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.


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