Solution of the Ulam stability problem for Euler Lagrange k-quintic mappings

The "oldest quartic" functional equation 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), no. 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-beta-normed spaces, J. Inequal. Appl. 2010 (2010), Article ID 368981, in the following form: 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.