Stability of a System of Euler-Lagrange Type Cubic Functional Equations in non-Archimedean 2-Normed Spaces

Document Type : Original Paper


Department of mathematics, Shahid Chamran university of Ahvaz, Ahvaz, Iran.


Freese and Cho have introduced the non-Archimedean 2-normed spaces and Eshaghi, et al. have introduced the Menger probabilistic non-Archimedean 2-normed spaces. In this paper, we prove the generalized Hyers-Ulam-Rassias stability for a system of Euler-Lagrange type cubic functional equations in the non-Archimedean 2-normed spaces. Also, we prove the generalized Hyers-Ulam-Rassias stability for this system in the Menger probabilistic non–Archimedean 2–normed spaces.


Main Subjects

[1] S. Gähler, 2-metrische Räume und ihre topologische Struktur, Math. Nachr. 26 (1963) 115-148.
[2] S. Gähler, Lineare 2-normierte Räumen, Math. Nachr. 28 (1964) 1-43.
[3] S. Gähler, Über 2-Banach-Räume, Math. Nachr. 42 (1969) 335-347.
[4] A. White, 2-Banach spaces, Doctorial Diss., St. Louis Univ., 1968.
[5] A. White, 2-Banach spaces, Math. Nachr. 42 (1969) 43-60.
[6] Z. Lewandowska, Linear operators on generalized 2-normed spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 42(4), 353-368 (1999).
[7] Z. Lewandowska, Generalized 2-normed spaces, Slupskie Prace Matematyczno- Fizyczne 1, 33-40(2001).
[8] Z. Lewandowska, On 2-normed sets, Glas. Mat. Ser. III 38 (1) 99-110 (2003).
[9] Z. Lewandowska, Banach-Steinhaus theorems for bounded linear operators with values in a generalized 2-normed space, Glas. Mat. Ser. III 38 (2) 329-340 (2003).
[10] Z. Lewandowska, Bounded 2-linear operators on 2–normed sets, Glas. Mat. Ser. III 39 (2) 301-312 (2004).
[11] W. -G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011) 193-202.
[12] K. Hensel, Über eine neue Begründung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein 6 (1897) 83-88.
[13] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-adic Analysis and Mathematical Physics, World Scientific, 1994.
[14] A. Khernikov, non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht, 1997.
[15] R. W. Freese, Y. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, 2001.
[16] A. N. Srstnev, On the motion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280-283 English translation in Soviet Math. Dokl. 4 (1963) 388-390.
[17] C. Alsina, B. Schweizer and A. Sklar, On the definition of a probabilistic normed space, Aequationes Math. 46 (1993) 91-98.
[18] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, NewYork, 1983.
[19] M. Eshaghi Gordji, M. B. Ghaemi, Y. J. Cho and H. Majani, A General System of Euler–Lagrange-Type Quadratic Functional Equations in Menger Probabilistic Non-Archimedean 2-Normed Spaces, Abs. Appl. Anal., Volume 2011, Article ID 208163, 21 pages.
[20] O. Hadzić , A fixed point theorem in Menger spaces, Publ. Inst. Math. (Beograd) T 20 (1979) 107-112.
[21] O. Hadzić, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Fuzzy Set. Syst. 88 (1997) 219-226.
[22] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York,1964.
[23] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222-224.
[24] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950) 64-66.
[25] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.
[26] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431-436.
[27] L. M. Arriola and W. A. Beyer, Stability of the Cauchy functional equation over p-adic fields, Real Anal. Exchange 31 (2005/2006), 125-132.
[28] M. Eshaghi Gordji, M. B. Ghaemi and H. Majani, Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces, Discrete Dyn. Nat. Soc., Volume 2010, Article ID 162371, 11 pages.
[29] M. Eshaghi Gordji, M. B. Ghaemi, H. Majani and C. Park, Generalized Ulam–Hyers Stability of Jensen Functional Equation in Serstnev PN Spaces, J. Ineq. Appl., Volume 2010, Article ID 868193, 14 pages.
[30] M. Eshaghi and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009) 5629-5643.
[31] H. Khodaei, On the stability of additive, quadratic, cubic and quartic set-valued functional equations, Results Math. 68 (2015) 1-10.
[32] K. -W. Jun and H. -M. Kim, On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl. 332 (2007) 1335-1350.