Boundedness of solutions for semilinear reversible systems

Abstract

In this paper we will study the boundedness of all solutions for second-order differential equations \[ x ¨ + f ( x ) x ˙ + λ 2 x + g ( x ) = p ( t ) , \ddot {x} + f(x)\dot {x} +\lambda ^2 x+ g(x)=p(t), \] where λ ∈ R \lambda \in R and g ( x ) g(x) satisfies the sublinear growth condition. Since the system in general is non-Hamiltonian, we have to introduce reversibility assumptions to apply the twist theorem for reversible mappings. Under some suitable conditions we then obtain the existence of invariant tori and thus the boundedness of all solutions.