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Existence of periodic solutions for a \(2n\)th-order nonlinear difference equation. (English) Zbl 1153.39302

Summary: The authors consider the \(2n\)th-order difference equation \[ \Delta^n(r_{t-n}\Delta^n x_{t-n})+f(t,x_t)=0,\quad n \in \mathbb{Z}(3), t\in \mathbb{Z} \] where \(f: \mathbb{Z} \times \mathbb{R} \to \mathbb{R}\) is a continuous function in the second variable, \(f(t+T,z)=f(t,z)\) for all \((t,z)\in \mathbb{Z}\times \mathbb{R}, r_{t+T}=r_t\) for all \(t \in \mathbb{Z}\), and \(T\) a given positive integer. By the Linking Theorem, some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation.

MSC:

39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
39B22 Functional equations for real functions
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