be a complete metric space with a partial order,
be a monotonically nondecreasing operator. A fixed point
means that there exists some
. The present paper discusses the existence and uniqueness of the fixed points of
under the conditions that
is order-contractive and the fixed point equation
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.