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Download PDFABSTRACT.
In this paper, we determine some stability results concerning the quartic functional equation as of the form
\[
\sum_{b=1}^{s} \phi\!\left(-v_b + \sum_{a=1,\,a\ne b}^{s} v_a\right)
– 4 \sum_{1 \le a < b < c \le s} \phi(v_a + v_b + v_c)
- \sum_{b=1}^{s} \phi(2v_b)
\]
\[
= (-4s+14)\sum_{a=1,\,a\ne b}^{s} \phi(v_a + v_b)
+ (s-8)\phi\!\left(\sum_{a=1}^{s} v_a\right)
\]
\[
\quad + 2 \left[
\sum_{a=1,\,a\ne b}^{s} \phi(v_a - v_b)
+ (s^{2}-7s+7)\sum_{a=1}^{s} \phi(v_a)
\right],
\]
where any positive integers \( s \ge 3 \), in Banach algebra via direct and fixed point approaches.
