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 systemΔx(n)=α(n)x(n+1)+β(n)|y(n)|μ-2y(n),Δy(n)=-γ(n)|x(n+1)|ν-2x(n+1)-α(n)y(n)has no solution(x(n),y(n))satisfying condition0<∑n=-∞+∞[|x(n)|ν+(1+β(n))|y(n)|μ]<+∞, whereμ,ν>1and1/μ+1/ν=1andα(n),β(n),andγ(n)are real-valued functions defined onℤ.