Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 119, pp. 1-14.

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

Authors: Hujun Yang   (Northwest Normal Univ., Lanzhou, China)
         Xiaoling Han (Northwest Normal Univ., Lanzhou, China)

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
 $$\displaylines{
 u^{iv}-pu'' -a(x)u^{n}+b(x)u^{n+2}=0, \cr
 u^{iv}-pu'' +a(x)u^{n}-b(x)u^{n+2}=0, 
 }$$
 where $n$ is a positive integer, $p \leq1$, and a(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.

Submitted October 17, 2019. Published November 14, 2019.
Math Subject Classifications: 34C25, 34G20, 35A01.
Key Words: Fourth-order nonlinear differential equations; multiplicity;
           positive periodic solutions; Mawhin's continuation theorem.