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Nontrivial periodic solutions for asymptotically linear resonant difference problem. (English) Zbl 1103.39005

By combining minimax methods and Morse theory, the authors obtain results for the existence of nontrivial periodic solutions of second order difference equations of the form \[ \Delta^2 x_{n-1}+F^\prime(n,x_n)=0,\quad n\in \mathbb{Z}. \] where \(\Delta\) is the forward difference operator, \(F\in C^2\left(\mathbb{Z}\times R^N, \mathbb{R}\right)\) such that \(F(n,z)\) is \(T\)-periodic with respect to the first variable, and \(F^\prime(n,z)\) denotes the gradient of \(F\) with respect to the second variable. Furthermore, they assume that \(F^\prime\) satisfies the following asymptotically linear growth conditions. \[ \begin{aligned} &F^\prime(n,z)=A_\infty(n)z+o(| z| )\quad \text{as} \quad | z| \rightarrow \infty, \tag{H1}\\ &F^\prime(n,z)=A_0(n)z+o(| z| )\quad \text{as} \quad | z| \rightarrow 0,\tag{H2}\end{aligned} \] where \(A_\infty(n),~A_0(n)\) are \(N\times N\) symmetric and \(T\)-periodic matrices.

MSC:

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