Existence and multiplicity of positive periodic solutions for second-order functional differential equations with infinite delay

Li, Qiang ; Li, Yongxiang

Electronic journal of differential equations, Tome 2014 (2014)

Résumé

In this article, the existence and multiplicity results of positive periodic solutions are obtained for the second-order functional differential equation with infinite delay

$ u''(t)+b(t)u'(t)+a(t)u(t)=c(t)f(t,u_t),\quad t\in \mathbb{R} $

where $a, b, c$ are continuous $\omega$-periodic functions, $u_t\in C_B$ is defined by $u_t(s)=u(t+s)$ for $s\in(-\infty,0], C_{B}$ denotes the Banach space of bounded continuous function $\phi:(-\infty,0]\to\mathbb{R}$ with the norm $\|\phi\|_B=\sup_{s\in(-\infty,0]}|\phi(s)|$, and $f: \mathbb{R}\times C_B\to [0,\infty)$ is a nonnegative continuous functional. The existence conditions concern with the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed point index theory in cones.

Détail

Classification : 34C25, 47H10
Keywords: second-order functional differential equations, first eigenvalue, positive periodic solution, cone, fixed point index