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Fixed points and Hyers–Ulam–Rassias stability of Cauchy–Jensen functional equations in Banach algebras. (English) Zbl 1167.39018

The author, using the fixed point method, proves the Hyers-Ulam-Rassias stability of homomorphisms and of generalized derivations in real Banach algebras for a Cauchy-Jensen type functional equation.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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References:

[1] Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
[2] Hyers DH: On the stability of the linear functional equation.Proceedings of the National Academy of Sciences of the United States of America 1941,27(4):222-224. 10.1073/pnas.27.4.222 · Zbl 0061.26403
[3] Rassias ThM: On the stability of the linear mapping in Banach spaces.Proceedings of the American Mathematical Society 1978,72(2):297-300. 10.1090/S0002-9939-1978-0507327-1 · Zbl 0398.47040
[4] Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.Journal of Mathematical Analysis and Applications 1994,184(3):431-436. 10.1006/jmaa.1994.1211 · Zbl 0818.46043
[5] Park C-G: On the stability of the linear mapping in Banach modules.Journal of Mathematical Analysis and Applications 2002,275(2):711-720. 10.1016/S0022-247X(02)00386-4 · Zbl 1021.46037
[6] Park C-G: Modified Trif’s functional equations in Banach modules over a[InlineEquation not available: see fulltext.]-algebra and approximate algebra homomorphisms.Journal of Mathematical Analysis and Applications 2003,278(1):93-108. 10.1016/S0022-247X(02)00573-5 · Zbl 1046.39022
[7] Park C-G: On an approximate automorphism on a[InlineEquation not available: see fulltext.]-algebra.Proceedings of the American Mathematical Society 2004,132(6):1739-1745. 10.1090/S0002-9939-03-07252-6 · Zbl 1055.47032
[8] Park C-G: Lie[InlineEquation not available: see fulltext.]-homomorphisms between Lie[InlineEquation not available: see fulltext.]-algebras and Lie[InlineEquation not available: see fulltext.]-derivations on Lie[InlineEquation not available: see fulltext.]-algebras.Journal of Mathematical Analysis and Applications 2004,293(2):419-434. 10.1016/j.jmaa.2003.10.051 · Zbl 1051.46052
[9] Park C-G: Homomorphisms between Lie[InlineEquation not available: see fulltext.]-algebras and Cauchy-Rassias stability of Lie[InlineEquation not available: see fulltext.]-algebra derivations.Journal of Lie Theory 2005,15(2):393-414. · Zbl 1091.39006
[10] Park C-G: Homomorphisms between Poisson[InlineEquation not available: see fulltext.]-algebras.Bulletin of the Brazilian Mathematical Society 2005,36(1):79-97. 10.1007/s00574-005-0029-z · Zbl 1091.39007
[11] Park C-G: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between[InlineEquation not available: see fulltext.]-algebras.Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(4):619-632. · Zbl 1125.39027
[12] Park C: Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between -algebras. to appear in Mathematische Nachrichten · Zbl 1142.39023
[13] Park C, Hou J: Homomorphisms between[InlineEquation not available: see fulltext.]-algebras associated with the Trif functional equation and linear derivations on[InlineEquation not available: see fulltext.]-algebras.Journal of the Korean Mathematical Society 2004,41(3):461-477. · Zbl 1058.39025
[14] Rassias ThM: Problem 16; 2; Report of the 27th International Symposium on Functional Equations.Aequationes Mathematicae 1990,39(2-3):292-293, 309.
[15] Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings.Journal of Mathematical Analysis and Applications 2000,246(2):352-378. 10.1006/jmaa.2000.6788 · Zbl 0958.46022
[16] Rassias ThM: On the stability of functional equations in Banach spaces.Journal of Mathematical Analysis and Applications 2000,251(1):264-284. 10.1006/jmaa.2000.7046 · Zbl 0964.39026
[17] Rassias JM: On approximation of approximately linear mappings by linear mappings.Bulletin des Sciences Mathématiques 1984,108(4):445-446. · Zbl 0599.47106
[18] Rassias JM: Solution of a problem of Ulam.Journal of Approximation Theory 1989,57(3):268-273. 10.1016/0021-9045(89)90041-5 · Zbl 0672.41027
[19] Rassias JM: On approximation of approximately linear mappings by linear mappings.Journal of Functional Analysis 1982,46(1):126-130. 10.1016/0022-1236(82)90048-9 · Zbl 0482.47033
[20] Cădariu L, Radu V: Fixed points and the stability of Jensen’s functional equation.Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):7. · Zbl 1043.39010
[21] Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space.Bulletin of the American Mathematical Society 1968, 74: 305-309. 10.1090/S0002-9904-1968-11933-0 · Zbl 0157.29904
[22] Baak C: Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces.Acta Mathematica Sinica 2006,22(6):1789-1796. 10.1007/s10114-005-0697-z · Zbl 1118.39012
[23] Ara P, Mathieu M: Local Multipliers of C∗-Algebras, Springer Monographs in Mathematics. Springer, London, UK; 2003:xii+319. · Zbl 1015.46001
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