Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

Abstract

We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system $\mathrm{\Delta }x(n)=\alpha (n)x(n+1)+\beta (n)|y(n){|}^{\mu -2}y(n),\mathrm{\Delta }y(n)=-\gamma (n)|x(n+1){|}^{\nu -2}x(n+1)-\alpha (n)y(n)$ has no solution $(x(n),y(n))$ satisfying condition , where $\mu ,\nu >1$ and $1/\mu +1/\nu =1$ and $\alpha (n),\beta (n),$ and $\gamma (n)$ are real-valued functions defined on $\mathbb{Z}$.