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Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems. (English) Zbl 1208.39008
The article deals with homoclinic solutions to the second-order self-adjoint discrete Hamiltonian system
\[ \Delta[p(n) \Delta u(n - 1)] - L(n)u(n) + \nabla W(n,u(n)) =0, \]
(\(n \in {\mathbb Z}\), \(u \in {\mathbb R}^N\), \(\Delta u(n) = u(n + 1) - u(n)\), \(p, L: \;{\mathbb Z} \to {\mathbb R}^{N \times N}\), \(W:\;{\mathbb Z} \times {\mathbb R}^N \to {\mathbb R}\)). It is assumed that \(p(n)\) and \(L(n)\) are real symmetric positive definite matrices for all \(n \in {\mathbb Z}\), and there exists a function \(l: \;{\mathbb Z} \to (0,\infty)\), \(l(n) \to \infty\) as \(n \to \pm \infty\), such that \((L(n)x,x) \geq l(n)|x|^2\) for all \((n,x) \in {\mathbb Z} \times {\mathbb R}^N\), \(W(n,-x) = W(n,x)\) and \(W(n) = W_1(n,x) - W_2(n,x)\) where \(W_1\) and \(W_2\) are continuously differentiable in \(x\) satisfying one of the following systems of additional assumptions:9mm
(I)
There is a bounded set \(J \subset {\mathbb Z}\) such that \(W_2(n,x) \geq 0\), \((n,x) \in J \times {\mathbb R}^N\), \(|x| \leq 1\) and \((1 / l(n)) |\nabla W(n,x)| = o(|x|)\) as \(x \to 0\) uniformly in \(n \in {\mathbb Z} \setminus J\); moreover, there is a constant \(\mu > 2\) such that \(0 < \mu W_1(n,x) \leq (\nabla W_1(n,x),x)\) and there is a constant \(\varrho \in (2,\mu)\) such that \((\nabla W_2(n,x),x) \leq \varrho W_2(n,x)\).
(II)
\((1/l(n)) |\nabla W(n,x)| = o(|x|)\) as \(x \to 0\); moreover, there is a constant \(\mu > 2\) such that \(0 < \mu W_1(n,x) \leq (\nabla W_1(n,x),x)\) and there is a constant \(\varrho \in (2,\mu)\) such that \((\nabla W_2(n,x),x) \leq \varrho W_2(n,x0\).
(III)
\((1/l(n)) |\nabla W(n,x)| = o(|x|)\) as \(x \to 0\); moreover, for any \(r > 0\), there exist \(a = a(r)\), \(b = b(r) > 0\) and \(\nu < 2\) such that \[ 0 \leq \bigg(2 + \frac1{a + b|x|^\nu}\bigg)W(n,x) \leq (\nabla W(n,x),x) \] for \(|x| \geq r\) and, further, for any \(n \in {\mathbb Z}\) \[ \lim_{s \to \infty} \;\bigg(s^{-2} \min_{|x| = 1} \;W(n,sx)\bigg) = + \infty. \]
Under these assumptions there exists an unbounded sequence of homoclinic solutions. At the end of the article three concrete examples are given.

MSC:
39A12 Discrete version of topics in analysis
39A05 General theory of difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
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