Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

Xiaoyan Lin; Qi-Ming Zhang; X. H. Tang

doi:10.1155/2013/547682

2013 Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

Xiaoyan Lin, Qi-Ming Zhang, X. H. Tang

Abstr. Appl. Anal. 2013: 1-10 (2013). DOI: 10.1155/2013/547682

Abstract

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

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Xiaoyan Lin. Qi-Ming Zhang. X. H. Tang. "Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems." Abstr. Appl. Anal. 2013 1 - 10, 2013. https://doi.org/10.1155/2013/547682