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Sharp conditions for oscillation and nonoscillation of functional equations. (English) Zbl 1020.39004

The authors derive oscillation and nonoscillation criteria for the functional equation \[ x(g(t))=P(t)x(t)+\sum_{i=1}^m Q_i(t)x(g^{k_i+1}(t)) \tag{*} \] where \(k_i\) are positive integers, \(P,Q_i,g\) are nonnegative continuous functions and \(g^i\) denotes the \(i\)th iteration of \(g\), i.e. \(g^0(t)=t\), \(g^{i+1}(t)=g(g^i(t))\). A typical result is the following statement.
Theorem: Assume that for some \(t_0>0\) \[ \inf_{t\geq t_0,0<\lambda<1}\left\{ \frac{1}{\lambda} \sum_{i=1}^m \frac{1}{(1-\lambda)^{k_i}}Q_i(t)\prod_{j=1}^{k_i} P(g^j(t))\right\}>1. \] Then every solution of (*) oscillates.
Nonoscillatory counterparts of the previous statement are given as well.

MSC:

39A11 Stability of difference equations (MSC2000)
39A10 Additive difference equations
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