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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.

MSC:
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
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References:
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