In A Class With The Ambrosetti-rabinowitz, Super Quadratic Potential Fourth-order Ordinary Differential Equations Periodic Solution

Pulse propagation through optical fibers involving fourth order dispersion leads to a generalized nonlinear Schrodinger equationIn practice, people are interested in founding the solutions of equations (1) of the form and the equation (1) can be converted into the formIn this paper, we mainly discus the existence of 2T-periodic solutions of a class of the more general fourth order ordinary differential equation than equation (3) where A>0,B>0, f (t,u)∈C(RxR,R),and F(x,u)= f f(x,s)ds is a potential with Ambrosetti-Rabinowitz type superquadratic nonlinearity, there exits a constantθ> 2, such that 0<θF(t,u)≤uf(t,u), (?)t∈R, u∈R\{0}.For that, we first consider the following boundary problem If u is a solution of the above problem, by(FO) f(t,0)= 0,f(t+2T,u)= f(t,u), f(t,-u)=-f(t,u), (?)t∈R, u∈R, we take the odd extension in [-T,T] obviously, the 2T-extention u=u(t) in R is a 2T-periodic solution of (Ⅰ) on R.To study the existence of the solution of the boundary problem (P), we look for the critical point of the function in X(T)= H2(0,T)∩H01(0,T). It is easy to prove that the critical point of I(u;T) is the classical solution of the boundary value problem (P).when T> T1, the equation (Ⅰ) has at least one nontrivial solution with the Linking Theorem; while 0