Approximation of the Quadratic and Cubic Functional Equations in RN–spaces

[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ.
Press, 1989.
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc.
Japan. 2 (1950), 64–66.
[3] E. Baktash, Y.J. Cho, M. Jalili, R. Saadati and S. M. Vaezpour, On the stability of cubic
mappings and quadratic mappings in random normed spaces, To appear in JIA.
[4] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer.
Math. Soc. 57 (1951) 223-˝U237.
[5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27
(1984) 76-˝U86.
[6] S. S. Chang, Y. J. Cho and S. M. Kang, Nonlinear operator theory in probabilistic metric
spaces, Nova Science Publishers, Inc. New York, 2001.
[7] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math.
Sem. Univ. Hamburg. 62 (1992), 59–64.
[8] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive
mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
[9] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991),
431–434.
[10] A. Grabiec, The generalized Hyers˝UUlam stability of a class of functional equations,
Publ. Math. Debrecen 48 (1996) 217˝U-235.
[11] O. Hadži´c and E. Pap, Fixed Point Theory in PM Spaces, Kluwer Academic Publishers,
Dordrecht, 2001.
[12] O. Hadži´c, E. Pap and M. Budincevi´c, Countable extension of triangular norms and their
applications to the fixed point theory in probabilistic metric spaces, Kybernetica, 38 (3)
(2002), 363–381.
[13] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA
27 (1941), 222˝U-224.
REFERENCES 506
[14] K. Jun and Y. Lee, On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality,
Math. Inequal. Appl. 4 (2001), no. 1, 93–118.
[15] K. W. Jung and H. M. Kim, The generalized Hyers-Ulam-Russias stability of a cubic
functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267–278.
[16] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math.
27 (1995), no. 3-4, 368–372.
[17] D. Mihe¸t and V. Radu, On the stability of the additive Cauchy functional equation in
random normed spaces, J. Math. Anal. Appl. 343 (2008), 567˝U-572.
[18] D. Mihe¸t, The probabilistic stability for a functional equation in a single variable, Acta
Math. Hungar. 2008 DOI: 10.1007/s10474-008-8101-y.
[19] D. Mihe¸t, The fixed point method for fuzzy stability of the Jensen functional equation,
Fuzzy Sets and Systems, doi:10.1016/j.fss.2008.06.014.
[20] M. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen
functional equation, Fuzzy Sets and Systems, 159 (2008), 730–738.
[21] A. K. Mirmostafee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem,
Fuzzy Sets and Systems, 159 (2008), 720–729.
[22] A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Inform.
Sci. 178 (2008), 3791–3798.
[23] C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal. TMA 57
(2004), 713–722.
[24] J. M. Rassias, Solution of a stability problem of Ulam, Discuss. Math. 12 (1992), 95˝U-
103.
[25] J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glas. Mat. Ser.
III 36 (56) (2001), no. 1, 63-˝U72.
[26] J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, An. Univ. Timi
c soara Ser. Mat.-Inform. 38 (2000), no. 1, 121-˝U132.
[27] J. M. Rassias, On approximation of approximately linear mappings by linear mappings.
J.Funct. Anal. 46 (1982), no. 1, 126–130.
[28] J. M. Rassias, Solution of a problem of Ulam. J. Approx. Theory 57 (1989), no. 3, 268–
REFERENCES 507
273.
[29] J. M. Rassias, On a new approximation of approximately linear mappings by linear
mappings, Discuss. Math.7 (1985), 193-196.
[30] J. M. Rassias, On approximation of approximately linear mappings by linear mappings,
Bull. Sci. Math. (2) 108 (1984), no.4, 445-446.
[31] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer.
Math. Soc. 72 (1978), 297–300.
[32] Th. M. Rassias, On the stability of the quadratic functional equation and its applications,
Studia Univ. Babes-Bolyai (1998), 89–124.
[33] Th. M. Rassias and K. Shibata, Variational problem of some quadratic functionals in
complex analysis, J. Math. Anal. Appl. 228 (1998), 234–253.
[34] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier, North Holand, New York,
1983.
[35] A. N. Sherstnev, On the notion of a random normed space, Dokl. Akad. Nauk SSSR 149
(1963), 280˝U-283 (in Russian).
[36] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53
(1983), 113–129.
[37] S. M. Ulam, Problems in modern mathematics, Wiley, New York, 1964.