Result: Quartic functional equations

In analogy to the ``quadratic functional equation'' \[ f(x+y)+f(x-y)=2f(x)+2f(y), \] that is, \(_s\Delta^2_y f(x)=2f(y),\) the authors call \[ f(2x+y)-4f(x+y)+6f(y)-4f(x-y)+f(2x-y)=4! f(x) \] (rather than \(_s\Delta^4_y f(x):= f(x+2y)-4f(x+y)+6f(x)-4f(x-y)+f(x-2y)=4! f(y)\)) ``quartic functional equation''. They offer its general solution from the real vector space into a real vector space (using solutions of the quadratic equation and four pages of calculations including up to 18-line equations) and a stability theorem for functions from a real normed linear space into a real Banach space.