Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term

Abstract

In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi-periodic solutions for the second order Hamiltonian system: $$\frac{d[P(t)\dot{u}(t)]}{dt}=\nabla F(t,u(t))+ e(t),$$ which generalize and improve the corresponding results in recent literature [J. Kuang, Abstr. Appl. Anal. 2012, Art. ID 271616]. Moreover, when the assumptions $F(t,x)=F(t,-x)$ and $e(t)\equiv 0$ are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.