| Citation: |
Ali Ebadian, Somaye Zolfaghari, Saed Ostadbashi, Yongqiao Wang, Choonkil Park. SUPERSTABILITY OF GENERALIZED DERIVATIONS ON NON-ARCHIMEDEAN RANDOM BANACH ALGEBRAS VIA FIXED POINT METHOD[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3162-3174. doi: 10.11948/20230480
|
SUPERSTABILITY OF GENERALIZED DERIVATIONS ON NON-ARCHIMEDEAN RANDOM BANACH ALGEBRAS VIA FIXED POINT METHOD
-
Abstract
Using the fixed point method, we prove the superstability of generalized derivations on non-Archimedean random Banach algebras associated with the Cauchy functional equation.
-
-
References
[1] M. R. Abdollahpour, R. Aghayari and M. Th. Rassias, Hyers-Ulam stability of associated Laguerre differential equations in a subclass of analytic functions, J. Math. Anal. Appl., 2016, 437(1), 605–612. doi: 10.1016/j.jmaa.2016.01.024 [2] M. R. Abdollahpour and M. Th. Rassias, Hyers-Ulam stability of hypergeometric differential equations, Aequationes Math., 2019, 93(4), 691–698. doi: 10.1007/s00010-018-0602-3 [3] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989. [4] L. M. Arriola and W. A. Beyer, Stability of the Cauchy functional equation over $p$-adic fields, Real Analysis Exchange, 2005/2006, 31, 125–132. [5] A. R. Aruldass, D. Pachaiyappan and C. Park, Kamal transform and Ulam stability of differential equations, J. Appl. Anal. Comput., 2021, 11(3), 1631–1639. [6] R. Badora, On approximate derivations, Math. Inequal. Appl., 2006, 9, 167–173. [7] A. R. Baias, D. Poap and M. Th. Rassias, Set-valued solutions of an equation of Jensen type, Quaest. Math., 2023, 46(6), 1237–1244. doi: 10.2989/16073606.2022.2072249 [8] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 1949, 16, 385–397. [9] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 1951, 57, 223–237. doi: 10.1090/S0002-9904-1951-09511-7 [10] R. Chaharpashlou and A. M. Lopes, Hyers-Ulam-Rassias stability of a nonlinear stochastic fractional Volterra integro-differential equation, J. Appl. Anal. Comput., 2023, 13(5), 2799–2808. DOI: 10.11948/20230005. [11] S. S. Chang, Y. Cho and S. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers Inc., New York, 2001. [12] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 1992, 62, 59–64. doi: 10.1007/BF02941618 [13] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientic, Singapore, 2002. [14] J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 1968, 74, 305–309. doi: 10.1090/S0002-9904-1968-11933-0 [15] I. EL-Fassi, Generalized hyperstability of a Drygas functional equation on a restricted domain using Brzdek's fixed point theorem, J. Fixed Point Theory Appl., 2017, 19, 2529–2540. doi: 10.1007/s11784-017-0439-8 [16] I. El-Fassi, E. El-Hady and W. Sintunavarat, Hyperstability results for generalized quadratic functional equations in $(2, \alpha)$-Banach spaces, J. Appl. Anal. Comput., 2023, 13(5), 2596–2612. DOI: 10.11948/20220462. CrossRef $(2, \alpha)$-Banach spaces" target="_blank">Google Scholar
[17] E. Elqorachi and M. Th. Rassias, Generalized Hyers-Ulam stability of trigonometric functional equations, Math., 2018, {6}(5), Paper No. 83. [18] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci., 1991, 14, 431–434. doi: 10.1155/S016117129100056X [19] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 1994, 184, 431–436. doi: 10.1006/jmaa.1994.1211 [20] K. Hensel, Uber eine neue Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein, 1897, 6, 83–88. [21] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 1941, 27, 222–224. doi: 10.1073/pnas.27.4.222 [22] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhaüser, Basel, 1998. [23] D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math., 1992, 44, 125–153. doi: 10.1007/BF01830975 [24] G. Isac and Th. M. Rassias, Stability of $\psi$-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci., 1996, 19, 219–228. doi: 10.1155/S0161171296000324 CrossRef $\psi$-additive mappings: Applications to nonlinear analysis" target="_blank">Google Scholar
[25] Y. F. Jin, C. Park and M. Th. Rassias, Hom-derivations in $C. *$-ternary algebras, Acta Math. Sin. (Eng. Ser. ), 2020, 36(9), 1025–1038. doi: 10.1007/s10114-020-9323-3 CrossRef $C.*$-ternary algebras" target="_blank">Google Scholar
[26] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2001. [27] S. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011. [28] S. Jung, K. Lee, M. Th. Rassias and S. Yang, Approximation properties of solutions of a mean valued-type functional inequality, $Ⅱ$, Math., 2020, 8, Paper No. 1299. [29] S. Jung, D. Popa and M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim., 2014, 59(1), 165–171. doi: 10.1007/s10898-013-0083-9 [30] S. Jung and M. Th. Rassias, A linear functional equation of third order asssociated with the Fibonacci numbers, Abstr. Appl. Anal., 2014, 2014, Art. ID 137468. [31] S. Jung, M. Th. Rassias and C. Mortici, On a functional equation of trigonometric type, Appl. Math. Comput., 2015, 252, 294–303. [32] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009. [33] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht, 1997. [34] Y. Lee, S. Jung and M. Th. Rassias, On an $n$-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), 13–16. [35] Y. Lee, S. Jung and M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 2018, 12(1), 43–61. [36] D. Miheƫ and D. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 2006, 343, 567–572. [37] A. K. Mirmostafaee, Perturbation of generalized derivations in fuzzy Menger normed algebras, Fuzzy Sets Syst., 2012, 195, 109–117. doi: 10.1016/j.fss.2011.10.015 [38] C. Mortici, M. Th. Rassias and S. Jung, On the stability of a functional equation associated with the Fibonacci numbers, Abstr. Appl. Anal., 2014, 2014, Art. ID 546046. [39] R. Murali, C. Park and A. Ponmana Selvan, Hyers-Ulam stability for an $n$th order differential equation using fixed point approach, J. Appl. Anal. Comput., 2021, 11(2), 614–631. $n$th order differential equation using fixed point approach" target="_blank">Google Scholar
[40] S. Paokanta, M. Dehghanian, C. Park and Y. Sayyari, A system of additive functional equations in complex Banach algebra, Demonstr. Math., 2023, 56(1), Paper No. 20220165. doi: 10.1515/dema-2022-0165 [41] C. Park and M. Th. Rassias, Additive functional equations and partial multipliers in $C. *$-algebras, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 2019, 113(3), 2261–2275. doi: 10.1007/s13398-018-0612-y [42] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 2003, 4, 91–96. [43] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 1978, 72, 297–300. doi: 10.1090/S0002-9939-1978-0507327-1 [44] Th. M. Rassias, Problem 16; 2, Report of the 27th International Symp. on Functional Equations, Aequationes Math., 1990, 39, 292–293; 309. [45] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 2000, 62, 23–130. doi: 10.1023/A:1006499223572 [46] Th. M. Rassias, Functional Equations and Inequalities, Kluwer Academic Publ., Dordrecht, 2000. [47] Th. M. Rassias and P. Šemrl, On the Hyers–Ulam stability of linear mappings, J. Math. Anal. Appl., 1993, 173, 325–338. doi: 10.1006/jmaa.1993.1070 [48] P. K. Sahoo and P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, FL, 2011. [49] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, North Holand and New York, 1983. [50] A. N. Šerstnev, On the notion of a random normed space (in Russian), Doklady Akademii Nauk SSSR, 1963, 149, 280–283. [51] W. Smajdor, Note on a Jensen type functional equation, Publ. Math. Debrecen, 2003, 163, 703–714. [52] A. Thanyacharoen and W. Sintunavarat, On new stability results for composite functional equations in quasi-$\beta$-normed spaces, Demonstr. Math., 2021, 54(1), 68–84. doi: 10.1515/dema-2021-0002 CrossRef $\beta$-normed spaces" target="_blank">Google Scholar
[53] T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex function, J. Math. Anal. Appl., 2002, 272, 604–616. doi: 10.1016/S0022-247X(02)00181-6 [54] A. Turab, N. Rosli, W. Ali and J. J. Nieto, The existence and uniqueness of solutions to a functional equation arising in psychological learning theory, Demonstr. Math., 2023, 56(1), Paper No. 20220231. doi: 10.1515/dema-2022-0231 [55] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1940. [56] A. C. M. Van Rooij, Non-Archimedean Functional Analysis, in: Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1978, 51. [57] J. Wang, Some further generalization of the Ulam-Hyers-Rassias stability of functional equations, J. Math. Anal. Appl., 2001, 262, 406–423. -
-
Export File
Citation
Ali Ebadian, Somaye Zolfaghari, Saed Ostadbashi, Yongqiao Wang, Choonkil Park. SUPERSTABILITY OF GENERALIZED DERIVATIONS ON NON-ARCHIMEDEAN RANDOM BANACH ALGEBRAS VIA FIXED POINT METHOD[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3162-3174. doi: 10.11948/20230480
DownLoad: