984. · Zbl 0599.47106
[9] F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113-129, 1983. · Zbl 0599.39007 · doi:10.1007/BF02924890
[10] P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1, pp. 76-86, 1984. · Zbl 0549.39006 · doi:10.1007/BF02192660
[11] St. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59-64, 1992. · Zbl 0779.39003 · doi:10.1007/BF02941618
[12] K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a pexiderized quadratic inequality,” Mathematical Inequalities & Applications, vol. 4, no. 1, pp. 93-118, 2001. · Zbl 0976.39031
[13] S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic property,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126-137, 1998. · Zbl 0928.39013 · doi:10.1006/jmaa.1998.5916
[14] J. M. Rassias, “Solution of a quadratic stability Hyers-Ulam type problem,” Ricerche di Matematica, vol. 50, no. 1, pp. 9-17, 2001. · Zbl 1221.39039
[15] J. M. Rassias, “On approximation of approximately quadratic mappings by quadratic mappings,” Annales Mathematicae Silesianae, no. 15, pp. 67-78, 2001. · Zbl 1087.39518
[16] M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361-376, 2006. · Zbl 1118.39015 · doi:10.1007/s00574-006-0016-z
[17] Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,” Studia Universitatis Babe\cs-Bolyai, vol. 43, no. 3, pp. 89-124, 1998. · Zbl 1009.39025
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