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Quartic functional equations. (English) Zbl 1072.39024
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.

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
Full Text: DOI
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