Volume 51 | Number 6 | Year 2017 | Article Id. IJMTT-V51P555 | DOI : https://doi.org/10.14445/22315373/IJMTT-V51P555
In this paper, the authors discussed the generalized UH stability of two types of n-dimensional quadratic functional equation of the form.
1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.
3. M. Arunkumar, S. Karthikeyan, Solution and stability of n-dimensional mixed Type additive and quadratic functional equation, Far East Journal of Applied Mathematics, Volume 54, Number 1, (2011), 47-64.
4. M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International Journal Mathematical Sciences and Engineering Applications, Vol. 7 No. I (January, 2013), 383-391.
5. M. Arunkumar, P. Agilan, Additive Quadratic functional equation are Stable in Banach space: A Direct Method, Far East Journal of Applied Mathematics, Volume 80, No. 1, (2013), 105 – 121.
6. Arunkumar, M., Solution and Stability of Arun-Additive functional equations. International Journal Mathematical Sciences and Engineering Applications, 4(3) (2010), 33-46.
7. Baak, C, Boo, D., Rassias. Th.M., Generalized additive mapping in Banach modules and isomorphism between C*-algebras, J. Math. Anal. Appl., 314 (2006), 150-161.
8. Bae, J. H., On the stability of n..Dimensional quadratic functional equations, Comm. Kor. Math. Soc, 16(1) (2001), 103-113.
9. Borelli, C, Forti, G. L., On a general Hyers-Ulam stability, Internat J. Math. Math. Sci, 18 (1995), 229-236.
10. Cholewa, P. W., Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.
11. Czerwik, S., On the stability of the quadratic mappings in normed spaces, Abh. "Math. Sem.Univ Hamburg., 62 (1992), 59-64.
12. Czerwik, S., Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, 2002.
13. Gavruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
14. Hyers, D. H., On the stability of ` the linear functional equation, Proc. Nat. Acad. Sci., U.S.A., 27 (1941), 222-224.
15. Hyers, D. H., Isac, G., Rassias, Th.M., Stability of functional equations in several variables, Birkhauser, Basel, 1998.
16. Jung, S. M., Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
17. K.W. Jun, H.M. Kim, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive type functional equation, Bull. Korean Math. Soc. 42 (1) (2005), 133-148.
18. K.W. Jun and H.M. Kim, On the stability of an n-dimensional quadratic and additive type functional equation, Math. Ineq. Appl 9 (1) (2006), 153-165.
19. S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137.
20. Y.S. Jung, The Hyers-Ulam-Rassias Stability of module left derivations, J. Math. Anal. Appl., doi: 10.1016/j.jmaa.2007.07.003, 1-9.
21. Jun, K. W. and Kim, H. M., The Generalized Hyers-Ulam-Rassias Stability of a Cubic Functional Equation, J. Math. Anal. Appl., 274 (2002), 867-878.
22. Jung, S. M., On the Hyers-Ulam-Rassias stability of the quadratic functional equations, J. Math. Anal. Appl., 232 (1998), 384-339.
23. Jung, Y. S., The Ulam-Gavruta-Rassias stability of module left derivations.
24. Katsaras, A. K., Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143-154.
25. Kramosil, I., Michalek, J., Fuzzy metric and statistical metric spaces, Kyber-netica, 11 (1975), 326-334.
26. Krishna, S. V., Sarma, K. K. M., Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems, 63 (1994), 207-217.
27. K.W. Jun and D.W. Park, Almost derivations on the Banach algebra Cn[0,1], Bull. Korean Math.Soc.vol 33, No.3 (1996), 359-366.
28. K.W. Jun, H.M. Kim, On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive type functional equation, Bull. Korean Math. Soc. 42 (1) (2005), 133-148.
29. K.W. Jun and H.M. Kim, On the stability of an n-dimensional quadratic and additive type functional equation, Math. Ineq. Appl 9 (1) (2006), 153-165.
30. S.M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137.
31. S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Plan Harbor, 2001.
32. Y.S. Jung, The Hyers-Ulam-Rassias Stability of module left derivations, J. Math. Anal. Appl., doi: 10.1016/j.jmaa.2007.07.003, 1-9.
33. Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368-372.
34. Park. C, Linear derivations on Banach algebras, Nonlinear Funct. Anal. Appl., 9(3) (2004), 359-368.
35. Park, C, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equation in Banach algebras, Fixed Point Theory and Applications, 2007, Art ID 50175.
36. Park. C. and Hou, J., Homomorphism and derivations in C*-algebras, Abstract Appl. Anal. 2007, Art. Id 80630.
37. Park, C. and Rassias, Th.M., Fixed points and stability of the Cauchy functional equation, Australian J. Math. Anal. Appl. 6(1). 14 (2009), 1-9.
38. Park, C. and Gordji, M. E., Comment on Approximate ternary Jordan derivations on Banach ternary algebras [ Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)], J. Math. Phys. 51, 044102 (2010) (7 pages).
39. Ravi, K., Arunkumar, M. and Rassias, J. M., On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, 3(8) (Autumn 2008), 36-47.
40. Rassias, J. M., On approximately of approximately linear mappings by linear mappings, Bull. Sc. Math, 108 (1984), 445-446.
41. Rassias, J. M., On the stability of the Euler- Lagrange functional equation, Chinese J. Math., 20 (1992), 185-190.
42. Rassias, J. M., On the stability of the Euler-Lagrange functional equation, C. R. Acad. Bulgare Sci., 45(6) (1992), 17-20.
43. Rassias, J. M., On the stability of the general Euler- Lagrange functional equation, Demonstratio Math., 29 (1996), 755-766.
44. Rassias, J. M., Solution of the Ulam stability problem for Euler- Lagrange quadratic mappings, J. Math. Anal. Appl., 220 (1998), 613-639.
45. Ravi, K., Arunkumar, M. and Rassias, J. M., On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation. International Journal of Mathematical Sciences, 3(8) (Autumn 2008), 36-47.
46. Ulam, S. M., Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964.
V.Govindan, S.Murthy, M.Saravanan, "Generalized Ulam-Hyers Stability of two types of n-dimensional Quadratic functional equation in Banach Space: Direct and Fixed Point Methods," International Journal of Mathematics Trends and Technology (IJMTT), vol. 51, no. 6, pp. 396-403, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V51P555