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.

MSC:

 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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References:

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