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Existence of homoclinic solutions for a class of nonlinear difference equations. (English) Zbl 1208.39003

The authors consider the nonlinear difference equation
\[ \Delta(p(n)(\Delta u(n-1))^\delta)-q(n)(x(n))^\delta=f(n,u(n)),\quad n\in\mathbb{Z}, \]
where \(\Delta\) is the usual forward difference operator, \(\delta\) is the ratio of odd positive integers, \(p,q\) are real sequences with \(p(n)\neq 0\), and \(f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{R}\). They are interested in obtaining conditions guaranteeing the existence of a nontrivial homoclinic solution. Another of the main purposes is to develop a new approach to this problem by using critical point theory. Various types of sufficient existence conditions are obtained. In particular, the conditions on the nonlinear term are rather relaxed, and so generalize some existing results in the literature; the expression \(\int_0^x f(n,s)ds\) satisfies a kind of new superquadratic condition.
Reviewer: Pavel Rehak (Brno)

MSC:

39A10 Additive difference equations
39A12 Discrete version of topics in analysis
37C29 Homoclinic and heteroclinic orbits for dynamical systems
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References:

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