## 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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### References:

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