Boundedness For Two Class Of Differential Equations

In this paper, we are concerned with the boundedness of all the solutions for the following two class of differential equations:boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials; boundedness for the general semilinear Duffing equations at resonance.Under some suitable conditions, we come to the conclusion that all the solutions are bounded using canonical transformation and Moser’s small twist theorem.The first chapter is an introduction, here we give a brief introduction of the problem and give some basic knowledge which will be used in the study of this article. And then we introduce the main idea of the method.In the second chapter, we are concerned with the boundedness of all the solutions for a class of impact oscillators with time dependent polynomial potentials: where n∈N+, pi(t+1)=pi(t) and pi(t)∈C5.Since the solution of impact problem is only defined in the right half-plane, and the derivative changes when occurring impacts, we introduce a new area-preserving map by which we extent it to the whole plane. Then the problem is transformed into the existence of periodic-solutions. After that, by constructing a series canonical transformations, a large number of calculations and Moser small twist theorem, we prove that the transformed system is Lagrange stability, and then we obtain all the solutions of the original system is also bounded. Thus we extent the solution in [1].In the third chapter, we investigate the boundedness of all the solutions for the following general Duffing equations x+n2x+φ(x)=Gx (x, t)+p(t), where n∈N, p(t)∈C9(R/23πZ),φ(x)∈C7(R) and G(x,t)∈C10(RxR/2πZ).Since the potential function of the equation neither satisfy the asymptotic limit condition nor satisfy the growth condition, combining the idea in [2] and [3], we construct a series canonical transformations so that the P map of the transformed system satisfying the conditions of the Moser’s small twist theorem. Finally, we prove that all solutions of the general Duffing equations are bounded.