Existence and multiplicity of positive periodic solutions for fourth-order nonlinear differential equations

Abstract

In this article we study the existence and multiplicity of positive periodic solutions for two classes of non-autonomous fourth-order nonlinear ordinary differential equations uiv - pu″ - α(x)un + b(x)un+2 = 0, uiv - pu″ + α(x)un - b(x)un+2 = 0 where n is a positive integer, p ≤ 1, and α(x), b(x) are continuous positive T-periodic functions. These equations include particular cases of the extended Fisher-Kolmogorov equations and the Swift-Hohenberg equations. By using Mawhin's continuation theorem, we obtain two multiplicity results these equations.