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Existence of nontrivial homoclinic orbits for fourth-order difference equations. (English) Zbl 1171.39005
The authors study the existence of nontrivial homoclinic orbits emanating from \(0\) of the fourth-order difference equation
\[ \Delta ^{4}u\left( t-2\right) -q\left( t\right) u\left( t\right) +f\left( t,u\left( t+1\right) ,u\left( t\right) ,u\left( t-1\right) \right) =0,\;\;t\in \mathbb{Z}, \] where \(\Delta \) is the forward difference operator \(\Delta u\left( t\right) =u\left( t+1\right) -u\left( t\right) \). The main result is based on Mountain Pass Lemma of D. Smets and M. Willem [J. Funct. Anal. 149, No. 1, 266–275 (1997; Zbl 0889.34059)], a weak convergence argument and a discrete version of E. H. Lieb’s lemma [Invent. Math. 74, 441–448 (1983; Zbl 0538.35058)]. The used method is different from those of M. J. Ma and Z. M. Guo [Nonlinear Anal., Theory Methods Appl. 67, No. 6 (A), 1737–1745 (2007; Zbl 1120.39007), J. Math. Anal. Appl. 323, No. 1, 513–521 (2006; Zbl 1107.39022)].

MSC:
39A12 Discrete version of topics in analysis
37C29 Homoclinic and heteroclinic orbits for dynamical systems
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