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Homoclinic orbits and subharmonics for nonlinear second order difference equations. (English) Zbl 1120.39007
The authors derive the existence of a nontrivial homoclinic orbit as limit of subharmonic solutions for scalar nonlinear second order self-adjoint difference equations of the form $\Delta[p(t)\Delta u(t-1)]+q(t)u(t)=f(t,u(t)),$ where $$p,q$$ and $$f$$ are $$T$$-periodic functions (in time). Moreover, the precise assumptions read as follows: (1) $$p(t)>0$$, (2) $$q(t)<0$$, (3) $$\lim_{x\to 0}\frac{f(t,x)}{x}=0$$, and (4) $$xf(t,x)\leq\beta\int_0^xf(t,s)\,ds<0$$ for some constant $$\beta>2$$.
The basic proof technique is to embed the above problem into a variational framework based on Ekeland’s principle and the application of an appropriate Mountain Pass lemma.

##### MSC:
 39A14 Partial difference equations 37C29 Homoclinic and heteroclinic orbits for dynamical systems 49J40 Variational inequalities
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