Asia Mathematika

An International Journal. ISSN: 2457-0834 (online)

Volume 2, Issue 2, August 2018, Pages: 61-66

Generalized Functional inequalities in non-Archimedean Banach spaces

Y. Aribou* and S. Kabbaj

Department of Mathematics, University of Ibn Tofail, B.P. 133, Kenitra, Morocco.

Abstract

In this work, we prove the generalized Hyers-Ulam stability of the following functional inequality:

||f(x1)+f(x2)+f(x3)+¼+f(xn) || £ ||kf(
x1+x2+¼ xn

k
) ||,
  n   >   k  
in non-Archimedean Banach spaces.

Keywords

Non-Archimedean Banach space Generalized Hyers{Ulam stability Jordan{von Neumann functional equationFunctional inequality.

Preview

Reference

1 N. Anitha, M. Latha A study on T-Fuzzysoft subhemirings of a Hemiring, Asia mathematika, 1, 1-6.

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

3 C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Mathematica Sinica, English Series, Vol.22(2006), 1789-1796.

4 L. Cˇadariu, V. Radu, Fixed points and the stability of Jensens functional equation, Journal of Inequalities in Pure and Applied Mathematics, 4(2003).

5 Y.J.Cho, C.Park, R.Saadati Functional inequalities in non-Archimedean Banach spaces, Applied Mathematics Letters 23(2010), 1238-1242.

6 P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore, London, 2002.

7 P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436.

8 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27 (1941), 222-224.

9 D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, 1998.

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

11 S. M. Jung, M. S. Moslehian, P. K. Sahoo, Stability of a generalized Jensen equation on restricted domains, J. Math. Ineq., 4 (2010), 191-206.

12 Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), 499-507.

13 Y. H. Lee, K. W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen-s equation, J. Math. Anal. Appl., 238 (1999), 305-315.

14 A.K. Mirmostafaee, Stability of quartic mappings in non-Archimedean normed spaces, Kyungpook Math. J., 49 (2009) 289-297.

15 M.S. Moslehian, Th.M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007) 325-334.

16 M.S. Moslehian, Gh. Sadeghi, Stability of two types of cubic functional equations in non-Archimedean spaces, Real Anal. Exchange 33 (2) (2007-2008)

17 M.S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.; 69 (2008) 3405-3408.

18 A. Najati, F. Moradlou, Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in nonArchimedean spaces, Tamsui Oxf. J. Math. Sci. 24 (2008) 367-380.

19 C. Park, Y. Cho, M. Han, Stability of functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007 (2007) Art. ID 41820.

20 C. Park, J. Cui, Generalized stability of C∗ ternary quadratic mappings, Abstr. Appl. Anal. 2007 (2007) Art. ID 23282.

21 C. Park, A. Najati, Homomorphisms and derivations in C∗ algebras, Abstr. Appl. Anal. 2007 (2007) Art. ID 80630.

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

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

24 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.

25 J. M. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281 (2003), 516-524.

26 J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982) 126-130.

27 J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984) 445-446.

28 J.M. Rassias, Refined Hyers-Ulam approximation of approximately Jensen type mappings, Bull. Sci. Math. 131 (2007) 89-98.

29 J.M. Rassias, M.J. Rassias, Asymptotic behavior of alternative Jensen and Jensen type functional equations, Bull. Sci. Math. 129 (2005) 545-558.

30 Th.M. Rassias, Problem 16; 2, in: Report of the 27th International Symp. on Functional Equations, in: Aequat. Math., vol. 39, 1990, pp. 292-293. 309.

31 Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babe 3 -Bolyai XLIII (1998) 89-124.

32 Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000) 352-378.

33 Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000) 264-284.

34 Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 23-130.

35 Th.M. Rassias, P.S’emrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. ˇ Math. Soc. 114 (1992) 989-993.

36 Th.M. Rassias, P.S’emrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl., 173 (1993), 325-338.

37 S. M. Ulam, Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, 1960.

Visitors count


Join us