Existence of homoclinic orbits for \(2n\)th-order nonlinear difference equations containing both many advances and retardations. (English) Zbl 1228.39005
This paper is concerned with the \(2n\)-th order difference equation
\[ \Delta ^{n}\left( r(t-n)\Delta ^{n}u(t-n)\right) +q(t)u(t)=f\left( t,u(t+n\right) ,\dots,u(t),\dots,u(t-n)),\quad t\in\mathbb Z. \]
A solution \(\left\{ u(n)\right\} _{n\in\mathbb Z}\) is said to be homoclinic if \(\lim_{t\to \pm \infty}u(t)=0\). Assuming the function \(f\) is a potential (i.e., a derived function) satisfying several sets of appropriate conditions, existence of one or infinitely many homoclinic solutions are derived. The proofs make use of variational setups and critical point theory. Examples are also given for illustration.
It should be remarked that if the variable \(t\) in the above equation is interpreted as a spatial variable, then it is not necessary to make use of the terms advances and retardations in the title (which are usually accompanied with time variables).
\[ \Delta ^{n}\left( r(t-n)\Delta ^{n}u(t-n)\right) +q(t)u(t)=f\left( t,u(t+n\right) ,\dots,u(t),\dots,u(t-n)),\quad t\in\mathbb Z. \]
A solution \(\left\{ u(n)\right\} _{n\in\mathbb Z}\) is said to be homoclinic if \(\lim_{t\to \pm \infty}u(t)=0\). Assuming the function \(f\) is a potential (i.e., a derived function) satisfying several sets of appropriate conditions, existence of one or infinitely many homoclinic solutions are derived. The proofs make use of variational setups and critical point theory. Examples are also given for illustration.
It should be remarked that if the variable \(t\) in the above equation is interpreted as a spatial variable, then it is not necessary to make use of the terms advances and retardations in the title (which are usually accompanied with time variables).
Reviewer: Sui Sun Cheng (Hsinchu)
MSC:
| 39A12 | Discrete version of topics in analysis |
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
| 37C29 | Homoclinic and heteroclinic orbits for dynamical systems |
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