Volume 10, Issue 8, pp 4071--4076
Publication Date: 2017-08-06
Krzysztof Ciepliński - AGH University of Science and Technology, Faculty of Applied Mathematics, Mickiewicza 30, 30-059 Krakow, Poland.
We prove the generalized Ulam stability of ternary homomorphisms from commutative ternary semigroups into \(n\)-Banach spaces as well as into complete non-Archimedean normed spaces. Ternary algebraic structures appear in various domains of theoretical and mathematical physics, and \(p\)-adic numbers, which are the most important examples of non-Archimedean fields, have gained the interest of physicists for their research in some problems coming from quantum physics, \(p\)-adic strings and superstrings.
Ulam stability, (commutative) ternary semigroup, ternary homomorphism, n-Banach space, (complete) non-Archimedean normed space, p-adic numbers.
 M. Amyari, M. S. Moslehian, Approximate homomorphisms of ternary semigroups, Lett. Math. Phys., 77 (2006), 1–9.
 N. Bazunova, A. Borowiec, R. Kerner, Universal differential calculus on ternary algebras, Lett. Math. Phys., 67 (2004), 195–206.
 N. Brillouët-Belluot, J. Brzdęk, K. Ciepliński, On some recent developments in Ulam’s type stability, Abstr. Appl. Anal., 2012 (2012), 41 pages.
 H.-Y. Chu, A. Kim, J. Park, On the Hyers-Ulam stabilities of functional equations on n-Banach spaces, Math. Nachr., 289 (2016), 1177–1188.
 H. Dutta, On some n-normed linear space valued difference sequences, J. Franklin Inst., 348 (2011), 2876–2883.
 G. P. Gehér, On n-norm preservers and the Aleksandrov conservative n-distance problem, ArXiv, 2015 (2015), 9 pages.
 H. Gunawan, M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci., 27 (2001), 631–639.
 S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Science and Business Media, New York, (2011).
 R. Kerner, Ternary and non-associative structures, Int. J. Geom. Methods Mod. Phys., 5 (2008), 1265–1294.
 A. Khrennikov, Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Kluwer Academic Publishers, Dordrecht, (1997).
 Y. Ma, The Aleksandrov-Benz-Rassias problem on linear n-normed spaces, Monatsh. Math., 180 (2016), 305–316.
 A. Misiak, n-inner product spaces, Math. Nachr., 140 (1989), 299–319.
 M. S. Moslehian, Ternary derivations, stability and physical aspects, Acta Appl. Math., 100 (2008), 187–199.
 M. S. Moslehian, T. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math., 1 (2007), 325–334.
 C. Park, M. E. Gordji, Comment on ”Approximate ternary Jordan derivations on Banach ternary algebras” [Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)], J. Math. Phys., 2010 (2010), 7 pages.
 J. M. Rassias, H.-M. Kim, Approximate homomorphisms and derivations between \(C^*\)-ternary algebras, J. Math. Phys., 2008 (2008), 10 pages.
 M. L. Santiago, S. Sri Bala, Ternary semigroups, Semigroup Forum, 81 (2010), 380–388.
 T. Z. Xu, Stability of multi-Jensen mappings in non-Archimedean normed spaces, J. Math. Phys., 2012 (2012), 9 pages.
 T. Z. Xu, J. M. Rassias, On the Hyers-Ulam stability of a general mixed additive and cubic functional equation in n-Banach spaces, Abstr. Appl. Anal., 2012 (2012), 23 pages.