Global uniqueness of homoclinic orbits for a class of fourth order equations

Abstract

In this paper we show the global existence and uniqueness of certain orbits homoclinic to the zero stationary solution of the fourth order equation $$\alpha x\prime \prime \prime \prime + \beta x\prime \prime + yx + k(x) = 0,x > 0,$$ whenα, γ>0>β,dk/dx 0 andK(0)=0. The existence problem is approached via the general theory of [1] and uniqueness follows from the Maximum Principle and some geometrical observations about the role of convexity. There are no small amplitude assumptions.