Order Hamilton Systems Of Second Order Ordinary Differential Equations The Existence Of Homoclinic Orbits

This paper is devoted to the study of homoclinic orbits of both Hamiltonian systems and ordinary differential equations. It consists of two chapters.In chapter 1 we first consider the asymptotically periodic Hamiltonian system possessing superquadratic potentials-u +l(t)u = (l + g(t)Vu(t,u),(HS)where L C(R,RN x RN) , V C2(R x RN,R) , N 1 and g C1(R,R) . Assuming that L and V are periodic in t and that g(t)-->0 (t-->∞), we prove the existence of a nontrivial homoclinic orbit of (HS) . This homoclinic is obtained by using the concentration-compactness method and Ekeland's variational principle. Next we study the asymptotically periodic singular Hamiltonian system in R2where W C2(R X (R2\R) , h C2(R,R) and h(t) --> 0 (t-->∞) .When W has the strong force singularity at and is periodic int , we can show that there exist two nontrivial homoclinic orbits of (HSS) . The proof of the existence relies on the topological degree and the comparison method.In chapter 2 we deal with positive homoclinic orbits of the differential equationWe present two existence results. In the first one, assuming that 1 < q < p,α(x), β(x) andγ(x) are asymptotically periodic andby means of the minmax method and Ekeland's variational principle, we prove that there is a positive homoclinic orbit of (ODE). In our second result, we suppose that 0 < q < 1 < p , α(x) , β(x) and γ(x) are periodic functions. Assuming moreover that0<α(x) , β(x)<0 , 0<γ(x) , we show the existence of a positive homoclinic orbit of (ODE) by using the mountain pass theorem and the approximating method.