Fourth-order nonlinear differential equations; multiplicity; positive periodic solutions; Mawhin's continuation theorem

• Main Authors: Hujun Yang, Xiaoling Han
• Format: Article
• Language: English
• Published: Texas State University 2019-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: fourth-order nonlinear differential equations; multiplicity; positive periodic solutions; mawhin's continuation theorem
• Online Access: http://ejde.math.txstate.edu/Volumes/2019/119/abstr.html

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.