## What is/are Cubic Functional?

Cubic Functional - In this article, we introduce and discuss the general solution of a new n-dimensional cubic functional equation.^{[1]}In this paper, we study the generalized Hyers–Ulam stability of Euler–Lagrange-type cubic functional equation of the form $$\begin{aligned} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{aligned}$$ 2 m f ( x + m y ) + 2 f ( m x - y ) = ( m 3 + m ) [ f ( x + y ) + f ( x - y ) ] + 2 ( m 4 - 1 ) f ( y ) for all $$x,y \in X$$ x , y ∈ X , where m is a fixed scalar such that $$m \ne 0,1$$ m ≠ 0 , 1 , and f is a map from a quasi-normed space X to a quasi-Banach space Y over the same field with X by applying the alternative fixed point theorem.

^{[2]}In the second stage, although both an inverted U-shaped curve and an N-shaped curve were obtained, the cubic functional form model is better fitted.

^{[3]}The long-run estimations of the “Dynamic Seemingly Unrelated-co-integration Regression” (DSUR) signify that the effect of economic growth on ecological footprint is not stable and validate N-shaped relationship for cubic functional form between per capita income and ecological footprint (environmental quality).

^{[4]}We start with the cubic functional form to rule out any misleading results that can be caused by misspecification.

^{[5]}Nevertheless, an N-shape relationship observe in cubic functional, thus there is no guarantee that long-term levels of pollution emissions will continue to fall as countries shifting to services-based economy.

^{[6]}

## cubic functional form

In the second stage, although both an inverted U-shaped curve and an N-shaped curve were obtained, the cubic functional form model is better fitted.^{[1]}The long-run estimations of the “Dynamic Seemingly Unrelated-co-integration Regression” (DSUR) signify that the effect of economic growth on ecological footprint is not stable and validate N-shaped relationship for cubic functional form between per capita income and ecological footprint (environmental quality).

^{[2]}We start with the cubic functional form to rule out any misleading results that can be caused by misspecification.

^{[3]}

## cubic functional equation

In this article, we introduce and discuss the general solution of a new n-dimensional cubic functional equation.^{[1]}In this paper, we study the generalized Hyers–Ulam stability of Euler–Lagrange-type cubic functional equation of the form $$\begin{aligned} 2mf(x + my) + 2f(mx - y) = (m^3 + m)[f(x+ y) + f(x - y)] + 2(m^4 - 1)f(y) \end{aligned}$$ 2 m f ( x + m y ) + 2 f ( m x - y ) = ( m 3 + m ) [ f ( x + y ) + f ( x - y ) ] + 2 ( m 4 - 1 ) f ( y ) for all $$x,y \in X$$ x , y ∈ X , where m is a fixed scalar such that $$m \ne 0,1$$ m ≠ 0 , 1 , and f is a map from a quasi-normed space X to a quasi-Banach space Y over the same field with X by applying the alternative fixed point theorem.

^{[2]}