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)


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