## Bounded solutions of Schilling’s problem.(English)Zbl 0860.39031

The author considers Schilling’s functional equation $f(qx) = {1\over 4q} \bigl(f(x-1) + f(x+1) + 2f(x) \bigr),$ for functions $$f:R \to R$$ such that $$f(x)=0$$ for $$|x |> q/(1-q)$$, where $$q$$ is a fixed number in (0,1). Fixing an integer $$n$$ and $$q$$ in $$(0,q_n]$$ where $$q_n$$ is the unique solution in $$(1/3,1/2)$$ of $$x^{n+1} - 3x+1 = 0$$, it is proved that there is a set $$A_q^n$$ in the real line such that any solution $$f$$ of Schilling’s functional equation which is bounded in a neighbourhood of a point of $$A^n_q$$ vanishes everywhere. Sets $$A^n_q$$ and their union are studied in detail.

### MSC:

 39B22 Functional equations for real functions

### Keywords:

bounded solutions; Schilling’s functional equation
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