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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:
39B52 Functional equations for functions with more general domains and/or ranges
39B42 Matrix and operator functional equations
39B82 Stability, separation, extension, and related topics for functional equations
46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
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