Document Type : Research Paper
Authors
1 Department of Mathematics, Sirjan University of Technology, P.O.Box 7813733385, Sirjan, Iran.
2 Teacher of Mathematics, Fars education, Shiraz, Iran.
Abstract
In this paper, we introduce and solve a system of bi-Drygas functional equations
\begin{equation}
\left\{
\begin{aligned}
&f(x+y,z)+f(x-y,z)=2f(x,z)+f(y,z)+f(-y,-z)\nonumber\\
&f(x,y+z)+f(x, y-z)=2f(x,y)+f(x,z)+f(-x,-z)\nonumber
\end{aligned}
\right.
\end{equation}
for all $x,y,z\in X$. We will also investigate the Hyers-Ulam stability of the system of bi-Drygas functional equations.
Keywords
Main Subjects
1. M.R. Abdollahpour and Th.M. Rassias, Hyers-Ulam stability of hypergeometric differential equations, Aequat. Math., 93 (4) (2019), pp. 691-698.
2. M.R. Abdollahpour, R. Aghayari and Th.M. Rassias, Hyers-Ulam stability of associated Laguerre differential equations in a subclass of analytic functions, J. Math. Anal. Appl., 437 (2016), pp. 605-612.
3. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 31 (1989).
4. D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), pp. 223-237.
5. J. Brzdek, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, Aust. J. Math. Anal. Appl., 6 (1) (2009).
6. J. Brzdek and K. Ciepliński, Remarks on the Hyers-Ulam stability of some systems of functional equations, Appl. Math. Comput., 219 (2012), pp. 4096-4105
7. J. Brzdek and K. Ciepliński, Hyperstability and superstability, Abst. Appl. Anal., 2013 (2013), Article ID 401756.
8. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, 2002.
9. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), pp. 59-64.
10. M. Dehghanian and S.M.S. Modarres, Ternary γ-homomorphisms and ternary γ-derivations on ternary semigroups, J. Inequal. Appl., 2012 (2012).
11. M. Dehghanian, S.M.S. Modarres, C. Park and D. Shin, C∗- Ternary 3-derivations on C∗-ternary algebras, J. Inequal. Appl., 2013 (2013).
12. M. Dehghanian and C. Park, C∗-Ternary 3-homomorphisms on C∗-ternary algebras, Results Math., 66 (3) (2014), pp. 385-404.
13. M. Dehghanian, C. Park and Y. Sayyari, Stability of ternary antiderivation in ternary Banach algebras via fixed point theorem, Cubo, 25 (2) (2023), pp. 273-288.
14. M. Dehghanian and Y. Sayyari, The application of Brzdek’s fixed point theorem in the stability problem of the Drygas functional equation, Turk. J. Math., 47 (6) (2023), pp. 1778-1790.
15. M. Dehghanian, Y. Sayyari and C. Park, Hadamard homomorphisms and Hadamard derivations on Banach algebras, Miskolc Math. Notes, 24 (1) (2023), pp. 129-137.
16. H. Drygas, Quasi-inner products and their applications, in: Advances in Multivariate Statistical Analysis, Reidel Publishing Co. (Dordrecht, 1987), pp. 13-30.
17. B.R. Ebanks, Pl. Kannappan and P.K. Sahoo, A common generalization of functional equations characterizing normed and quasiinner- product spaces, Canad. Math. Bull., 35 (1992), pp. 321-327.
18. I. El-Fassi, J. Brzdek, A. Chahbi and S. Kabbaj, On hyperstability of the biadditive functional equation, Acta Math. Sci., 37 (6) (2017), pp. 1727-1739.
19. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), pp. 431-434
20. D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), pp. 222-224.
21. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, 1998.
22. D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequat. Math. 44 (1992), pp. 125-153.
23. S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, 2011.
24. S.M. Jung, D. Popa and Th.M. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Glob. Optim., 59 (2014), pp. 165-171.
25. S.M. Jung and Th.M. Rassias, A linear functional equation of third order associated to the Fibonacci numbers, Abst. Appl. Anal., 2014 (2014), Article ID 137468.
26. S.M. Jung, Th.M. Rassias and C. Mortici, On a functional equation of trigonometric type, Appl. Math. Comput., 252 (2015), pp. 294-303.
27. S.M. Jung and P.K. Sahoo, Stability of a functional of Drygas, Aequationes Math., 64 (2002), pp. 263-273.
28. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, 2009.
29. P. Kaskasem, A. Janchada and C. Klin-eam, On Approximate Solutions of the Generalized Radical Cubic Functional Equation in Quasi-β-Banach Spaces, Sahand Commun. Math. Anal., 17 (1) (2020), pp. 69-90.
30. Y.H. Lee, S.M. Jung and Th.M. Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), pp. 13-16.
31. Y.H. Lee, S.M. Jung and Th.M. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Ineq., 12 (1) (2018), pp. 43-61.
32. C. Mortici, Th.M. Rassias and S.M. Jung, On the stability of a functional equation associated with the Fibonacci numbers, Abst. Appl. Anal., 2014 (2014), Article ID 546046.
33. A. Najati and Y. Khedmati Yengejeh, Functional inequalities associated with additive, quadratic and Drygas functional equations, Acta Math. Hungar., 168 (2022), pp. 572-586.
34. A. Najati, B. Noori and M.B. Moghimi, On Approximation of Some Mixed Functional Equations, Sahand Commun. Math. Anal., 18 (1) (2021), pp. 35-46.
35. A. Najati and M.A. Tareeghee, Drygas functional inequality on restricted domains, Acta Math. Hungar., 166 (2022), pp. 115-123.
36. M. Nazarianpoor and G. Sadeghi, On the stability of the Pexiderized cubic functional equation in multi-normed spaces, Sahand Commun. Math. Anal., 9 (1) (2018), pp. 45-83.
37. S. Paokanta, M. Dehghanian, C. Park and Y. Sayyari, A system of additive functional equations in complex Banach algebras, Demonstr. Math., 56 (1) (2023), Article ID 20220165.
38. W.G. Park and J.H. Bae, On a bi-quadratic functional equation and its stability, J. Nonlin. Anal., 62 (2005), pp. 643-654.
39. C. Park and Th.M. Rassias, Additive functional equations and partial multipliers in C∗-algebras, Revista de la Real Academia de Ciencias Exactas, Serie A. Matemáticas, 113 (3)(2019), pp. 2261-2275.
40. Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, 2000.
41. P.K. Sahoo and P. Kannappan, Introduction to Functional Equations, CRC Press, 2011.
42. Y. Sayyari, M. Dehghanian and Sh. Nasiri, Solution of some irregular functional equations and their stability, J. Lin. Topol. Alg., 11 (4) (2022), pp. 271-277.
43. Y. Sayyari, M. Dehghanian and C. Park, A system of biadditive functional equations in Banach algebras, Appl. Math. Sci. Eng., 31 (1) (2023), Article ID 2176851.
44. Y. Sayyari, M. Dehghanian and C. Park, Stability and solution of two functional equations in unital algebras, Korean J. Math., 31 (3) (2023), pp. 363-372.
45. Y. Sayyari, M. Dehghanian and C. Park, Some stabilities of system of differential equations using Laplace transform, J. Appl. Math. Comput., 69 (4) (2023), pp. 3113-3129.
46. Y. Sayyari, M. Dehghanian, C. Park and J. Lee, Stability of hyper homomorphisms and hyper derivations in complex Banach algebras, AIMS Math., 7 (6) (2022), pp. 10700-10710.
47. T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl., 272 (2002), pp. 604-616.
48. S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
49. J. Wang, Some further generalization of the Ulam-Hyers-Rassias stability of functional equations, J. Math. Anal. Appl., 263 (2001), pp. 406-423.
)