Research On Homoclinic And Heteroclinic Orbits For Several Classes Of Differential Equations

In this paper we are concerned with the existence and multiplicity of homoclinic and hete-roclinic orbits for several classes of differential equations. Under certain assumptions, we obtain the existence and multiplicity of homoclinic and heteroclinic orbits by using variational meth-ods. The main contents are summarized as follows:In chapter 1, a brief introduction is given to the background and the structure of the disser-tation together with some preliminaries.In chapter 2, we are concerned with the existence of homoclinic orbits for the first order Hamiltonian systems, z= JHz?t,z?, where H?t, z? depends on t, but is not periodic in t. Under certain superquadratic assumptions on H?t, z?, we get the existence of homoclinic orbits by using linking theorem in critical point theory. Here, we do not assume that H?t, z? satisfies ?Ambrosetti-Rabinowitz? condition ??AR? condition for short?. Moreover, multiplicity of homoclinic orbits is considered under some sub-rquadratic conditions. These results improve and generalize some results in the literature.In chapter 3, we investigate the existence and multiplicity of homoclinic orbits for the second order damped differential equations ü+cu-L?t?u+Wu_{t, u}= 0, where L?t? and W?t, u? are dependent on t, but nonperiodic in t. Under certain assumptions on L and W, we get infinitely many homoclinic orbits or quasi-homoclinic orbits in the cases where W?t, u? is subquadratic or superquadratic by using some critical point theorems. Some recent results in the literature are generalized or significantly improved. Especially, an open problem proposed by Zhang and Yuan is solved. In addition, with the help of the Nehari manifold, we consider the case where W?t, u? is indefinite and prove the existence of at least one nontrivial quasi-homoclinic orbit for the equations.In chapter 4, we investigate the existence and multiplicity of homoclinic orbits for the second order damped differential equations ü+g?t?u-L?t?u+Wu_t,u?0, where g?t?, L?t? and W?t, u? depend on t, but are nonperiodic in t. Under certain assumptions on g, L and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave-convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results improve and generalize some results in the literature.In chapter 5, we study the existence of homoclinic orbits and heteroclinic orbits for the following second order damped equation ü+Au-L?t?u+Wu_{t, u}= f?t?, where A is an antisymmetric constant matrix, L?t? is a symmetric matrix for all t?R. Existence and multiplicity of homoclinic orbits for strongly indefinite problems is considered under some periodic and asymptotically quadratic conditions. Furthermore, by variational approach, under certain assumptions on L?t?, W?t, u? and f?t?, we show that for every x ? m, there exists at least one heteroclinic orbit u such that u?-??= x and u?+?? ? m\{x}, where m??? RN is a set isolated points and #m? 2.