## 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 $$f(x)\neq x$$ 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 $$\varphi: X\to\mathbb{R}$$ 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 $$\varphi (x_n) \varphi(x_{n+1}) \leq 0$$ for every $$n\in \mathbb{N}$$.
Reviewer: K.Baron (Katowice)

### MSC:

 39B12 Iteration theory, iterative and composite equations 39B22 Functional equations for real functions 39B72 Systems of functional equations and inequalities
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