Boundedness For Equations With Jumping P-Laplacian Term

In this paper,we are concerned with the boundednes s of all the solutions for two kinds of second order differential equations with p-Laplacian and an oscillating term. 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 main contents summed up as follows: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.The second and the third chapter are the principal part of the article.There we describe separately the two kinds of equations and give the process of proof.There into:In the second chapter,we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian term (φp(x’))’+αφ(x+)-bφp(x-)+f(x=e(t),where x+=max(x,0), x-=max(-x,0), φp(s)=|s|p-2s, p≥2, a and b are positive constants (a≠b),and satisfy1/β1/p+1/b1/p=2ω-1,where ω∈R+\Q,the perturbation f is unbounded, e(t)∈C6is a smooth2πp-periodic function on t,where πp=2π(p-1)1/p/p sin π/pIn the third chapter,we investigate the boundedness of all the solutions for the more general equations(φp(x’))’+αφp(x+)-bφp(x-)=Gx(x,t)+f (t), where x+=max(x,0),x=max(-x,0),φp(s)=|s|p-2s,p≥2,a and b are positive constants(a≠b),the perturbation,f(t)∈C23(R/3πp Z),the oscillating term G∈C21(R×R/2πp Z)satisfying|Dxi Dtj G(t,t)|≤C,0≤i+j≤21.