Approximation of Random Homomorphisms and Random Derivations on Banach Algebras via Direct and Fixed Point Methods
Abstract
In this paper, we prove the approximation of homomorphisms and derivations related to the following functional equation:
\[
f(2x+y) + f(2x-y)
= f(x+y) + f(x-y) + 2f(2x) - 2f(x).
\]
The results are obtained in random Banach algebras by means of the direct method and the fixed point method.
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References
Alshybani, S., Vaezpour, S. M., & Mahmood, S. J. (2019). Generalized Hyers-Ulam stability of mixed-type additive-quadratic functional equation of random homomorphisms in random normed algebras. Journal of Physics: Conference Series, 1294, 032004. https://doi.org/10.1088/1742-6596/1294/3/032004
Al-Shybani, S., Vaezpour, S. M., & Saadati, R. (2018). Generalized Hyers-Ulam stability of a sextic functional equation in random normed space. Journal of Computational Analysis and Applications, 24 (2), 370-381.
Al-Shybani, S., Vaezpour, S. M., & Saadati, R. (2017). Generalized Hyers-Ulam stability of mixed-type additive-quadratic functional equation in random normed spaces. Journal of Mathematical Analysis, 5, 12-25.
Aoki, T. (1950). On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan, 2, 64-66. https://doi.org/10.2969/jmsj/00210064
Cho, Y. J., Rassias, T. M., & Saadati, R. (2013). Stability of functional equations in random normed spaces. Springer. https://doi.org/10.1007/978-1-4614-8477-6
Hsdžić, O., & Pap, E. (2001). Fixed point theory in probabilistic metric spaces. Kluwer Academic Publishers. https://doi.org/10.1007/978-94-017-1560-7
Hyers, D. H. (1941). On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224. https://doi.org/10.1073/pnas.27.4.222
Sabah, M. S., & Al-Shybani, S. (2025). Approximation of fuzzy homomorphisms and fuzzy derivations in Banach algebras. Journal of Al-Qadisiyah for Computer Science and Mathematics, 17(1), 50-60. https: //doi.org/10.29304/jqcsm.2025.17.12009
Sabah, M. S., & Al-Shybani, S. (2025). Approximation of homomorphisms and derivations in Banach algebras via direct and fixed point methods. Nonlinear Functional Analysis and Applications, 30(2), 625-646.
Park, C., Lee, J. R., & Shin, D. Y. (2012). Generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation. Applied Mathematics Letters, 25 (2), 200-205. https://doi.org/10.1016/j.aml.2011.08.018
Park, C., Gordji, M. E., & Saadati, R. (2012). Random homomorphisms and random derivations in random normed algebras via fixed point method. Journal of Inequalities and Applications, 2012, 194. https://doi. org/10.1186/1029-242X-2012-194
Rassias, T. M. (1978). On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society, 72, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
Rheaf, N. D., & Al-Shybani, S. (2024). Some new topological structures in lattice random normed spaces. Journal of Nonlinear Functional Analysis and Applications, 1095-1107.
Ulam, S. M. (1960). A collection of mathematical problems. Interscience Publishers.
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