# zbMATH — the first resource for mathematics

Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of an additive functional equation in several variables. (English) Zbl 1148.39027
Let $$n>1$$ be an integer and $$X,Y$$ be real vector spaces, $$\delta,\theta\geq 0$$ and $$p>0$$, $$p\neq 1$$ with $$\delta=0$$ if $$p>1$$. Based on Hyers’ method, the author investigates the stability problem of a functional equation in the following settings: $\Bigl\|nf\Bigl(\sum_{i=1}^nx_i\Bigr)-\sum_{i=1}^nf(x_i)- \sum_{1\leq i<j\leq n}f(x_i+x_j)\Bigr\|\leq \delta+\theta\sum_{i=1}^n\|x_i\|^p$ $\Bigl\|nf\Bigl(\sum_{i=1}^nx_i\Bigr)-\sum_{i=1}^nf(x_i)- \sum_{1\leq i<j\leq n}f(x_i+x_j)\Bigr\|\leq \delta+\theta\sum_{1\leq i<j\leq n}\|x_i\|^{p/2}\|x_j\|^{p/2}$ The main results state, that, in each cases, the solutions of the inequalities are “close” to a unique additive mapping. In particular, the solutions of the corresponding functional equations are additive.
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:
  S. M. Ulam, Problems in Modern Mathematics, chapter 6, John Wiley & Sons, New York, NY, USA, 1964. · Zbl 0137.24201  D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United State of America, vol. 27, no. 4, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222  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  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  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  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  J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005  P. G\uavruta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hadronic Math. Ser., pp. 67-71, Hadronic Press, Palm Harbor, Fla, USA, 1999.  B. Bouikhalene and E. Elqorachi, “Ulam-Gavruta-Rassias stability of the Pexider functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 27-39, 2007. · Zbl 1130.39022  K. Ravi and M. Arunkumar, “On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, no. Fe07, pp. 143-156, 2007.  P. Nakmahachalasint, “On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 63239, 10 pages, 2007. · Zbl 1148.39026 · doi:10.1155/2007/63239 · eudml:54655  S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen/s equation and its application,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3137-3143, 1998. · Zbl 0909.39014 · doi:10.1090/S0002-9939-98-04680-2  K.-W. Jun and H.-M. Kim, “Stability problem of Ulam for generalized forms of Cauchy functional equation,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 535-547, 2005. · Zbl 1082.39023 · doi:10.1016/j.jmaa.2005.03.052  W.-G. Park and J.-H. Bae, “On a Cauchy-Jensen functional equation and its stability,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 634-643, 2006. · Zbl 1104.39027 · doi:10.1016/j.jmaa.2005.09.028
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.