A Six-order Nonlinear Periodic Solutions And Homoclinic Orbit Solution Study

In the studies of physiology, ecology, population genetics, medicine, and pharmacology, there has been a class of six-order nonlinear differential equation model. For instance, Gardner and Jones and Caginalp and Fife researched six-order differential equationin which A and B are constants when they studied phase-field models. At this point, this type of equation can be transformed into the form of ordinary differential equations. In the field, people use the the rapid development of the critical point theory (that is, large-scale variational method) in most recent decades, made a number of very deep resultFirstly, the existence of periodic solutions of the Sixth Order ordinary differential equationwill be discussed in this paper. We study the solvability of the two-point boundary value problem for the Eq.(â… ) with the boundary conditionsusing variational methods. We obtain 2T-periodic solutions u of Eq.(â… ) antisymmetric with respect to 0 and T taking the 2T -periodic extension of the odd extension of the solution u of the problem Eq.(â… ) and (3).To study the solution of the boundary problem Eq.(â… ) and (3), we look for the critical point of the functionalin X = H³(0,T) (?) H₀¹(0,T).We also study the existence of homoclinic orbits for a class of semilinear sixth order periodic differential equations with superquadratic potentials.The above equation has a variational stmcture and its homoclinic solution is the critical point of the following functionalφ(u): H³(R)â†'RWe shall apply variational methods to prove the above results.