Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation

We find the general solution of the functional equation $$\begin{array}{l} D_f {\rm{(}}x,y{\rm{)}}\,\,{\rm{: = }}f{\rm{(}}x + {\rm{2}}y{\rm{)}} + f{\rm{(}}x - {\rm{2}}y{\rm{)}} - {\rm{4[}}f{\rm{(}}x + y{\rm{)}} - f{\rm{(}}x - y{\rm{)]}} - f{\rm{(4}}y{\rm{)}} + {\rm{4}}f{\rm{(3}}y{\rm{)}} \\ - {\rm{6}}f{\rm{(2}}y{\rm{)}} + {\rm{4}}f{\rm{(}}y{\rm{)}} + {\rm{6}}f{\rm{(}}x{\rm{) }} = {\rm{ 0}}{\rm{.}} \\ \end{array}$$ in the context of linear spaces. We prove that if a mapping f from a linear space X into a Banach space Y satisfies f(0)=0 and $$\|D_f(x,y)\|\leq\epsilon \quad (x,y\in X),$$ where e > 0, then there exist a unique additive mapping \(A:X\to Y,\) a unique quadratic mapping \(Q_1:X\to Y,\) a unique cubic mapping \(C:X\to Y\) and a unique quartic mapping \(Q_2:X\to Y\) such that $$\|f(x)-A(x)-Q_1(x)-C(x)-Q_2(x)\|\leq\frac{1087 \epsilon}{140}\quad \forall x\in X.$$