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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

Δ[p(t)Δu(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 x0 f(t,x) x=0, and (4) xf(t,x)β 0 x f(t,s)ds<0 for some constant β>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:
39A14Partial difference equations
37C29Homoclinic and heteroclinic orbits
49J40Variational methods including variational inequalities