Let

$E$ be a complete metric space with a partial order,

$T:C\left(\right[a,b],E)\to E$ be a monotonically nondecreasing operator. A fixed point

$\varphi \in C\left(\right[a,b],E)$ of

$T$ means that there exists some

$c\in [a,\phantom{\rule{0.166667em}{0ex}}b]$ such that

$T\varphi =\varphi \left(c\right)$. The present paper discusses the existence and uniqueness of the fixed points of

$T$ under the conditions that

$T$ is order-contractive and the fixed point equation

$\varphi \left(c\right)=T\varphi $ has a lower solution. The obtained fixed point theorem is applied to a periodic boundary value problem of a delay ordinary differential equation, and a unique existence result for periodic solutions is obtained.