A Class Of Generalized Fisher-kolmogorov Equation And The Swift-hohenberg, Periodic Solutions Existence Of A Homoclinic Orbits

This paper discusses the existence of periodic solutions for one classes of fourth order Extended Fisher-Kolmogorov equation and Swift-Hohenberg equationWe considered the boundary value problemIf u(x) is classical solution of (P) and (?)(x) is its antisymmetricextension with respect to x = 0:then the 2T periodic extension of (?) over R is classical 2T periodic solution of (Ⅰ). Consider the functionalon the space X = H²(0,T) (?) H₀¹(0,T) .The critical points ofΦis weak solutions of (Ⅰ).As for the existence of homoclinic solutions we studyed the equationwhere V(x,u) is a positive and super-quadratic potential function.We suppose that b(x)∈C(R,R) is 1-periodic function , 0 < b₁≤b(x)≤b₂ and q <2(?).The homoclinic solution can be found as critical points of the functional We also study the existence of homoclinic solutions for one classes 2n order differential equationswith the same methods.