Jordan k-∗-türevlerinin Γ*-Banach cebirlerinde sabit nokta yöntemiyle stabilitesi

Abstract

Sabit nokta yöntemlerini kullanarak, μf((x+y)/2)+μf((x-y)/2)=f(μx) ile tanımlı Jensen tipi fonksiyonel denklem için Jordan k-∗-türevlerinin Γ*-Banach cebirleri üzerindeki stabilitesini ve süper stabilitesini kanıtlıyoruz. Burada μ sayısı, |μ| = 1 şartını sağlayan bir karmaşık sayıdır. Ayrıca, Γ*-Banach cebirleri üzerindeki f(2μx+μy)+f(μx+2μy)=μ[f(3x)+f(3y)] fonksiyonel denklemi ile Jordan k-∗-türevlerinin stabilitesini ve süper stabilitesini araştırıyoruz.

Keywords

• İnvolüsyonlu Γ-halkaları
• Hyers-Ulam-Rassias stabilitesi
• Jordan k-*-türevler.

The stability of Jordan k-*-derivations on Γ∗-Banach algebras by fixed point method

Abstract

Using fixed point methods, we prove the stability and the superstability of Jordan k-∗-derivations on Γ∗-Banach algebras for the following Jensen-type functional equation μf((x+y)/2)+μf((x-y)/2)=f(μx) where μ is a complex number such that |μ| = 1. We also investigate the stability and the superstability of Jordan k-∗-derivations with the functional equation f(2μx+μy)+f(μx+2μy)=μ[f(3x)+f(3y)] on Γ∗-Banach algebras.

Keywords

• Γ-rings with involution
• Hyers-Ulam-Rassias stability
• Jordan k-∗-derivations.

References

1. Ulam, S.M., Problems in modern mathematics, Chapter VI, science ed. Wiley, New York, (1940).
2. Hyers, D.H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224, (1941).
3. Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 297-300, (1978).
4. Hyers, D.H., Isac, G. ve Rassias, Th.M., Stability of functional equations in several variables, Birkhauser, Basel, (1998).
5. Czerwik, S., Functional equations and inequalities in several variables, River Edge, NJ: World Scientific, (2002).
6. Jung, S.M., Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Palm Harbor: Hadronic Press, (2001).
7. Semrl, P., The functional equation of multiplicative derivation is superstable on standard operator algebras, Integral Equations and Operator Theory, 18, 118-122, (1994).
8. An, J., Cui, J. ve Park, C., Jordan ∗-derivations on C∗-algebras and JC∗-algebras, Abstract Applied Analysis, Article ID 410437, 1-12, (2008).

9. Jang, S. ve Park, C., Approximate ∗-derivations and approximate quadratic ∗-derivations on C∗-algebras, Journal of Inequalities and Applications, Article ID 55, 1-13 pages, (2011).
10. Park, C. ve Bodaghi, A., On the stability of ∗-derivations on Banach ∗-algebras, Advances in Differential Equations, 55, 1-10, (2012).
11. Nobusawa, N. On the generalization of the ring theory, Osaka Journal of Mathematics, 1, 81-89, (1964).
12. Barnes, W.E., On the Γ-rings of Nobusawa, Pacific Journal of Mathematics, 18, 3, 411-422, (1966).
13. Sapanci, M. ve Nakajima, A., Jordan derivations on completely prime Γ-rings, Mathematica Japonica, 46, 1, 47-51, (1997).
14. Kandamar, H., The k-derivation of a Gamma ring, Turkish Journal of Mathematics, 24, 221-231, (2000).
15. Bhattacharya, D.K. ve Maity, A.K., Semilinear tensor product of Gamma Banach algebras, Ganita, 40, 78-80, (1989).
16. Hoque, M.F., Alshammari, F.S. ve Paul, A.C., Left centralizers of semiprime Γ-rings with involution, Applied Mathematical Sciences, 95, 4713-4722, (2014).
17. Park, C., Homomorphisms between Poisson JC∗-algebras, Bulletin of the Brazilian Mathematical Society, 36, 1, 79-97, (2005).
18. Caˇdariu, L. ve Radu, V., On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte, 346, 43-52, (2004).
19. Arslan, B. ve Arslan, O., On the stability of homomorphisms and k-derivations on Gamma-Banach algebras, University Politehnica of Bucharest Scientific Bulletin Series A, 80, 2, 69-78, (2018).
20. Najati, A. ve Park, C., Stability of homomorphisms and generalized derivations on Banach algebras, Journal of Inequalities and Applications, 2009, 1-12, (2009).