A Few Resonance Problem Of Ordinary Differential Equations

The existence of Periodic Solution of Differential Equation’s Resonance has become one of the most challenge research subjects in recent years. Based on the researches of the differential equation from domestic and foreign scholars, this article introduces the present situation and prospects about the existence of solutions of differential equations. The main works of this paper as follows.Chapter two proves that the first-order differential equation with the resonance meets the Landesman-Lazer condition and the existence of the periodic solution of the equation under this condition. This chapter also obtains and proves several equivalent conditions of Landesman-Lazer condition. As for the first-order differential equation with the simple resonance, this chapter gets the same conclusion.Chapter three mainly researches the second-order differential equation. This chapter not only obtains the second-order differential equation without damping satisfies the Landesman-Lazer condition, but also proves the existence of periodic solution of the equation under this condition. In addition, this chapter gives the existence of periodic solution of the damping second-order scalar differential equation under the Landesman-Lazer condition. Finally, taking an example explain that not all the existence of periodic solution of the damping second-order scalar differential equation must satisfy the Landesman-Lazer condition.Chapter four studies a special kind of first-order plane systems of Ordinary Differential Equations. Some plane systems with double resonance and the same form of two Hamiltonian functions had been studied to meet the Landesman-Lazer condition and proved the existence of periodic solution of systems in other literatures. However, for compactness, this paper proves that the system with simple resonance or without resonance satisfies the Landesman-Lazer condition and the existence of periodic solution.