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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.

39A14Partial difference equations
37C29Homoclinic and heteroclinic orbits
49J40Variational methods including variational inequalities
Full Text: DOI
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