##
**Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras.**
*(English)*
Zbl 1218.46050

Summary: We prove the generalized Hyers-Ulam stability of homomorphisms and derivations on non-Archimedean Banach algebras. Moreover, we prove the superstability of homomorphisms on unital non-Archimedean Banach algebras and we investigate the superstability of derivations in non-Archimedean Banach algebras with bounded approximate identity.

### MSC:

46S10 | Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis |

39B82 | Stability, separation, extension, and related topics for functional equations |

PDF
BibTeX
XML
Cite

\textit{M. Eshaghi Gordji}, Abstr. Appl. Anal. 2010, Article ID 393247, 12 p. (2010; Zbl 1218.46050)

### References:

[1] | K. Hensel, “Über eine neue Begründung der Theorie der algebraischen Zahlen,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 6, pp. 83-88, 1897. · JFM 30.0096.03 |

[2] | A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, vol. 427 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1997. · Zbl 0920.11087 |

[3] | L. M. Arriola and W. A. Beyer, “Stability of the Cauchy functional equation over p-adic fields,” Real Analysis Exchange, vol. 31, no. 1, pp. 125-132, 2005/2006. · Zbl 1099.39019 |

[4] | M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of cubic and quartic functional equations in non-Archimedean spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1321-1329, 2010. · Zbl 1192.39018 |

[5] | M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1198-1202, 2010. · Zbl 1204.39028 |

[6] | M. Eshaghi Gordji, H. Khodaei, and R. Khodabakhsh, “General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces,” University Politechnica of Bucharest Scientific Bulletin Series A, vol. 72, no. 3, pp. 69-84, 2010. · Zbl 1249.39030 |

[7] | L. Narici and E. Beckenstein, “Strange terrain-non-Archimedean spaces,” The American Mathematical Monthly, vol. 88, no. 9, pp. 667-676, 1981. · Zbl 0486.46054 |

[8] | C. Park, D. H. Boo, and T. M. Rassias, “Approximately addtive mappings over p-adic fields,” Journal of Chungcheong Mathematical Society, vol. 21, pp. 1-14, 2008. |

[9] | V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic analysis and Mathematical Physics, vol. 1 of Series on Soviet and East European Mathematics, World Scientific, River Edge, NJ, USA, 1994. · Zbl 0864.46048 |

[10] | N. Shilkret, Non-archimedian Banach algebras, Ph.D. thesis, Polytechnic University, 1968, ProQuest LLC. · Zbl 0198.47501 |

[11] | S. M. Ulam, Problems in Modern Mathematics, Chapter VI, John Wiley & Sons, New York, NY, USA, 1940. · Zbl 0137.24201 |

[12] | 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, pp. 222-224, 1941. · Zbl 0061.26403 |

[13] | 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 |

[14] | T. 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 |

[15] | D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223-237, 1951. · Zbl 0043.32902 |

[16] | P. G\uavru\cta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043 |

[17] | R. Badora, “On approximate derivations,” Mathematical Inequalities & Applications, vol. 9, no. 1, pp. 167-173, 2006. · Zbl 1093.39024 |

[18] | M. Eshaghi Gordji, M. Bavand Savadkouhi, and M. Bidkham, “Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces,” Journal of Computational Analysis and Applications, vol. 12, no. 2, pp. 454-462, 2010. · Zbl 1198.39037 |

[19] | R. Farokhzad and S. A. R. Hosseinioun, “Perturbations of Jordan higher derivations in Banach ternary algebras: an alternative fixed point approach,” International Journal of Nonlinear Analysis and its Applications, vol. 1, no. 1, pp. 42-53, 2010. · Zbl 1281.39037 |

[20] | S.-M. Jung, “On the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 221-226, 1996. · Zbl 0888.46018 |

[21] | A. Najati and F. Moradlou, “Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces,” Tamsui Oxford Journal of Mathematical Sciences, vol. 24, no. 4, pp. 367-380, 2008. · Zbl 1170.39017 |

[22] | A. Najati and C. Park, “Stability of homomorphisms and generalized derivations on Banach algebras,” Journal of Inequalities and Applications, vol. 2009, Article ID 595439, 12 pages, 2009. · Zbl 1187.39046 |

[23] | A. Najati and T. M. Rassias, “Stability of a mixed functional equation in several variables on Banach modules,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1755-1767, 2010. · Zbl 1191.39027 |

[24] | A. Najati and T. M. Rassias, “Stability of homomorphisms and (\theta ,\varphi )-derivations,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 2, pp. 264-281, 2009. · Zbl 1186.39035 |

[25] | A. Najati and G. Z. Eskandani, “A fixed point method to the generalized stability of a mixed additive and quadratic functional equation in Banach modules,” Journal of Difference Equations and Applications, vol. 16, no. 7, pp. 773-788, 2010. · Zbl 1197.39018 |

[26] | C. Park and M. E. Gordji, “Comment on “Approximate ternary Jordan derivations on Banach ternary algebras” [Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)],” Journal of Mathematical Physics, vol. 51, no. 4, Article ID 044102, 7 pages, 2010. · Zbl 1310.46047 |

[27] | C. Park and A. Najati, “Generalized additive functional inequalities in Banach algebras,” Journal of Nonlinear Analysis and its Applications, vol. 1, no. 2, pp. 54-62, 2010. · Zbl 1281.39032 |

[28] | 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 |

[29] | 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 |

[30] | J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268-273, 1989. · Zbl 0672.41027 |

[31] | 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 0789.46036 |

[32] | J. M. Rassias, “Solution of a stability problem of Ulam,” Discussiones Mathematicae, vol. 12, pp. 95-103, 1992. · Zbl 0779.47005 |

[33] | J. M. Rassias, “Complete solution of the multi-dimensional problem of Ulam,” Discussiones Mathematicae, vol. 14, pp. 101-107, 1994. · Zbl 0819.39012 |

[34] | P. G\uavru\cta, “An answer to a question of John. M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hardronic Mathematical Series, pp. 67-71, 1999. |

[35] | 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 |

[36] | K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,” International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36-46, 2008. · Zbl 1144.39029 |

[37] | B. Bouikhalene, E. Elqorachi, and J. M. Rassias, “The superstability of d’Alembert’s functional equation on the Heisenberg group,” Applied Mathematics Letters, vol. 23, no. 1, pp. 105-109, 2010. · Zbl 1195.39007 |

[38] | H.-X. Cao, J.-R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. I,” Journal of Inequalities and Applications, vol. 2009, Article ID 718020, 10 pages, 2009. · Zbl 1185.46033 |

[39] | H.-X. Cao, J.-R. Lv, and J. M. Rassias, “Superstability for generalized module left derivations and generalized module derivations on a Banach module. II,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, article 85, 8 pages, 2009. · Zbl 1211.39015 |

[40] | M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, and A. Ebadian, “On the stability of J\ast -derivations,” Journal of Geometry and Physics, vol. 60, no. 3, pp. 454-459, 2010. · Zbl 1188.39029 |

[41] | M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetapeh, “Approximately n-Jordan homomorphisms on Banach algebras,” Journal of Inequalities and Applications, Article ID 870843, 8 pages, 2009. · Zbl 1162.39017 |

[42] | M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, “Generalized Hyers-Ulam stability of generalized (n,k)-derivations,” Abstract and Applied Analysis, vol. 2009, Article ID 437931, 8 pages, 2009. · Zbl 1177.39032 |

[43] | M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, “Solution and stability of a mixed type additive, quadratic, and cubic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 826130, 17 pages, 2009. · Zbl 1177.39031 |

[44] | M. Eshaghi Gordji and A. Najati, “Approximately J\ast -homomorphisms: a fixed point approach,” Journal of Geometry and Physics, vol. 60, no. 5, pp. 809-814, 2010. · Zbl 1192.39020 |

[45] | M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, “Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces,” Abstract and Applied Analysis, vol. 2009, Article ID 417473, 14 pages, 2009. · Zbl 1177.39034 |

[46] | P. G\uavru\cta and L. G\uavru\cta, “A new method for the generalized Hyers-Ulam-Rassias stability,” Journal of Nonlinear Analysis and its Applications, vol. 1, no. 2, pp. 11-18, 2010. |

[47] | J. M. Rassias, “Two new criteria on characterizations of inner products,” Discussiones Mathematicae, vol. 9, pp. 255-267, 1988. · Zbl 0701.46011 |

[48] | J. M. Rassias, “Four new criteria on characterizations of inner products,” Discussiones Mathematicae, vol. 10, pp. 139-146, 1990. · Zbl 0753.46018 |

[49] | G. A. Tabadkan and A. Rahmani, “Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stability of generalized quadratic functional equations,” Advances in Applied Mathematical Analysis, vol. 4, no. 1, pp. 31-38, 2009. |

[50] | H. Khodaei and T. M. Rassias, “Approximately generalized additive functions in several variables,” Journal of Nonlinear Analysis and its Applications, vol. 1, pp. 22-41, 2010. · Zbl 1281.39041 |

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.