Two classes of asymptotically different positive solutions to advanced differential equations via two different fixed-point principles

The paper considers a system of advanced-type functional differential equations
$$
\dot{x}(t) = F(t,x^t)
$$
where $F$ is a given functional, $x^t \in C([0,r],{\mathbb R}^n)$, $r>0$
and $x^t(\theta)=x(t+\theta)$, $\theta \in [0,r]$.
Two different results on the existence of solutions, with coordinates
bounded above and below by the coordinates of the given vector functions
if $t\to\infty$,
are proved using two different fixed-point principles.
It is illustrated by examples that, applying both results simultaneously to the same equation
yields
two positive solutions asymptotically different for $t\to\infty$.
The equation
$$
\dot{x}(t) = \left(a+{b}/{t}\right)\,x(t+\tau)
$$
where $a, \tau \in (0,\infty)$, $a<1/(\tau\e)$, $b \in {\mathbb R}$ are constants
can serve as a linear example.
The existence of a pair of positive solutions
asymptotically different for $t\to\infty$ is proved
and their asymptotic behavior is investigated.
The results are also illustrated by a nonlinear equation.