Electron. J. Diff. Equ., Vol. 2014 (2014), No. 93, pp. 1-14.

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

Qiang Li, Yongxiang Li

Abstract:
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.

Submitted September 22, 2013. Published April 4, 2014.
Math Subject Classifications: 34C25, 47H10.
Key Words: Second-order functional differential equations; first eigenvalue; positive periodic solution; cone; fixed point index.

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Qiang Li
Department of Mathematics
Northwest Normal University
Lanzhou 730070, China
email: lznwnuliqiang@126.com
Yongxiang Li
Department of Mathematics
Northwest Normal University
Lanzhou 730070, China
email: liyxnwnu@163.com, Phone 86-0931-7971111

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