Volume 2, Issue 3, December 2018, Pages: 24-31
Youssef Aribou* and Samir Kabbaj
Department of Mathematics, University of Ibn Tofail, B. P. 133, Kenitra, Morocco
Let R be the set of real numbers, Y is a non-Archimedean Banach space. In this work, we prove the generalized Hyers-Ulam stability of functional inequality.
Non-Archimedean Banach space Generalized Hyers{Ulam stability Jordan{von Neumann functional equation Functional inequality.
1. N. Anitha, M. Latha A study on T-Fuzzy soft subhemirings of a Hemiring, Asia mathematika, 1, 1-6.
2. L. Aiemsomboon, W. Sintunavarat, On a new type of stability of a radical quadratic functional equation using Brzd¸ek’s fixed point theorem. Acta Math. Hungar. 151(2017) 35{46.
3. Z. Alizade, A.G. Ghazanfari, On the stability of a radical cubic functional equation in quasi-β -spaces. J. Fixed Point Theory Appl. 18 (2016) 843{853.
4. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950) 64-66.
5. J. Brzd¸ek, Remarks on solutions to the functional equations of the radical type. Adv. Theory Nonlinear Anal. Appl.1(2017) 125{135.
6. J. Brzd¸ek, Remark 3. In: Report of Meeting of 16th International Conference on Functional Equations and Inequalities (B¸edlewo, Poland, May (2015) 17{23. p. 196, Ann. Univ. Paedagog. Crac. Stud. Math.14(2015) , 163{202 .
7. J. Brzd¸ek, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, The Australian Journal of Mathematical Analysis and Applications, 6 (2009), 1-10.
8. Y.J.Cho, C.Park, R.Saadati Functional inequalities in non-Archimedean Banach spaces, Applied Mathematics Letters 23 (2010) 1238-1242.
9. J. Chung , Stability of a conditional Cauchy equation on a set of measure zero, Aequationes mathematicae, 87(2014), 391-400.
10. J. Chung and J. M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, Journal of Mathematical Analysis and Applications, 419 (2014), 1065-1075.
11. J. Chung and J. M. Rassias, On a measure zero Stability problem of a cyclic equation, Bulletin of the Australian Mathematical Society, 93 (2016), 1-11.
12. K. Ciepli`nski, Aplications of fixed point theorems to the Hyers{Ulam stability of functional equations-a survey. Ann. Funct. Anal.3 (2012),no. 1, 151-164.
13. P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore, London, 2002.
14. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, 62 (1992), 59-64.
15. P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994) 431-436.
16. Gordji, M.E. Khodaei, H., Ebadian, A., Kim, G. Nearly radical quadratic functional equations in p-2-normed spaces. Abstr. Appl. Anal. 2012, Art. ID 896032 (2012)
17. Gordji, M.E., Parviz, M. On the Hyers{Ulam stability of the functional equation f(px2 + x2 = f(x) + f(y). Nonlinear Funct. Anal. Appl.14 (2009) 413{420
18. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27 (1941) 222-224. 30Y. Aribou and S. Kabbaj
19. D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkh¨auser, Basel, 1998.
20. D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992) 125-153.
21. D.H. Hyers, Transformations with bounded n-th differences, Pacific J. Math. 11 (1961) 591-602.
22. K. W. Jun, H. M. Kim, J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings, J. Diference Equ. Appl., 13(12) (2007) 1139-1153.
23. S. M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, Journal of Mathematical Analysis and Applications, 222 (1998), 126-137.
24. A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models., Kluwer Academic Publishers, Dordrecht, (1997).
25. H. Khodaei, M.E. Gordji, S. Kim, Y. Cho, Y. Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 395 (2012), 284{297
26. S. Kim, Y. Cho, M.E. Gordji, M.E. On the generalized Hyers{Ulam{Rassias stability problem of radical functional equations. J. Inequal. Appl. 186 (2012)
27. Y. Li, L. Hua, Hyers-Ulam stability of polynomial equation. Banach J. Math. Anal. 3(2009), 86-90.
28. A.K. Mirmostafaee, Stability of quartic mappings in non-Archimedean normed spaces, Kyungpook Math. J. 49 (2009) 289-297.
29. M.S. Moslehian, Th.M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007) 325-334.
30. M.S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal. 69 (2008) 3405-3408.
31. A. Najati, S. M. Jung, Approximately quadratic mappings on restricted domains, Journal of Inequalities and Applications, 2010 (2010).
32. 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.
33. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
34. Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106-113.
35. 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.
36. Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 23-130.
37. J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, Journal of Mathematical Analysis and Applications, 276 (2002), 747-762.
38. J. M. Rassias, Assymptotic behavior of mixed type functional equations, Austral. Journal of Mathematical Analysis and Applications, 1 (2004), 1-21.
39. Th.M. Rassias, P.Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993) 325-338.
40. F. Skof, Approssimazione di funzioni δ -quadratic su dominio restretto, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118 (1984) 58-70.
41. S. M. Ulam, A collection of mathematical Problems , Interscience publishers, New York, 1960. (Reprinted as: Problems in Modern Mathematics 1964. New York: Wiley)