Cubic Functional Equation Research Articles

Solution of a 3-D cubic functional equation and its stability

In this paper, we define and find the general solution of the following 3-D cubic functional equation $ \begin{eqnarray*} &&\left(2x_{1}+x_{2}+x_{3}\right) = 3f\left(x_{1}+x_{2}+x_{3}\right)+f\left(-x_{1}+x_{2}+x_{3}\right) +2f\left(x_{1}+x_{2}\right)+2f\left(x_{1}+x_{3}\right) \nonumber \\&& \hspace{3.0cm}-6f\left(x_{1}-x_{2}\right)-6f\left(x_{1}-x_{3}\right)-3f\left(x_{2}+x_{3}\right)+2f\left(2x_{1}-x_{2}\right)\nonumber \\&& \hspace{3.0cm} +2f\left(2x_{1}-x_{3}\right)-18f\left(x_{1}\right)-6f\left(x_{2}\right)-6f\left(x_{3}\right). \end{eqnarray*} $ We also prove the Hyers-Ulam stability of this functional equation in fuzzy normed spaces by using the direct method and the fixed point method.

Generalized Ulam–Hyers stability of (a, b; k > 0)-cubic functional equation in intuitionistic fuzzy normed spaces

In this paper, we proved the general solution in vector space and established the generalized Ulam–Hyers stability of $$(a,b;k>0)-$$ cubic functional equation $$\begin{aligned} \frac{a+\sqrt{k}~b}{2} f\left( ax+\sqrt{k}~by\right)&+\frac{a-\sqrt{k}~b}{2} f\left( ax-\sqrt{k}~by\right) +k(a^2-kb^2)b^2f(y)\\&=k(ab)^2f(x+y)+(a^2-kb^2)a^2f(x) \end{aligned}$$ where $$a\ne \pm 1, 0; b\ne \pm 1, 0; k>0$$ in Banach space and Intuitionistic fuzzy normed spaces using both direct and fixed point methods.

THE GENERALIZED HYERS-ULAM FUZZY STABILITY OF CUBIC FUNCTIONAL EQUATION

In this paper, we prove the generalized Hyers-Ulam stability for the generalized cubic functional equation in matrix fuzzy normed space using fixed point method.

Stability results in non-Archimedean "Equation missing" -fuzzy normed spaces for a cubic functional equation

Abstract We establish some stability results concerning the functional equation n f ( x + n y ) + f ( n x − y ) = n ( n 2 + 1 ) 2 [ f ( x + y ) + f ( x − y ) ] + ( n 4 − 1 ) f ( y ) , where n ≥ 2 is a fixed integer in the setting of non-Archimedean L -fuzzy normed spaces. MSC:39B52, 46S10, 46S40, 47S10, 47S40.

Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces

Consider the functional equation \({\Im_1(f ) = \Im_2(f )\,\,(\Im)}\) in a certain general setting. A function g is an approximate solution of \({(\Im)}\) if \({\Im_1(g)}\) and \({\Im_2(g)}\) are close in some sense. The Ulam stability problem asks whether or not there is a true solution of \({(\Im)}\) near g. In this paper, we achieve the general solution and the stability of the following functional equation $$\begin{array}{ll}f\left(\sum\limits^{n}_{i=1}x_{i} \right)+f\left(\sum\limits^{n-1}_{i=1} x_{i}-x_{n} \right)\\\quad=2f\left(\sum\limits^{n-1}_{i=1}x_{i} \right)+\sum\limits^{n-1}_{i=1}(f(x_{i}+x_{n}) +f(x_{i}-x_{n})-2f(x_{i}))\end{array}$$ for all xi (i = 1,2, . . . , n), in non-Archimedean spaces.