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In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed additive-cubic functional equation f(kx + y) + f(kx - y) = kf(x + y) + kf(x - y) + 2f(kx) - 2kf(x) in the quasi-Banach spaces. © 2012 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg.

We determine the general solution of the functional equation f(x+ky)+f(x−ky) = g(x+y)+g(x−y)+ +h(x)+h˜(y) for fixed integers k with k 6= 0, ±1 without assuming any regularity condition on the unknown functions f, g, h, h˜. The method used for solving these functional equations is elementary but exploits an important result due to Hosszu. The solution of this functional equation can also be determined in certain type ´ of groups using two important results due to Szekelyhidi Визначено загальний розв’язок функцiонального рiвняння f(x + ky) + f(x − ky) = g(x + y) + + g(x − y) + h(x) + h˜(y) для фiксованих цiлих k при k 6= 0, ±1 без припущення наявностi будь-якої умови регулярностi для невiдомих функцiй f, g, h, h˜. Метод, що використано для розв’язку цих функцiональних рiвнянь, елементарний, але базується на важливому результатi Хозу. Розв’язок цього функцiонального рiвняння може бути визначений у певному типi груп з використанням двох важливих результатiв Чекелiхiдi.

We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and h̃. The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú. The solution of this functional equation can also be obtained in groups of certain type by using two important results due to Székelyhidi. © 2011 Springer Science+Business Media, Inc.

We achieve the general solution of the quintic functional equation f (x + 3 y) - 5 f (x + 2 y) + 10 f (x + y) - 10 f (x) + 5 f (x - y) - f (x - 2 y) = 120 f (y) and the sextic functional equation f (x + 3 y) - 6 f (x + 2 y) + 15 f (x + y) - 20 f (x) + 15 f (x - y) - 6 f (x - 2 y) + f (x - 3 y) = 720 f (y). Moreover, we prove the stability of the quintic and sextic functional equations in quasi-β-normed spaces via fixed point method. Copyright 2011 Tian Zhou Xu et al.

We establish some stability results concerning the general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. In addition, we establish some results of approximately general mixed additive-cubic mappings in non-Archimedean fuzzy normed spaces. The results improve and extend some recent results. © 2010 American Institute of Physics.