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

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 quasiperiodic solutions for the second order Hamiltonian system: d[P(t)u(t)] dt = ∇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) ≡ 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.