Abstract
This investigation is carried out by the inspiration of noteworthy results pertaining to approximate Lie \(\star \)-derivations connected with various additive, quadratic, cubic, quartic functional equations. For the first time, we study the stabilities of reciprocal Lie \(\star \)-derivations and reciprocal-quadratic Lie \(\star \)-derivations in the setting of normed \(\star \)-algebras through direct method. The stabilities associated with different upper bounds are also discussed. In addition, we present the relationship of reciprocal derivations dealt in this study with automorphism. The comparative study of stability results obtained in this investigation is also discussed at the end of this paper.
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References
Abellanas L, Alonso LM (1975) A general setting for Casimir invariants. J Math Phys 16:1580–1584
Alshybani S, Vaezpour SM, Saadati R (2018) Stability of the sextic functional equation in various spaces. J Inequ Special Func 9(4):8–27
Aoki T (1950) On the stability of the linear transformation in Banach spaces. J Math Soc Japan 2:64–66
Bodaghi A, Alias IA, Gordji ME (2021) On the stability of quadratic double centralizers and quadratic multipliers: a fixed point approach. J Inequ Appl 957541:1–9
Bodaghi A, Senthil Kumar BV (2017) Estimation of inexact reciprocal-quintic and reciprocal-sextic functional equations. Mathematica 49(82):3–14
Bodaghi A, Zabandan G (2014) On the stability of quadratic \(\star \)-derivations on \(\star \)-Banach algebras. Thai J Math 12(2):343–356
Dutta H, Senthil Kumar BV (2019a) Geometrical elucidations and approximation of some functional equations in numerous variables. Proc Indian Natl Soc Acad 85(3):603–611
Dutta H, Senthil Kumar BV (2019b) Classical stabilities of an inverse fourth power functional equation. J Interdisciplinary Math 22(7):1061–1070
Ebadian A, Zolfaghari S, Ostadbashi S, Park C (2015) Approximation on the reciprocal functional equation in several variables in matrix non-Archimedean random normed spaces. Adv Differ Equ 314:1–13
Forti GL, Shulman E (2020) A comparison among methods for proving stability. Aequq Math 94:547–574
Fošner A, Fošner M (2013) Approximate cubic Lie derivations. Abst Appl Anal 425784:1–15
Gajda Z (1991) On the stability of additive mappings. Int J Math Math Sci 14:431–434
Găvruta P (1994) A generalization of the Hyers-Ulam-Rassias stability of approximately additive mapppings. J Math Anal Appl 184:431–436
Gordji ME, Habibian F (2009) Hyers-Ulam-Rassias stability of quaratic derivations on Banach algebras. Nonlinear Func Anal Appl 14:759–766
Gordji ME, Gharetapeh SK, Savadkouhi MB, Aghaei M, Karimi T (2010) On cubic derivations. Int J Math Anal 4(49–52):2501–2514
Hyers DH (1941) On the stability of the linear functional equation. Proc Natl Acad Sci USA 27:222–224
Jacobson N (1979) Lie algebras. Dover Publications, New York
Jang S, Park C (2011) Approximate \(\star \)-derivations and approximate quadratic \(\star \)-derivations on \(C^{\star }\)-algebra. J Inequal Appl 55:1–13
Kang D, Koh H (2016) A fixed point approach to the stability of quartic Lie \(\star \)-derivations. Korean J Math 24(4):587–600
Kang D, Koh H (2017) A fixed point approach to the stability of sextic Lie \(\star \)-derivations. Filomat 31(15):4933–4944
Kim GH, Shin HY (2017) Hyers-Ulam stability of quadratic functional equations on divisible square-symmetric groupoid. Int J Pure Appl Math 112(1):189–201
Kim SO, Senthil Kumar BV, Bodaghi A (2017) Stability and non-stability of the reciprocal-cubic and reciprocal-quartic functional equations in non-Archimedean fields. Adv Differ Equ 77:1–12
Koh H (2015) Approximate quartic Lie \(\star \)-derivations. J Korean Soc Math Educ Ser B Pure Appl Math 22(4):389–401
Najati A, Lee JR, Park C, Rassias TM (2018) On the stability of a Cauchy type functional equation. Demonstr Math 51:323–331
Park C, Bodaghi A (2012) On the stability of \(\star \)-derivations on Banach \(\star \)-algebras. Adv Differ Equ 138:1–11
Popovych RO, Boyko VM, Nesterenko MO, Lutfullin MW (2003) Realizations of real low-dimensional Lie algebras. J Phys A 36(26):7337–7360
Rand D, Winternitz P, Zassenhaus H (1988) On the identification of a Lie algebra given by its structure constants. I. Direct decompositions, Levi decompositions, and nilradicals. Linear Algebra Appl 109:197–246
Rassias JM (1982) On approximately of approximately linear mappings by linear mappings. J Funct Anal USA 46:126–130
Rassias JM (2001) Solution of the Ulam stability problem for cubic mappings. Glasnik Matematicki Ser III 36(56):63–72
Rassias JM, Arunkumar M, Karthikeyan S (2015) Lagrange’s quadratic functional equation connected with homomorphisms and derivations on Lie \(C^{\star }\)-algebras: direct and fixed point methods, Malaya. J Math S(1):228–241
Rassias TM (1978) On the stability of the linear mapping in Banach spaces. Proc Am Math Soc 72:297–300
Ravi K, Arunkumar M, Rassias JM (2008) On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation. Int J Math Sci 3(8):36–47
Ravi K, Rassias JM, Pinelas S, Narasimman P (2014) The stability of a generalized radical reciprocal quadratic functional equation in Felbin’s. Pan Am Math J 24(1):75–92
Ravi K, Rassias JM, Senthil Kumar BV (2010) A fixed point approach to the generalized Hyers-Ulam stability of reciprocal difference and adjoint functional equations. Thai J Math 8(3):469–481
Ravi K, Rassias JM, Senthil Kumar BV (2011) Ulam stability of reciprocal difference and adjoint functional equations. Aust J Math Anal Appl 8(1):1–18
Ravi K, Senthil Kumar BV (2010) Ulam-Gavruta-Rassias stability of Rassias reciprocal functional equation. Global J Appl Math Sci 3(1–2):57–79
Ravi K, Senthil Kumar BV (2015) Generalized Hyers-Ulam Stability of a system of bi-reciprocal functional equations. Eur J Pure Appl Math 8(2):283–293
Ravi K, Thandapani E, Senthil Kumar BV (2011) Stability of reciprocal type functional equations. Pan Am Math J 21(1):59–70
Senthil Kumar BV, Dutta H (2018) Non-Archimedean stability of a generalized reciprocal-quadratic functional equation in several variables by direct and fixed point methods. Filomat 32(9):3199–3209
Senthil Kumar BV, Dutta H (2019a) Fuzzy stability of a rational functional equation and its relevance to system design. Int J General Syst 48(2):157–169
Senthil Kumar BV, Dutta H (2019b) Approximation of multiplicative inverse undecic and duodecic functional equations. Math Method Appl Sci 42:1073–1081
Senthil Kumar BV, Dutta H, Sabarinathan S (2019) Approximation of a system of rational functional equations of three variables. Int J Appl Comput Math 5(3):1–16
Senthil Kumar BV, Dutta H, Sabarinathan S (2020a) Fuzzy approximations of a multiplicative inverse cubic functional equation. Soft Comput 24:13285–13292
Senthil Kumar BV, Al-Shaqsi K, Dutta H (2020b) Classical stabilities of multiplicative inverse difference and adjoint functional equations. Adv Differ Equ 215:1–9
Ulam SM (1964) Problems in modern mathematics, Chapter VI. Wiley, New York
Wang Z, Saadati R (2018) Approximation of additive functional equations in NA Lie \(C^{\star }\)-algebras. Demonstr Math 51:37–44
Yang SY, Bodaghi A, Atan KAM (2012) Approximate cubic \(\star \)-derivations on Banach \(\star \)-algebra. Abst Appl Anal 684179:1–12
Funding
The first two authors are supported by the Research Council, Oman (Under Project Proposal ID: BFP/RGP/CBS/18/099). The third author is supported by the Science and Engineering Research Board, India, under MATRICS Scheme (F. No.: MTR/2020/000534).
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Kumar, B.V.S., Al-Shaqsi, K. & Dutta, H. Approximate reciprocal Lie \(\star \)-Derivations. Soft Comput 25, 14969–14977 (2021). https://doi.org/10.1007/s00500-021-06395-9
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DOI: https://doi.org/10.1007/s00500-021-06395-9