zbMATH — the first resource for mathematics

On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations. (English) Zbl 1148.39026
Let $$n>2$$ be an integer, and let $$X,Y$$, $$f:X\to Y$$ be vector spaces. One of the corollaries of the main result is that the functional equation below is stable in the sense of Hyers and Ulam: $f\biggl(\sum_{i=1}^nx_i\biggr)+(n-2)\sum_{i=1}^nf(x_i)= \sum_{1\leq i<j\leq n}f(x_i+x_j)\qquad (x_1,\ldots,x_n\in X).$ The author also gives its general solutions, concluding that the even part of $$f$$ satisfies the classical quadratic equation, while the odd part of $$f$$ satisfies the Cauchy functional equation. The main tool of the proofs is based on Hyers’ method.
Consult also the papers T. Aoki [J. Math. Soc. Japan 2, 64–66 (1950; Zbl 0040.35501)] and L. Maligranda [Aequationes Math., 75, No. 3, 289–296 (2008; Zbl 1158.39019)] for interesting and important historical remarks.

MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
Keywords:
Hyers-Ulam stability
Full Text:
References:
 [1] S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1964. · 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] J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126-130, 1982. · Zbl 0482.47033 · doi:10.1016/0022-1236(82)90048-9 [4] 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 [5] J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989. · Zbl 0672.41027 · doi:10.1016/0021-9045(89)90041-5 [6] J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005 [7] J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185-190, 1992. · Zbl 0753.39003 [8] J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101-107, 1994. · Zbl 0819.39012 [9] J. M. Rassias, “Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 220, no. 2, pp. 613-639, 1998. · Zbl 0928.39014 · doi:10.1006/jmaa.1997.5856 [10] J. M. Rassias, “On the Ulam stability of mixed type mappings on restricted domains,” Journal of Mathematical Analysis and Applications, vol. 276, no. 2, pp. 747-762, 2002. · Zbl 1021.39015 · doi:10.1016/S0022-247X(02)00439-0
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.