Invariant Set Theroy Of Descending Flow With Applications To Ordinary Differential Equations

In paper [4], Zhaoli Liu and Jingxian Sun constructed systemicly and integrately the invariant set theory of descending flows, which give an effective method to study multi-solutions of differential equations: transform multi-solutions problem of a differential equation into critical points problem of its corresponding variational functional, use the (pseudo-)gradient vector field of such variational functional to produce descending flows, study the property of these descending flows to get critical points, construct disjoint invariant sets of descending flows and search critical points in each invariant set for more different critical points. This method is not only convenient and applied but also exact and intuitionistic .In paper [4], when study second order differential equations, we find the gradient vector field of the variational functional f must meet such form: df = u - KGu, which is critial to overcome the following two difficulties: no inner points in invariant sets of Hilbert space and no compactness condition in the imbeded Banach space. But we cann't validate whether the gradient vector field of the variational funcional corresponded by the Hamilton equation meet such form or not. In this paper, to use the method of invariant sets of descending flows, we appropriately modify the Theorem3.3 in [4], that is , we replace the gradient vector field with a pseudogradient vector field. we get the existence of four critical points.When use this result to study Hamiltonian systems, combining with the dual least action principle, we choose an appropriate pseudogradient vector field to produce descending flows, construct invariant sets and get four periodic solutions.As an application of invariant set method, we also study a class of subquadratic second order differential equations.