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.


39A14 Partial difference equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
49J40 Variational inequalities
Full Text: DOI


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