On stability of the monomial functional equation in normed spaces over fields with valuation. (English) Zbl 1101.39018

Let \(X\) and \(Y\) be linear spaces and let \(n\) be a positive integer. A mapping \(g:X\to Y\) is called monomial of degree \(n\) if \(\Delta^n_y g(x)-n! g(y)=0\) \((x,y\in X)\), where \(\Delta_yg(x):=g(x+y) - g(x)\) and \(\Delta_{y_1, \dots, y_{n-1}, y_n}g(x):=\Delta_{y_1, \dots, y_{n-1}}(\Delta_{y_n}g(x))\) \((n \geq 2)\). The following type of stability result is shown for the above monomial functional equation.
Suppose that \((X, \| .\| _1)\) is a normed space over a field \(F\) of characteristic zero with a valuation \(| .| _F\), \((Y, \| .\| _2)\) is a Banach space over a field \(K\) of characteristic zero with a valuation \(| .| _K\), \(n\) is a positive integer and \(\alpha\) is a real number. If the function \(f\) satisfies \[ \| \Delta_y^nf(x) - n!f(y)\| _2 \leq \epsilon (\| x\| _1^\alpha+\| y\| _1^\alpha) \qquad (x, y \in X), \] and there exists a positive integer \(s\) such that \(| s| _F^\alpha \neq | s| _K^n\), then there exists a unique monomial mapping \(g: X \to Y\) for which \(\| f(x)-g(x)\| _2 \leq \epsilon \kappa \| x\| _1^\alpha \quad (x \in X)\) for some \(\kappa=\kappa(n,s,\alpha) \in {\mathbb R}\).
Some regularity properties of the monomial mappings are also discussed.


39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
12J20 General valuation theory for fields
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