Homoclinic Solutions For Second-order Asymptotically Linear Indeterminate Equation

In this paper,we study the existence of homoclinic orbits for the second-order asymptotically linear indeterminate equation:where I is the N-identity matrix,V(t,q)is periodic with respect to t and Vq(t,q)is asymptotically linear with respect to q as |q|??.The second-order asymptotically linear indeterminate equation is discussed in two Chapters.The first Chapter gives some basic concepts and the main results.The second Chapter adopts the method of reference[11]and combines the monotonicity trick of reference[22].By constructing a new topology,a proper topological degree is defined,and the mountain pass geometry of a functional family is proved by using pseudogradient vector field,and then it is concluded that the functional family has a bounded(PS)sequence.Therefore it is proved that the functional family corresponding to the homoclinic solutions of the system(V)has a bounded(PS)sequence by using the conclusions of reference[27].In the end,the compactness principle of reference[23]is used to prove the existence of weakly convergent(PS)sequence,and then theorem 1.2.1 is proved.