Generalized Hyers-Ulam-Rassisa Stability of An Additive (Î²1,Î²2)-Functional Inequalities With n- Variables In Complex Banach Space
Keywords:Additive (Î²1, Î²2)-functional inequality, fixed point method, direct method, Banach space, Hyersâˆ’Ulam stability.
In this paper we study to solve the o f additive (Î²1,Î²2)-f unctional inequality with nâˆ’variables and their Hyers-Ulam stability. First are investigated in complex Banach spaces with a fixed point method and last are investigated in complex Banach spaces with a direct method. I will show that the solutions of the additive (Î²1,Î²2)-f unctional inequality are additive mapping. Then Hyersâˆ’Ulam stability o f these equation are given and proven. T hese are the main results o f this paper .
 T.Aoki, On the stability of the linear transformation in Banach space, J. Math. Soc. Japan 2(1950), 64-66.
 A.Bahyrycz, M. Piszczek, Hyers stability of the Jensen function equation, Acta Math. Hungar.,142 (2014),353-365.
 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 Vol.13, 2019, no. 11. 519-53014), 296-310..https://doi.org/10.12988/ijma.2019.9954.
 Ly Van An, Hyers-Ulam stability of Î²-functional inequalities with three variable in non-Archemdean Banach spaces and complex Banach spaces Inter-
national Journal of Mathematical Analysis Vol. 14, 2020, no. 5, 219 - 239 .HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2020.91169.
 M.Balcerowski, On the functional equations related to a problem of z Boros and Z. Dr Ìoczy, Acta Math. Hungar.,138 (2013), 329-340.
 L. CADARIU ? , V. RADU, Fixed points and the stability of Jensenâ€™s functional equation, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003).
 J. DIAZ, B. MARGOLIS, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Am. Math. Soc. 74
 W Fechner, On some functional inequalities related to the logarithmic mean, Acta Math., Hungar., 128 (2010,)31-45, 303-309 .
 W. Fechner, Stability of a functional inequlities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149-161.
 Pascus. Ga Ì†vruta, A generalization of the Hyers-Ulam -Rassias stability of approximately additive mappings, Journal of mathematical Analysis and
Aequations 184 (3) (1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211.
 Attila Gil Ìanyi, On a problem by K. Nikodem, Math. Inequal. Appl., 5 (2002), 707-710.
 Attila Gil Ìanyi, Eine zur parallelogrammleichung Ìˆaquivalente ungleichung, Aequations 5 (2002),707-710. https: //doi.org/ 10.7153/mia-05-71.
 Donald H. Hyers, On the stability of the functional equation, Proceedings of the National Academy of the United States of America, 27 (4) (1941),
 Jung Rye Lee, Choonkil Park, and Dong Yun Shin. Additive and quadratic functional in equalities in Non-Archimedean normed spaces, International
Journal of Mathematical Analysis, 8 (2014), 1233-1247. https://doi.org/10.12988/ijma.2019.9954.
 L.Maligranda.Tosio Aoki (1910-1989). International symposium on Banach and function spaces:14/09/2006-17/09/2006, pages 1â€“23. Yokohama
 D. MIHET Ì§, V. RADU, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572.
 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):1318â€“1331,
 C.Park, Y. Cho, M.Han. Functional inequalities associated with Jordan-von Newman-type additive functional equations , J. Inequality .Appl. ,2007(2007),
Article ID 41820, 13 pages.
 W. P and J. Schwaiger, A system of two in homogeneous linear functional equations, Acta Math. Hungar 140 (2013), 377-406 .
 Jurg Ìˆ Ra Ìˆtz On inequalities assosciated with the jordan-von neumann functional equation, Aequationes matheaticae, 66 (1) (2003), 191-200 https;//
 Choonkil. Park. The stability an of additive (Î²1,Î²2
-functional inequality in Banach space
Journal of mathematical Inequalities Volume 13, Number 1 (2019), 95-104.
 Choonkil. Park. Additive Î²-functional inequalities, Journal of Nonlinear Science and Appl. 7(2014), 296-310.
 Choonkil Park, functional in equalities in Non-Archimedean normed spaces. Acta Mathematica Sinica, English Series, 31 (3), (2015), 353-366.
 C. PARK, Additive Ï-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17-26.
 C. PARK, Additive Ï-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal. 9 (2015), 397-407.
 Themistocles M. Rassias, On the stability of the linear mapping in Banach space, proceedings of the American Mathematical Society, 27 (1978),
297-300. https: //doi.org/10.2307 /s00010-003-2684-8.
 F. SKOF, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983),113-129.
 S. M. ULam. A collection of Mathematical problems, volume 8, Interscience Publishers. New York,1960.