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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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