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Quartic functional equations. (English) Zbl 1072.39024

In analogy to the “quadratic functional equation”


that is, s Δ y 2 f(x)=2f(y), the authors call


(rather than s Δ y 4 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.

39B52Functional equations for functions with more general domains and/or ranges
39B42Matrix and operator functional equations
39B82Stability, separation, extension, and related topics
46B20Geometry and structure of normed linear spaces
46B25Classical Banach spaces in the general theory of normed spaces