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Zbl 1120.39007
Ma, Manjun; Guo, Zhiming
Homoclinic orbits and subharmonics for nonlinear second order difference equations.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 67, No. 6, A, 1737-1745 (2007). ISSN 0362-546X

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 $\beta>2$. \par 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.
[Christian Pötzsche (München)]
MSC 2000:
*39A14
37C29 Homoclinic and heteroclinic orbits
49J40 Variational methods including variational inequalities

Keywords: discrete variational method; diagonal method; homoclinic orbit; subharmonics; nonlinear second order self-adjoint difference equations; Ekeland's principle; Mountain Pass lemma

Cited in: Zbl 1171.39005

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