Generalized Hyers-Ulam Type Stability of the $2k$-Variables Quadratic $\beta$-Functional Inequalities And Function in $\gamma$-Homogeneous Normed Space

# Ly Van An1

1Faculty of Mathematics Teacher Education, Tay Ninh University, Ninh Trung, Ninh Son, Tay Ninh Province, Vietnam.

Abstract: In this paper, we study to solve two quadratic $\beta$-functional inequalities with $2k$-variables in $\gamma$-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of quadratic $\beta$-functional equations associated two the quadratic $\beta$-functional inequalities in $\gamma$-homogeneous complex Banach spaces. We will show that the solutions of the first and second inequalities are quadratic mappings.
Keywords: Hyers-Ulam stability $\gamma$-homogeneous space; quadratic $\beta$-functional equation; $\beta$-functional inequality.

Cite this article as: Ly Van An, Generalized Hyers-Ulam Type Stability of the $2k$-Variables Quadratic $\beta$-Functional Inequalities And Function in $\gamma$-Homogeneous Normed Space, Int. J. Math. And Appl., vol. 9, no. 3, 2021, pp. 81-93.

References
1. T. Aoki, On the stability of the linear transformation in Banach space, J. Math. Soc. Japan, 2(1995), 64-66.
2. Ly Van An, Hyers-Ulam stability of functional inequalities with three variable in Banach spaces and Non-Archemdean Banach spaces, International Journal of Mathematical Analysis, 13(11)(2019), 296-310.
3. Ly Van An, Hyers-Ulam stability additive $\beta$-functional inequalities with three variable in Banach spaces and Non-Archemdean Banach spaces, International Journal of Mathematical Analysis, 14(5-8)(2020), 296-310.
4. Ly Van An, Hyers-Ulam stability quadratic $\beta$-functional inequalities with three variable in $\gamma$-homogeneous normed spaces, International Journal of Mathematic Research, 12(1)(2020), 47-64.
5. J. Bae and W. Park, Approximate bi-homomorphisms and bi-derivations in $C^*$-ternary algebras, Bull. Korean Math. Soc, 47(2010), 195-209.
6. A. Bahyrycz and M. Piszczek, Hyers stability of the Jensen function equation, Acta Math. Hungar., 142(2014), 353-365.
7. M. Balcerowski, On the functional equations related to a problem of z Boros and Z. Dr$\acute{o}czy$, Acta Math. Hungar., 138(2013), 329-340.
8. P. W. Cholewa, Remarks on the stability of functional eqution, Aequationes Math., 27(1984), 76-86.
9. Chookil, Sang Og Kim, Jung Rye Lee and Dong Yun Shin, Quadraic $\rho$functional inequalities in $\beta$-homogeueous normed Space, Int. J. Nonlinear Anal. Appl., 6(2)(2015), 21-26.
10. Z. Dar$\acute{o}$czy and Gy. Mackasa, Afunction equation involving comparable weighted quasi-arthmetric means, Acta Math Hungar., 138(2013), 329-340.
11. I. -i. EL-Fassi, Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-$\beta$-Banach spaces, Rev. R. Acad. Cienc. Exactas F\'{i}s. Nat. Ser. A Mat., 2018(2018), 1-13.
12. W. Fechner, Stability of a functional inequlities associated with the Jordan-von Newmann functional equation, Aequtiones Math., 71(2006), 149-161.
13. A. Gil$\acute{a}nyi$, Eine zur Parallelogrammaleichung $\ddot{a}$ Ungleichung, Aeq. Math., 62(2001), 303-309.
14. A. Gil$\acute{a}nyi$, On u problemby $K$, Nikodem Math. Inequal. Appl., 5(2002), 707-710.
15. P. G$\check{a}$vruta, A generalization of the Hyers-Ulam -Rassias stability, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo, 18(10)(1942), 588-594.
16. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA., 27(1941), 222-224.
17. S. Jung, On the quadrtic functional equation modul modulo a subgroup, Indian J. Pure Appl. Math., 36(2005), 441-450.
18. P. Kaskasem, C. Klin-eam and Y. J. Cho, On the stability of the generalized Cauchy-Jensen set-valued functional equations, J. Fixed Point Theory Appl., 20(2018), 1-14.
19. C. Park, Quadrtic $\beta$-functional inequalites and equation, J. Nonlinear Anal. Appl., 2014(2014).
20. C. Park, Functional equation in Banach modules, Indian J. Pure Anal. Appl Math., 33(2002), 1077-1086.
21. C. Park, Multilinear mappings in Banach modules over a $C^{*}$-algebra, Indian J. Pure Appl Math., 35(2004), 183-192.
22. C. Park, Y. Cho and M. Han, Functional inequalities associiated with Jordan -von Neuman- type additive functional equational, J. Inequal. Appl., 2007(2007).
23. A. Najati and G. Z. Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl., 342(2)(2008), 1318-1331.
24. A. Najati and M. B. Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl., 337(1)(2008), 399-415.
25. Th. M. Rassias, On the stability of the linear mappings in Banach space, Proc. Amer. Math. So., 72(1978), 297-300.
26. J. R$\ddot{a}tz$, On inqualities associated with the Jordan Neumann functional equation, Aequationes Math., 66(2003), 191-200.
27. S. Rolewicz, Metric linear space, PWN-Polish Scientific Publishersm, Warsaw, (1972).
28. S. M. Ulam, Collection of the mathematical Problems, Interscience Publ New York, (1960).
29. S. M. Ulam, Problems in modern mathematics, Wiley, (1964).
30. T. Z. Xu, J. M. Rassias and W. X. Xu, Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces, Acta Math. Sin. (Engl. Ser.), 28(3)(2012), 529-560.

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