A fixed point method for proving the stability of ring $(\alpha, \beta, \gamma)$-derivations in $2$-Banach algebras

Journal of Linear and Topological Algebra

مقاله 2، دوره 06، شماره 04، اسفند 2017، صفحه 269-276   

نوع مقاله: Special Issue on Fixed Point Theory

نویسندگان

M. Eshaghi Gordji¹؛ S. Abbaszadeh^{* 2}

¹Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

²Department of Mathematics, Payame Noor University, P.O. BOX 19395-4697, Tehran, Iran

چکیده

In this paper, we first present the new concept of $2$-normed algebra. We investigate the structure of this algebra and give some examples. Then we apply a fixed point theorem to prove the stability and hyperstability of $(\alpha, \beta, \gamma)$-derivations in $2$-Banach algebras.

کلیدواژه‌ها

Fixed point theorem؛ $2$-normed algebras؛ $(\alpha, \beta, \gamma)$-derivations؛ hyperstability

مراجع

[1] S. M. Ulam, A Collection of Mathematical Problems, Inter science Publ, New York, 1960.

[2] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA. 27 (1941), 222-224.

[3] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

[4] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.

[5] R. P. Agarwal, B. Xu, W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl. 288 (2003), 852-869.

[6] S. Abbaszadeh, Intuitionistic fuzzy stability of a quadratic and quartic functional equation, Int. Journal Nonlinear Anal. Appl. 1 (2010), 100-124.

[7] M. Eshaghi-Gordji, S. Abbaszadeh, Stability of Cauchy-Jensen inequalities in fuzzy Banach spaces, Appl. Comput. Math. 11 (2012), 27-36.

[8] M. Eshaghi, S. Abbaszadeh, On the orthogonal pexider derivations in orthogonality Banach algebras, Fixed Point Theory. 17 (2016), 327-340.

[9] M. Eshaghi, S. Abbaszadeh, M. de la Sen, Z. Farhad, A fixed point result and the stability problem in Lie superalgebras, Fixed Point Theory Appl. (2015), 2015:219.

[10] M. Eshaghi, S. Abbaszadeh, Approximate generalized derivations close to derivations in Lie C∗-algebras, J. Appl. Anal. 21 (2015) 37-43.

[11] M. Eshaghi, S. Abbaszadeh, M. de la Sen, On the stability of conditional homomorphisms in Lie C∗-algebras, J. Generalized Lie Theory Appl. 9 (2015), 5 pages.

[12] M. E. Gordji, S. Abbaszadeh, Intuitionistic fuzzy almost Cauchy-Jensen mappings, Demo. Math. 49 (2016), 18-25.

[13] M. E. Gordji, S. Abbaszadeh, Theory of Approximate Functional Equations: In Banach Algebras, Inner Product Spaces and Amenable Groups, Academic Press, 2016.

[14] M. Khanehgir, F. Hasanvand, Asymptotic aspect of quadratic functional equations and super stability of higher derivations in multi-fuzzy normed spaces, J. Linear Topolog. Algebra. 5 (2016), 67-81.

[15] P. Kannappan, Functional Equations and Inequalities with Applications, Springer Science & Business Media, 2009.

[16] S. M. Jung, D. Popa, M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim. 59 (2014), 165-171.

[17] G. V. Milovanovi´c, Th. M. Rassias (eds.), Analytic Number Theory, Approximation Theory and Special Functions, Springer, 2014.

[18] S. G¨ahler, 2-metrische R¨aume und ihre topologische Struktur, Math. Nachr. 26 (1963), 115-148.

[19] S. Gahler, Lineare 2-normierte R¨aumen, Math. Nachr. 28 (1964), 1-43.

[20] S. A. Mohiuddine, M. Aiyub, Lacunary statistical convergence in random 2-normed spaces, Appl. Math. Inf. Sci. 6 (2012), 581-585.

[21] A. S¸ahiner, M. Gu¨rdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwan. J. Math. 11 (2007), 1477-1484.

[22] N. Srivastava, S. Bhattacharya, S. N. Lal, 2-normed algebras-I, Publ. Inst. Math. Nouv. ser. 88 (2010), 111-121.

[23] N. Srivastava, S. Bhattacharya, S. N. Lal, 2-normed algebras-II, Publ. Inst. Math. Nouv. ser. 90 (2011), 135-143.

[24] W. G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011), 193-202.

[25] J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc. 112 (1991), 729-732.

[26] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory. 4 (2003), 91-96.

[27] B. Margolis, J. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309.

[28] M. Eshaghi Gordji, H. Khodaei, A fixed point technique for investigating the stability of (α, β, γ)-derivations on Lie C∗-algebras, Nonlinear Anal. 76 (2013), 52-57.

[29] M. Akkouchi, Stability of certain functional equations via a fixed point of Ciric, Filomat. 25 (2011), 121-127.

[30] M. Eshaghi Gordji, H. Habibi, Fixed point theory in generalized orthogonal metric space, J. Linear Topolog. Algebra. 6 (3) (2017), 251-260.