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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

${\Delta }\left[p\left(t\right){\Delta }u\left(t-1\right)\right]+q\left(t\right)u\left(t\right)=f\left(t,u\left(t\right)\right),$

where $p,q$ and $f$ are $T$-periodic functions (in time). Moreover, the precise assumptions read as follows: (1) $p\left(t\right)>0$, (2) $q\left(t\right)<0$, (3) ${lim}_{x\to 0}\frac{f\left(t,x\right)}{x}=0$, and (4) $xf\left(t,x\right)\le \beta {\int }_{0}^{x}f\left(t,s\right)\phantom{\rule{0.166667em}{0ex}}ds<0$ 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 49J40 Variational methods including variational inequalities