System of AQC functional equations in non-Archimedean normed spaces
محورهای موضوعی : Functional analysis
H. Majani
1
(Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran)
کلید واژه: Additive&lrm, , &lrm, quadratic&lrm, , &lrm, cubic&lrm, , &lrm, functional equation&lrm, , &lrm, non--Archimedean spaces&lrm, , &lrm, generalized Hyers--Ulam--Rassias stability,
چکیده مقاله :
‎In 1897‎, ‎Hensel introduced a normed space which does‎‎not have the Archimedean property‎. ‎During the last three decades‎‎theory of non--Archimedean spaces has gained the interest of‎‎physicists for their research in particular in problems coming‎‎from quantum physics‎, ‎p--adic strings and superstrings‎. ‎In this paper‎, ‎we prove‎‎the generalized Hyers--Ulam--Rassias stability for a‎‎system of additive‎, ‎quadratic and cubic functional equations in‎‎non--Archimedean normed spaces‎.
[1] J. Aczel, J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989.
[2] L. M. Arriola, W. A. Beyer, Stability of the Cauchy functional equation over p-adic fields, Real Anal. Exchange. 31 (2005-2006), 125-132.
[3] K. Cieplinski, Stability of multi-additive mappings in non-Archimedean normed spaces, J. Math. Anal. Appl. 373 (2011), 376-383.
[4] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg. 62 (1992), 59-64.
[5] M. Eshaghi Gordji, Nearly ring homomorphisms and nearly ring derivations on non–Archimedean Banach algebras, Abs. Appl. Anal. 2010, 2010:393247.
[6] M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of a mixed type cubicquartic functional equation in non-Archimedean spaces, Appl. Math. Lett. 23 (10) (2010), 1198-1202.
[7] M. Eshaghi Gordji, H. Khodaei, On the generalized Hyers–Ulam–Rassias stability of quadratic functional equations, Abs. App. Anal. 2009, 2009:923476.
[8] M. Eshaghi Gordji, H. Khodaei, Stability of Functional Equations, LAP LAMBERT Academic Publishing, 2010.
[9] P. Gvruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.
[10] K. Hensel, Uber eine neue Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein. 6 (1897), 83-88.
[11] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941), 222-224.
[12] K. W. Jun, H. M. Kim, The generalized Hyers–Ulam–Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867-878.
[13] K. W. Jun, H. M. Kim, I. S. Chang, On the Hyers–Ulam stability of an Euler–Lagrange type cubic functional equation, J. Comput. Anal. Appl. 7 (2005), 21-33.
[14] S. M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg. 70 (2000), 175-190.
[15] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368-372.
[16] H. Khodaei, Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22-41.
[17] A. Khrennikov, non–Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht, 1997.
[18] L. Narici, E. Beckenstein, Strange terrain-non–Archimedean spaces, Amer. Math. Monthly 88 (1981), 667-676.
[19] C. Park, Multi-quadratic mappings in Banach spaces, Proc. Amer. Math. Soc. 131 (2002), 2501-2504.
[20] C. Park, D. H. Boo, Th. M. Rassias, Approximately addtive mappings over p-adic fields, J. Chungcheong. Math. Soc. 21 (2008), 1-14.
[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.
[22] F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. (1983), 113-129.
[23] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.
[24] V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-adic Analysis and Mathematical Physics, World Scienti c, 1994.
