Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 25, 2015

Approximate Higher Ring Derivations in Non-Archimedean Banach Algebras

  • Ick-Soon Chang EMAIL logo , Badrkhan Alizadeh , M. Eshaghi Gordji and Hark-Mahn Kim
From the journal Mathematica Slovaca

Abstract

In this paper, we prove the stability of higher ring derivations associated with a general Cauchy-Jensen functional inequality in the class of mappings from non-Archimedean normed algebras to non-Archimedean Banach algebras.

References

[1] AOKI, T.: On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.10.2969/jmsj/00210064Search in Google Scholar

[2] ASHRAF, M.-REHMAN, N.-ALI, S.-MOZUMDER, M. R.: On generalized (θ, φ)-derivations in semiprime rings with involution, Math. Slovaca 62 (2012), 451-460.10.2478/s12175-012-0021-1Search in Google Scholar

[3] BENYAMINI, Y.-LINDENSTRAUSS, J.: Geometric Nonlinear Functional Analysis, Vol. 1. Amer. Math. Soc. Colloq. Publ. 48, Amer. Math. Soc., Providence, RI, 2000.10.1090/coll/048Search in Google Scholar

[4] BOURGIN, D. G.: Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237.10.1090/S0002-9904-1951-09511-7Search in Google Scholar

[5] CHO, Y.-PARK, C.-SAADATI, R.: Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett. 23 (2010), 1238-1242.10.1016/j.aml.2010.06.005Search in Google Scholar

[6] GORDJI, M. E.: Nearly ring homomorphisms and nearly ring derivations on non-Archimedean Banach algebras, Abstr. Appl. Anal. 2010 (2010), Article ID 393247, 12 pp.Search in Google Scholar

[7] GORDJI, M. E.-GHAEMI, M. B.-ALIZADEH, B.: A fixed point method for perturbation of higher ring derivations in non-Archimedean Banach algebras, Int. J. Geom. Methods Mod. Phys. 8 (2011), N7 (To appear).10.1155/2011/417187Search in Google Scholar

[8] GORDJI, M. E.-GHAEMI, M. B.-ALIZADEH, B.: A fixed point method to superstability of generalized derivations on non-Archimedean Banach algebras, Abstr. Appl. Anal. 2011 (2011), Article ID 587097, 9 pp.Search in Google Scholar

[9] GAJDA, Z.: On the stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431-434.10.1155/S016117129100056XSearch in Google Scholar

[10] GǍVRUTA, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.10.1006/jmaa.1994.1211Search in Google Scholar

[11] GAO, Z-X.-CAO, H-X.-ZHENG, W-T.-XU, L.: Generalized Hyers-Ulam-Rassias stability of functional inequalities and functional equations, J. Math. Inequal. 3 (2009), 63-77.10.7153/jmi-03-06Search in Google Scholar

[12] GORDJI, M. E.-GHOBADIPOUR, N.: Nearly generalized Jordan derivations, Math. Slovaca 61 (2011), 55-62.10.2478/s12175-010-0059-xSearch in Google Scholar

[13] HYERS, D. H.: On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224.10.1073/pnas.27.4.222Search in Google Scholar

[14] JUNG, Y.-CHANG, I.: On approximately higher ring derivations, J. Math. Anal. Appl. 342 (2008), 636-643.10.1016/j.jmaa.2008.01.083Search in Google Scholar

[15] JUNG, S.: On the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 204 (1996), 221-226.10.1006/jmaa.1996.0433Search in Google Scholar

[16] KIM, H.-KANG, S.-CHANG, I.: On functional inequalities originating from module Jordan left derivations, J. Inequal. Appl. 2008 (2008), Article ID. 278505, 9 pp.Search in Google Scholar

[17] LEE, J.-PARK, C.-SHIN, D.: On the stability of generalized additive functional inequalities in Banach spaces, J. Inequal. Appl. 2008 (2008), Article ID. 210626, 13 pp.Search in Google Scholar

[18] MIHET¸ , D.-SAADATI, R.-VAEZPOUR, S. M.: The stability of an additive functional equation in Menger probabilistic ϕ-normed spaces, Math. Slovaca 61 (2011), 817-826.10.2478/s12175-011-0049-7Search in Google Scholar

[19] NAJATI, A.-MOGHIMI, M. B.: Stability of a functional equation deriving from quadratic and additive function in quasi-Banach spaces, J. Math. Anal. Appl. 337 (2008), 399-415.10.1016/j.jmaa.2007.03.104Search in Google Scholar

[20] PARK, C.: Fuzzy stability of additive functional inequalities with the fixed point alternative, J. Inequal. Appl. 2009 (2009), Article ID. 410576, 17 pp.Search in Google Scholar

[21] PARK, C.-AN, J.-MORADLOU, F.: Additive functional inequalities in Banach modules, J. Inequal. Appl. 2008 (2008), Article ID 592504, 10 pp.Search in Google Scholar

[22] RASSIAS, J.M.: On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130.10.1016/0022-1236(82)90048-9Search in Google Scholar

[23] RASSIAS, J.M.: On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), 445-446.Search in Google Scholar

[24] RASSIAS, T. M.: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.10.1090/S0002-9939-1978-0507327-1Search in Google Scholar

[25] RASSIAS, T. M.: The stability of mappings and related topics, in ‘Report on the 27th ISFE’, Aequationes Math. 39 (1990), 292-293.Search in Google Scholar

[26] RASSIAS, T. M.-ŠEMRL, P.: On the behaviour of mappings which do not satisfy Hyers- Ulam-Rassias stability, Proc. Amer. Math. Soc. 114 (1992), 989-993.10.1090/S0002-9939-1992-1059634-1Search in Google Scholar

[27] ROLEWICZ, S.: Metric Linear Spaces, PWN-Polish Sci. Publ./Reidel, Warszawa/Dordrecht, 1984.Search in Google Scholar

[28] ULAM, S. M.: A Collection of the Mathematical Problems, Interscience Publ., New York, 1960. Search in Google Scholar

Received: 2012-8-10
Accepted: 2012-9-18
Published Online: 2015-3-25
Published in Print: 2015-2-1

© 2015 Mathematical Institute Slovak Academy of Sciences

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/ms-2015-0013/html
Scroll to top button