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Oscillation of linear functional equations of higher order. (English) Zbl 0871.39008
Let $X$ be an unbounded subset of $[0,+\infty)$. Given $f:X\to X$ and $Q_0, Q_2, \dots, Q_N:X \to (0,+\infty)$ the authors consider the equation $$\varphi \bigl(f(x)\bigr) = Q_0(x) \varphi(x)+ Q_2(x)\varphi \bigl(f^2(x)\bigr) + \cdots+ Q_N(x)\varphi \bigl(f^N(x) \bigr) \tag *$$ and prove the following Theorem: Suppose $f(x)\ne x$ for $x\in X$ and $\lim_{x\to\infty} f(x)= +\infty$. If $$\liminf_{x\to\infty} \sum^N_{k=2} Q_k(x) \prod^{k-1}_{j=1} Q_0 \bigl(f^j (x)\bigr) > {1\over 4}$$ or $$\liminf_{x\to\infty} \sum^{N-2}_{k=0} G\bigl(f^k(x) \bigr) \prod^{N-1}_{j=1} Q_0 \bigl(f^{k+j} (x)\bigr) >\left(1- {1\over N} \right)^N$$ where $$G(x)= \sum^{N-1}_{k=2} Q_k(x) Q_{N+1-k} \bigl(f^{k-1} (x)\bigr) +Q_N(x),$$ and $\varphi: X\to\bbfR$ is a solution of (*) such that $\sup \{|\varphi (x)|:$ $x\in [x_0,+\infty) \cap X\} >0$ for any $x_0 \in [0,+\infty)$, then $\varphi$ oscillates, i.e. there exists a sequence $(x_n)$ of elements of $X$ such that $\lim_{n\to \infty} x_n= +\infty$ and $\varphi (x_n) \varphi(x_{n+1}) \le 0$ for every $n\in \bbfN$.

MSC:
 39B12 Iterative and composite functional equations 39B22 Functional equations for real functions 39B72 Systems of functional equations and inequalities
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