Existence of periodic solutions of pendulum-like ordinary and functional differential equations

• Main Author: László Hatvani
• Format: Article
• Language: English
• Published: University of Szeged 2020-12-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: leray–schauder degree; forced damped pendulum
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=8945

The equation \[x''(t)=a(t,x(t))+b(t,x)+d(t,x)e(x'(t))\] is considered, where $a:\mathbb{R}^2\to\mathbb{R}$, $b,d:\mathbb{R}\times C(\mathbb{R},\mathbb{R})\to\mathbb{R}$, $e:\mathbb{R}\to\mathbb{R}$ are continuous, and $a,b,d$ are $T$-periodic with respect to $t$. Using the Leray–Schauder degree theory we prove that a sign condition, in which $a$ dominates $b$, is sufficient for the existence of a $T$-periodic solution. The main theorem is applied to the equation of the forced damped pendulum.