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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.