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We study the following second-order differential system 0.1$$\begin{aligned} -\ddot{u}(t)-M\dot{u}(t)+L(t)u(t)=H_u(t,u(t)),\quad t\in \mathbb {R}, \end{aligned}$$-u¨(t)-Mu˙(t)+L(t)u(t)=Hu(t,u(t)),t∈R,which can be regarded as a second-order Hamiltonian system with a damped term. Here, the nonlinearity H(t, u) is… Click to show full abstract
We study the following second-order differential system 0.1$$\begin{aligned} -\ddot{u}(t)-M\dot{u}(t)+L(t)u(t)=H_u(t,u(t)),\quad t\in \mathbb {R}, \end{aligned}$$-u¨(t)-Mu˙(t)+L(t)u(t)=Hu(t,u(t)),t∈R,which can be regarded as a second-order Hamiltonian system with a damped term. Here, the nonlinearity H(t, u) is superquadratic as $$|u|\rightarrow \infty $$|u|→∞. We do not need any periodic conditions, and we obtain infinitely many nontrivial homoclinic orbits of this system by variational methods. Our result improves and extends the corresponding results existed.
Journal Title: Bulletin of the Malaysian Mathematical Sciences Society
Year Published: 2018
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