Abstract 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.

中文翻译：

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欧拉-拉格朗日 k-五次映射的 Ulam 稳定性问题的解

摘要 “最古老的四次”函数方程 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(xy)]-6f(x)+24f(y) ) 由本文的第二作者介绍并解决（参见 JM Rassias，四次映射的 Ulam 稳定性问题的解决方案，Glas. Mat. Ser. III 34(54) 1999, 2, 243–252）。类似地，IG Cho、D. Kang 和 H. Koh 介绍并研究了一个有趣的“五次”函数方程，准β 范数空间中五次映射的稳定性问题，J. Inequal。应用程序 2010 2010，文章 ID 368981，格式如下： 2 ⁢ f ⁢ ( 2 ⁢ x + y ) + 2 ⁢ f ⁢ ( 2 ⁢ x - y ) + f ⁢ ( x + 2 ⁢ y ) ( x f ) - 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(xy)]+90f(x)。