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The "oldest quartic"functional equation f ? (x + 2 ? y) + f ? (x-2 ? y) = 4 ? [ f ? (x + y) + f ? (x-y)[-6 ? f ? (x) + 24 ? f ? (y) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243-252). Similarly, an interesting "quintic"functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2 ? f ? (2 ? x + y) + 2 ? f ? (2 ? x-y) + f ? (x + 2 ? y) + f ? (x-2 ? y) = 20 ? [ f ? (x + y) + f ? (x-y)[ + 90 ? f ? (x). 2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this "Cho-Kang-Koh equation"by introducing pertinent Euler-Lagrange k-quintic functional equations, and investigate the "Ulam stability"of these new k-quintic functional mappings. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.

The “oldest cubic” functional equation was introduced and solved by the second author of this paper (see: Glas. Mat. Ser. III 36(56) (2001), no. 1, 63-72). which is of the form: f(x + 2y) = 3f(x + y) + f(x - y) - 3f(x) + 6f(y) For further research in various normed spaces, we are introducing new cubic functional equations, and establish fundamental formulas for the general solution of such functional equations and for the “Ulam stability” of pertinent cubic functional inequalities. © 2016, University of Nis. All rights reserved.

The purpose of this paper is to introduce and study the concepts of double ??-statistically convergent and double ??-statistically Cauchy sequences in probabilistic normed space. © 2012 Versita Warsaw and Springer-Verlag Wien.