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Linear almost periodic difference equations. (English) Zbl 1156.39002
The authors introduce the Fourier spectrum of almost periodic sequences on the nonnegative integers and they study its properties. Using this spectrum, they investigate the existence, uniqueness or nonexistence of almost periodic solutions of the scalar linear difference equation $$x_{n+1}=\alpha x_n+f(n)$$, $$n\in\mathbb{N}$$, where $$f\in{\mathcal A}{\mathcal P}(\mathbb{N},X)$$ (that is $$f$$ is a discrete almost periodic function with values in the Banach space $$X$$), $$f\not\equiv 0$$ and $$\alpha\in\mathbb{C}$$.
The obtained results are then generalized to the vector equation $$X_{n+1}=A X_n+F(n)$$, $$n\in\mathbb{N}$$, where $$F(n)=(f_1(n),f_2(n),\ldots,f_m(n))^t\not\equiv 0$$, $$f_j\in{\mathcal A}{\mathcal P}(\mathbb{N},X)$$ for all $$j=\overline{1,m}$$, and $$A$$ is an $$m\times m$$ constant matrix with eigenvalues $$\alpha_1,\ldots,\alpha_m$$.
In the last part of the paper, by using the spectrum and the geometric mean of almost periodic sequences, the authors study the scalar equation $$x_{n+1}=f(n)x_n$$, $$n\in\mathbb{N}$$, where $$f\in {\mathcal A}{\mathcal P}(\mathbb{N},X)$$ is nonconstant, and the equation $$x_{n+1}=f(n)x_n+g(n)$$, $$n\in\mathbb{N}$$, with $$f\in{\mathcal A}{\mathcal P}(\mathbb{N},\mathbb{R})$$, $$f\not\equiv {\text{const.}}$$, and $$g\in{\mathcal A}{\mathcal P}(\mathbb{N},\mathbb{R})$$, $$g\not\equiv 0$$.

MSC:
 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series