Stability of Cauchy-Jensen Type Functional Equation in $(2,\alpha)$-Banach Spaces


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Authors: K. Y. N. SAYAR AND A. BERGAM

DOI: 10.46793/KgJMat2206.905S

Abstract:

In this paper, we investigate some stability and hyperstability results for the following Cauchy-Jensen functional equation

  (        )      (        )
    x-+--y-         x--−-y-
f            +  f             =  f(x )
       2               2
in (2)-Banach spaces using Brzdȩk and Ciepliński’s fixed point approach.

Keywords:

Stability, hyperstability, (2)-Banach space, Cauchy-Jensen functional equation.



References:

[1]   L. Aiemsomboon and W. Sintunavarat, On new stability results for generalized Cauchy functional equations on groups by using Brzdȩk’s fixed point theorem, J. Fixed Point Theory Appl. 18 (2016), 45–59.

[2]   L. Aiemsomboon and W. Sintunavarat, On generalized hyperstability of a general linear equation, Acta Math. Hungar. 149 (2016), 413–422.

[3]   M Almahalebi, Non-Archimedean hyperstability of a Cauchy-Jensen type functional equation, J. Class. Anal. 11(2) (2017), 159–170.

[4]   M. Almahalebi and A. Chahbi, Approximate solution of p-radical functional equation in 2-Banach spaces, Acta Math. Sci. 39(2) (2019), 551–566.

[5]   T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.

[6]   C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Math. Sin. (Engl. Ser.) 22(6) (2006), 1789–1796.

[7]   D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.

[8]   J. Brzdȩk, A note on stability of additive mappings, in: T. M. Rassias and J. Tabor, (Eds.) Stability of Mappings of Hyers-Ulam Type, Hadronic Press, Palm Harbor, 1994, 19–22.

[9]   J. Brzdȩk, Stability of additivity and fixed point methods, Fixed Point Theory Appl. 2013(9) (2013), Article ID 285.

[10]   J. Brzdȩk, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141(1-2) (2013), 58–67.

[11]   J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), 6861–6867.

[12]   J. Brzdȩk and K. Ciepliński, On a fixed point theorm in 2-Banach spaces and some of its applications, Acta Math. Sci. 38(2) (2018), 377–390.

[13]   J. Brzdȩk and K. Ciepliński, A fixed point theorm in n-Banach spaces and Ulam stability, J. Math. Anal. Appl. 470 (2019), 632–646.

[14]   J. Brzdȩk, W. Fechner, M. S. Moslehian and J. Sikorska, Recent developments of the conditional stability of the homomorphism equation, Banach J. Math. Anal. 9 (2015), 278–327.

[15]   E. El-hady, On stability of the functional equation of p-Wright affine functions in (2)-Banach spaces, J. Egyptian Math. Soc.27 (2019), Article ID 21.

[16]   R. W. Freese and Y. J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers Inc., Hauppauge, NY, 2001.

[17]   Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431–434.

[18]   M. E. Gordji, H. Khodaei and M. Kamar, Stability of Cauchy-Jensen type functional equation in generalized fuzzy normed spaces, Comput. Math. Appl. 62 (2011), 2950–2960.

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

[20]   K.-W. Jun, H.-M. Kim and J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings, J. Diference Equ. Appl. 13 (2007), 1139–1153.

[21]   C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory Appl. (2007), Article ID 50175, 15 pages.

[22]   C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C-algebras: a fixed point approach, Abs. Appl. Anal. (2009), Article ID 360432, 17 pages.

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

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

[25]   Th. M. Rassias, Problem 16, 2. Report of the 27th international symposium on functional equations, Aequationes Math. 39 (1990), 292–293.

[26]   Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106–113.

[27]   Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993.

[28]   K. Y. N. Sayar and A. Bergam, Some hyperstability results for a Cauchy-Jensen type functional equation in 2-Banach spaces, Proyecciones 39(1)(2020), 73–89.

[29]   K. Y. N. Sayar and A. Bergam, Approximate solutions of a quadratic functional equation in 2-Banach spaces using fixed point theorem, J. Fixed Point Theory Appl. 22 (2020), Article ID 3.

[30]   S. M. Ulam, Problems in Modern Mathematics, Science Editions, John-Wiley & Sons Inc., New York, 1964.