Infinitely Many Subharmonic Solutions For Second Order Sublinear Equations

In this article, We consider two important models for second order sublinear differential equations: the existence of infinitely many subharmonic solutions for Duffing equation with bounded restore force and for the sublinear impact oscillator. They have attracted lots of researchers' attention in mathematical, other scientific and engineering fields.As for Duffing equation with bounded restore force, firstly we will change the old equation in a neighborhood of (0, 0). We change it into a new plane Hamiltonian system and make sure that (0, 0) is a solution for the Hamiltonian system. Then we use a function to control the inner boundary and make the Poincarémapping have the property of boundary twist. We obtain the existence of infinitely many subharmonic solutions for the new system by using Ding's Poincaré-Birkhoff theorem; and because of the angles of rotation for these subharmonic solutions, they are just the subharmonic solutions for the old equation. At last, we get a harmonic solution for the old equation by Massera theorem.As for the sublinear impact oscillator, firstly we will introduce a new coordinate transformation. Ittransforms the old impact system from right half plane to the whole plane, and translates the system into a new equal system. Then we deal with the new system with a method similar to the one we have applied to the sublinear equation. At last, we can get infinitely many subharmonic bouncing solutions for the old impact system.