Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

Abstract

We give several sufficient conditions under which the first-order nonlinear Hamiltonian system $x\mathrm{\text{'}}(t)=\alpha (t)x(t)+f(t,y(t)),\hspace{0.17em}\hspace{0.17em}y\mathrm{\text{'}}(t)=-g(t,x(t))-\alpha (t)y(t)$ has no solution $(x(t),y(t))$ satisfying condition where $\mu ,\nu >\mathrm{1}$ and $(\mathrm{1}/\mu )+(\mathrm{1}/\nu )=\mathrm{1}$, $\mathrm{0}\le xf(t,x)\le \beta (t)|x{|}^{\mu }$, $xg(t,x)\le {\gamma }_{\mathrm{0}}(t)|x{|}^{\nu }$, $\beta (t),{\gamma }_{\mathrm{0}}(t)\ge \mathrm{0}$, and $\alpha (t)$ are locally Lebesgue integrable real-valued functions defined on $ℝ$.