Proof. Recall that condition (F3) implies condition (F2). As mentioned above, thanks to condition (F2), if $\phi \leq _B\psi$ with $\phi (0)\ll \psi (0)$, then $y(t,\,\omega,\,\phi )\ll y(t,\,\omega,\,\psi )$ for all $t\geq 0$ for which both are defined, so that $y_t(\omega,\,\phi )\ll y_t(\omega,\,\psi )$ for all $t\geq r$. Note that $y(t,\,\omega,\,\phi )$ and $y(t,\,\omega,\,\psi )$ are both differentiable, and in particular locally Lipschitz for $t>0$. Then, by condition (F3), for all $t\geq r$,

(2.5)\begin{align} & y'(t,\omega,\psi) -y'(t,\omega,\phi)-B\,(y(t,\omega,\psi) -y(t,\omega,\phi)) \nonumber\\ & = F(\omega{\cdot}t,y_t(\omega,\psi))-F(\omega{\cdot}t,y_t(\omega,\phi))-B\,(y(t,\omega,\psi) -y(t,\omega,\phi))\gg 0\,. \end{align}

By continuity, for each $t\geq 2r$ we can take an $\varepsilon >0$ (which depends on $t$) such that $y'(t+s,\,\omega,\,\psi ) -y'(t+s,\,\omega,\,\phi )-B\,(y(t+s,\,\omega,\,\psi ) -y(t+s,\,\omega,\,\phi ))\geq \bar \varepsilon$ for all $s\in [-r,\,0]$. All in all, $y_t(\omega,\,\psi )-y_t(\omega,\,\phi )\gg _B 0$ for all $t\geq 2r$ for which both are defined, as asserted in (i).

As for (ii), if $\phi \ll _B\psi$, we already know that $y_t(\omega,\,\phi )\leq _B y_t(\omega,\,\psi )$ and, since in particular $\phi \ll \psi$, also $y_t(\omega,\,\phi )\ll y_t(\omega,\,\psi )$ for each $t\geq 0$ for which both solutions are defined. Then, relation (2.5) holds for all $t\geq 0$. From this and the fact that $\phi \ll _B\psi$, the result follows easily. The proof is finished.

Proof. (i) The proof is inspired in that of [Reference Smith and Thieme27, proposition 3.1]. Let $F^\varepsilon =F + \bar \varepsilon \,$ for $\varepsilon >0$ and let $y^\varepsilon (t,\,\omega,\,a(\omega ))$ denote the solution of $y'(t)=F^\varepsilon (\omega {\cdot }t,\,y_t)$ with initial value $a(\omega )$. We claim that $y_t^\varepsilon (\omega,\,a(\omega ))\geq _B a(\omega {\cdot }t)$ for each $\omega \in \Omega$, $t\geq 0$ and $\varepsilon >0$, and hence the proof is finished by letting $\varepsilon \downarrow 0$.

In order to see it, we consider $x^\varepsilon (t)=y^\varepsilon (t,\,\omega,\,a(\omega ))- \widetilde a_\omega (t)$, which satisfies $x^\varepsilon (s)=0$ for each $s\in [-r,\,0]$, and the closed set $I=\{t\in [0,\,\infty )\mid x_t^\varepsilon \geq _B 0\}$. Notice that $0\in I$. Note also that we are implicitly assuming that $y^\varepsilon (t,\,\omega,\,a(\omega ))$ is extendable to $[0,\,\infty )$. If not, we work on its maximal interval of existence, but recall that, given any compact set $[0,\,T]$, $y^\varepsilon (t,\,\omega,\,a(\omega ))$ is defined on $[0,\,T]$ for every $\varepsilon$ small enough, since $y(t,\,\omega,\,a(\omega ))$ is assumed to be globally defined and $F^\varepsilon =F + \bar \varepsilon \,$. Next, we check that given $t\in I$ there is a $\delta >0$ such that $[t,\,t+\delta )\subset I$. Inequality (2.6) can be written as $(\widetilde a_\omega )'(t)\leq F(\omega {\cdot }t,\,(\widetilde a_\omega )_t)$ for each $t\geq 0$ and $\omega \in \Omega$. Then, from condition (F2) and $t\in I$, we deduce that

\begin{align*} (x^\varepsilon)'(t)- B\, x^\varepsilon(t)& =F(\omega{\cdot}t,y^\varepsilon_t(\omega, a(\omega)))+\bar\varepsilon- (\widetilde a_\omega)'(t)- B\, x^\varepsilon(t)\\ & \geq \bar\varepsilon + F(\omega{\cdot}t,y^\varepsilon_t(\omega, a(\omega)))- F(\omega{\cdot}t,a(\omega{\cdot}t)) - B\, x^\varepsilon(t)\geq \bar\varepsilon\gg 0. \end{align*}

Hence, there is a $\delta >0$ such that $(x^\varepsilon )'- B\, x^\varepsilon \geq 0$ on $[t,\,t+\delta )$. Note that in particular $x^\varepsilon (t)\geq 0$ and $(x^\varepsilon )'\geq B\, x^\varepsilon$ on $[t,\,t+\delta )$. Since $B$ is a cooperative matrix, we can deduce by comparing the solutions that $x^\varepsilon (s)\geq 0$ for all $s\in [t,\,t+\delta )$. As a result, $x_s^\varepsilon \geq _B 0$ for all $s\in [t,\,t+\delta )$, that is, $[t,\,t+\delta ) \subset I$, as wanted. Therefore, $I=[0,\,\infty )$, which finishes the proof of (i).

(ii) From (i) we know that $a$ is a sub-equilibrium for the exponential ordering, that is, $a(\omega {\cdot }t)\leq _B y_t(\omega,\,a(\omega ))$ for each $\omega \in \Omega$ and $t\geq 0$. Therefore, according to [Reference Novo, Núñez and Obaya14, proposition 4.2(i)], which can be easily generalized to this case, to show that $a$ is a strong sub-equilibrium for the exponential ordering, it is enough to check the existence of a time $s_\ast >0$ and a point $\omega _0\in \Omega$ which is simultaneously a continuity point of $a$ and $a\circ \sigma _{s_\ast }$ such that $a(\omega _0{\cdot }s_\ast )\ll _B y_{s_\ast }(\omega _0,\,a(\omega _0))$.

In order to check this, notice that condition (2.7) and the continuity of the maps involved imply the existence of an $\varepsilon \in (0,\,r)$ such that $(\widetilde a_{\omega _0})'(t)\ll F(\omega {\cdot }t,\,(\widetilde a_{\omega _0})_t)$ for each $t\in [0,\,\varepsilon )$, and we can take a time $t_0\in (0,\,\varepsilon )$ satisfying $\widetilde a_{\omega _0}(t_0)\ll y(t_0,\,\omega _0,\,a(\omega _0))$, i.e., $a(\omega _0{\cdot } t_0)(0)\ll y_{t_0}(\omega _0,\,a(\omega _0))(0)$. Thus, since $a(\omega _0{\cdot } t_0)\leq _B y_{t_0}(\omega _0,\,a(\omega _0))$, by proposition 2.1(i),

\[ y_t(\omega_0{\cdot}t_0, a(\omega_0{\cdot}t_0)) \ll_By_t(\omega_0{\cdot}t_0,y_{t_0}(\omega_0,a(\omega_0))) \quad \text{for each } t\geq 2r\,, \]

and then, $a(\omega _0{\cdot }(t+t_0))\ll _B y_{t+t_0}(\omega _0,\,a(\omega _0))$. Thus, choosing $t=s_\ast -t_0\geq 2r$, we conclude that $a(\omega _0{\cdot }s_\ast )\ll _B y_{s_\ast }(\omega _0,\,a(\omega _0))$, which finishes the proof.

Proof. Note that proposition 2.4(i) implies that $y_t(\omega,\,0)\geq _B 0$ as long as $y(t,\,\omega,\,0)$ is defined. Then, by monotonicity, we deduce that, if $\psi \geq _B0$, then $y_t(\omega,\,\psi )\geq _B y_t(\omega,\,0)\geq _B 0$ for all $t\geq 0$ in the domain of definition of $\psi$, so that (i) holds.

For (ii), take $\omega \in \Omega$ and $\psi \gg _B 0$. Then, by proposition 2.1(ii), $y_t(\omega,\,\psi )\gg _B y_t(\omega,\,0)\geq _B 0$, whenever defined. The proof is finished.

Proof. From proposition 2.4(ii) we deduce that the null map from $\Omega$ to $X$ is a continuous strong sub-equilibrium for $\leq _B$ and, reasoning as in [Reference Novo, Núñez and Obaya14, proposition 4.3] for the exponential ordering, there exist a $\psi \gg _B 0$ and a time $s>0$ such that $a_s(\omega ):=y_{s}(\omega {\cdot }(-s),\,0)\geq _B \psi$ for each $\omega \in \Omega$. In particular, since for all $\omega$, $a_t(\omega )\geq _B a_s(\omega )$ whenever $t\geq s$, evaluating at $\omega {\cdot }t$ we obtain that $y_t(\omega,\,0)\geq _B \psi$ for each $\omega \in \Omega$ and $t\geq s$. Finally, from proposition 2.1(ii), if $\varphi \gg _B 0$, we deduce that $y_t(\omega,\,\varphi )\gg _B y_t(\omega,\,0)$ for each $t\geq 0$. Hence, $y_t(\omega,\,\varphi )\geq _B \psi$ for each $\omega \in \Omega$ and $t\geq s$, which shows the claimed uniform persistence.

Proof. First of all, corollary 2.5 gives us the invariance of both the cone of positive elements and its interior, that is to say, of the sets $\Omega \times K_B$ and $\Omega \times \operatorname {Int} \widetilde K_B$, respectively. Also note that the condition imposed on $B$ guarantees that the constant map $\bar R\in \operatorname {Int} \widetilde K_B$ for all $R> 0$.

1. (i) Notice that proposition 2.4(ii) implies that the constant maps $\Omega \to K_B$, $\omega \mapsto \bar R$ are continuous strong super-equilibria for the exponential ordering for each $R\geq R_0$. In particular, this means that the sets $\{(\omega,\,\phi )\in \Omega \times K_B\mid \phi \leq _B \bar R\}$ are positively invariant. Thanks to the restriction imposed on $B$, given a $\phi \in K_B$ we can take a sufficiently big $R>0$ so that $\phi \leq _B \bar R$. Then, we can deduce that all the solutions are bounded and the semiflow is globally defined on $\Omega \times K_B$.

   To get the existence of a global attractor, it suffices to find an absorbing compact set (see, e.g., [Reference Kloeden and Rasmussen12, theorem 1.36]). In our skew-product setting, we search for a compact set $H\subset K_B$ such that for each bounded set $X_1\subset K_B$ there exists a $t_1=t_1(X_1)$ such that $\tau _t(\Omega \times X_1)\subset \Omega \times H$ for all $t\geq t_1$. Thanks to condition (F1), the map $y_r:\Omega \times X\to X$ is compact, meaning that it takes bounded sets into relatively compact sets. Then, it is immediate that the set

   \[ H=\operatorname{cls} \left\{y_r(\omega,\varphi) \mid \omega\in \Omega\,,\; 0 \leq_B \varphi \leq_B \bar R_0 \right\} \]
   is compact in $K_B$. Let us check that the compact set $\Omega \times H$ is absorbing. For each bounded set $X_1\subset K_B$, we can find an $R\geq R_0$ big enough so that $0\leq _B \phi \leq _B \bar R$ for all $\phi \in X_1$ and, by monotonicity, $0\leq _B y_t(\omega,\,\phi ) \leq _B y_t(\omega,\,\bar R)$ for all $\omega \in \Omega$, $\phi \in X_1$, and $t\geq 0$. Since the constant maps $\bar R$ are strong super-equilibria for $\leq _B$ for all $R\geq R_0$, it is not difficult to deduce that there exists a $t_0\geq 0$ such that $0\leq _B y_t(\omega,\,\phi ) \leq _B \bar R_0$ for all $(\omega,\,\phi )\in \Omega \times X_1$ and $t\geq t_0$. Now, it suffices to apply the semicocycle property (2.3) to get that, if $t\geq t_1:= t_0+r$, then $y_t(\omega,\,\phi )=y_r(\omega {\cdot }(t-r),\,y_{t-r}(\omega,\,\phi ))\in H$ for all $(\omega,\,\phi )\in \Omega \times X_1$, as we wanted to see.

2. (ii) First of all, recall that the interior of the positive cone $K_B$ is empty in $X$ but it is nonempty when restricted to $X_L$. For this reason we consider compact sets in $X_L$ when we look for a global attractor in $\Omega \times \operatorname {Int}\widetilde K_B$.

   Since $F(\omega _1,\,0)\gg 0$, $\tau$ is uniformly persistent in $\operatorname {Int}\widetilde K_B$ by proposition 2.7. Let $\psi \gg _B 0$ be the map in definition 2.6(ii) and consider the compact set $H=\operatorname {cls} \big \{y_r(\omega,\,\varphi ) \mid \omega \in \Omega \,,\,\; \psi \leq _B \varphi \leq _B \bar R_0 \big \}$ in $K_B$. Then, take $\widetilde H=\{y_r(\omega,\,\varphi ) \mid (\omega,\,\varphi )\in \Omega \times H\}$. $\widetilde H$ is a compact set in $X_L$ as it is the image by the continuous map $y_r$ of a compact set in $\Omega \times X$ (see (2.4)). It is easy to check that $\widetilde H\subset \operatorname {Int} \widetilde K_B$ because the map $\omega \in \Omega \mapsto y_r(\omega,\,\psi _0)\in \operatorname {Int}\widetilde K_B\subset X_L$ is continuous for each $\psi _0\gg _B 0$ and $\Omega$ is compact. Let us see that $\Omega \times \widetilde H$ is absorbing. Given a compact set $X_1\subset \operatorname {Int} \widetilde K_B$, there exist a $\varphi _0\gg _B 0$ and an $R\geq R_0$ big enough such that $\varphi _0\leq _B \phi \leq _B \bar R$ for every $\phi \in X_1$. By monotonicity, $y_t(\omega,\,\varphi _0)\leq _B y_t(\omega,\,\phi ) \leq _B y_t(\omega,\,\bar R)$ for all $(\omega,\,\phi )\in \Omega \times X_1$ and $t\geq 0$. In fact, as seen in the proof of proposition 2.7, for every $\varphi _0\gg _B 0$ we get a time $t_0=t_0(\varphi _0)$ such that $\psi \leq _B y_t(\omega,\,\varphi _0)$ for all $t\geq t_0$ and $\omega \in \Omega$. As in (i), by taking $t_0$ bigger if necessary, we can assert that $\psi \leq _B y_t(\omega,\,\phi )\leq _B \bar R_0$ for all $(\omega,\,\phi )\in \Omega \times X_1$ and $t\geq t_0$. Now, by writing $y_t(\omega,\,\phi )=y_r(\omega {\cdot }(t-r),\,y_r(\omega {\cdot }(t-2r),\,y_{t-2r}(\omega,\,\phi )))$ it is easy to see that $y_t(\omega,\,\phi )\in \widetilde H$ for all $(\omega,\,\phi )\in \Omega \times X_1$ and $t\geq t_1:=t_0+2r$. The proof is finished.

Proof. Thanks to condition (F4), in particular $F(\omega,\,0)\geq 0$ for all $\omega \in \Omega$ so that, if $\omega \in \Omega$ and $\psi \geq _B0$, then $y_t(\omega,\,\psi )\geq _B 0$ whenever defined, by corollary 2.5(i). Once we know that the solutions starting in $\Omega \times K_B$ remain in $\Omega \times K_B$ while defined, we only worry about boundedness above. Let us check that there exists a $c>0$ such that $F(\omega,\,\psi )\leq (1+\|\psi \|_\infty )\,\bar c$ for all $(\omega,\,\psi )\in \Omega \times K_B$, for the vector $\bar c\in \mathbb {R}^m$. To see it, by (F1), if $\|\psi \|_\infty \leq 1$, then we can choose a $c>0$ big enough so that $F(\omega,\,\psi )\leq \bar c$, for all $\omega \in \Omega$, and if $\|\psi \|_\infty >1$, then we can deduce from (F4) that

\[ F(\omega,\psi)=F\left(\omega, \|\psi\|_\infty\,\frac{\psi}{\|\psi\|_\infty}\right)\leq \|\psi\|_\infty\,F\left(\omega, \frac{\psi}{\|\psi\|_\infty}\right)\leq \|\psi\|_\infty\,\bar c\,, \]

so that we are done. Now note that the solutions of the family of systems $y'(t)=\widetilde F(\omega {\cdot }t,\,y_t)$, $\omega \in \Omega$, given by the globally Lipschitz map in the second variable $\widetilde F(\omega,\,\psi )=(1+\|\psi \|_\infty )\,\bar c$, are globally defined, and that $\widetilde F$ satisfies the quasimonotone condition (Q) when restricted to $\Omega \times K_B$. Therefore, we can compare solutions (see [Reference Smith25, theorem 5.1.1]) to deduce that $y(t,\,\omega,\,\psi )$ is also globally defined for all $(\omega,\,\psi )\in \Omega \times K_B$.

Next, we check the sublinearity for fixed $\omega \in \Omega$ and $\psi \geq _B 0$. Since $y_t(\omega,\,\psi )\geq _B 0$ for all $t\geq 0$, from (F4) we have that

(3.1)\begin{equation} F(\omega{\cdot}t,\lambda\, y_t(\omega,\psi))\geq \lambda\, F(\omega{\cdot}t,y_t(\omega,\psi))\quad\text{for each } \lambda\in[0,1] \text{ and }t\geq 0\,. \end{equation}

The proof now continues as that of proposition 2.4. We consider the function $F^\varepsilon =F + \bar \varepsilon \,$ for $\varepsilon >0$ and let $y^\varepsilon (t,\,\omega,\,\varphi )$ denote the solution of $y'(t)=F^\varepsilon (\omega {\cdot }t,\,y_t)$ with initial value $\varphi$. We claim that $y_t^\varepsilon (\omega,\,\lambda \,\psi )\geq _B \lambda \, y_t(\omega,\,\psi )$ for all $t\geq 0$ and then the proof is finished by letting $\varepsilon \downarrow 0$.

In order to prove the claim, we take $x^\varepsilon (t)=y^\varepsilon (t,\,\omega,\,\lambda \, \psi )-\lambda \, y(t,\,\omega,\,\psi )$, which satisfies $x^\varepsilon (s)=\lambda \,\psi (s)-\lambda \,\psi (s)=0$ for each $s\in [-r,\,0]$, and we consider the closed set $I=\{t\in [0,\,\infty )\mid x_t^\varepsilon \geq _B 0\}$. Notice that $0\in I$. Next, we check that given $t\in I$ there is a $\delta >0$ such that $[t,\,t+\delta )\subset I$. Since

\[ (x^\varepsilon)'(t)-B\, x^\varepsilon(t)=F(\omega{\cdot}t,y^\varepsilon_t(\omega,\lambda\,\psi))+\bar\varepsilon-\lambda\, F(\omega{\cdot}t,y_t(\omega,\psi))-B\, x^\varepsilon(t)\,, \]

we can apply (3.1) to get

\begin{align*} (x^\varepsilon)'(t)-B\, x^\varepsilon(t)& \geq F(\omega{\cdot}t,y^\varepsilon_t(\omega,\lambda\,\psi))-F(\omega{\cdot}t,\lambda\, y_t(\omega,\psi))\\ & \quad -B\,(y^\varepsilon(t,\omega,\lambda\, \psi)-\lambda\, y(t,\omega,\psi))+\bar\varepsilon\,. \end{align*}

Now, since $t\in I$ we know that $y^\varepsilon _t(\omega,\,\lambda \,\psi )\geq _B \lambda \, y_t(\omega,\,\psi )$, and from (F2) we conclude that $(x^\varepsilon )'(t)-B\, x^\varepsilon (t)\geq \bar \varepsilon$. Hence, there is a $\delta >0$ such that $(x^\varepsilon )'(s)-B\, x^\varepsilon (s)\geq 0$ for $s\in [t,\,t+\delta )$. Besides, since in particular $x^\varepsilon (t)\geq 0$ and $(x^\varepsilon )'\geq B\, x^\varepsilon$ on $[t,\,t+\delta )$ and $B$ is a cooperative matrix, by comparing the solutions we get that $x^\varepsilon (s)\geq 0$ for $s\in [t,\,t+\delta )$. Consequently, $x_s^\varepsilon \geq _B 0$ for each $s\in [t,\,t+\delta )$ and so $[t,\,t+\delta ) \subset I$. Therefore, $I=[0,\,\infty )$ and the proof is finished.

Proof. Once we check that $\{(\omega {\cdot }t,\, y_t(\omega,\,\phi ))\mid t\ge 0\}$ is bounded, it follows easily from condition (F1) that it is relatively compact in $\Omega \times X$. More precisely, given a sequence $\{t_n\}$ there is a subsequence (for simplicity, we take the whole sequence) for which $\omega {\cdot }t_n\to \omega _1$ and $y_{t_n}(\omega,\,\phi )\to \phi _1$ in $X$ as $n\to \infty$. Thus, from

\[ y_{t_n}(\omega,\phi)(s)=y_{t_n}(\omega,\phi)({-}r)+\int_{{-}r}^s f(\omega{\cdot}(t_n+u),y_{t_n+u}(\omega,\phi))\,{\rm d}u,\quad s\in[{-}r,0]\,, \]

we deduce that $\phi _1(s)=\phi _1(-r)+\int _{-r}^s f(\omega _1{\cdot }u,\,(\phi _1)_u)\,{\rm d}u$, i.e., $\phi _1'(s)=f(\omega _1{\cdot }{s},\,(\phi _1)_s)$, which together with $(y_{t_n}(\omega,\,\phi ))'(s)=f(\omega {\cdot }(t_n+s),\,y_{t_n+s}(\omega,\,\phi ))\to f(\omega _1{\cdot }{s},\,(\phi _1)_s)$ shows that $(\omega {\cdot }t_n,\,y_{t_n}(\omega,\,\phi ))\to (\omega _1,\,\phi _1)$ in $\Omega \times X_L$, as wanted.

In order to check the boundedness, from (F5) and the previous argument, we deduce that $\{(\omega _0{\cdot }t,\, y_t(\omega _0,\,\phi _0))\mid t\ge 0\}$ is relatively compact in $\Omega \times X_L$ and, therefore, the omega-limit set $\mathcal {O}(\omega _0,\,\phi _0)$ is a compact and invariant subset of $\Omega \times \operatorname {Int}\widetilde K_B$. Moreover, there are $e_1$ and $e_2\in \operatorname {Int}\widetilde K_B$ such that

(3.4)\begin{equation} 0\ll_B e_1\leq _B\varphi\leq_Be_2 \quad \text{for each } (\omega,\varphi)\in \mathcal{O}(\omega_0,\phi_0). \end{equation}

Then, given $\omega \in \Omega$ and $\phi \gg _B 0$, we can take a pair $(\omega,\,\varphi )\in \mathcal {O}(\omega _0,\,\phi _0)$ (thanks to the minimality of $\Omega$) and a $\mu >1$ such that $\mu ^{-1}\varphi \ll _B\phi \leq _B \mu \, \varphi$. By monotonicity and sublinearity, $\mu ^{-1}y_t(\omega,\,\varphi )\leq _B y_t(\omega,\,\phi )\leq _B \mu \, y_t(\omega,\,\varphi )$ for $t\geq 0$ and the boundedness of the semiorbit of $(\omega,\,\phi )$, both from above and below, follows from the invariance of the set $\mathcal {O}(\omega _0,\,\phi _0)$, relation (3.4) and the semimonotonicity of the norm $\| \cdot \|_L$. Besides, it follows that $\mathcal {O}(\omega,\,\phi )\gg _B 0$.

As for the uniform stability of the semiorbit of $(\omega,\,\phi )$, from the uniform continuity of the part metric $p$ over the relatively compact set $\{ y_s(\omega,\,\phi )\mid s\ge 0\}\times \{ y_s(\omega,\,\phi )\mid s\ge 0\}\subset \operatorname {Int}\widetilde K_B \times \operatorname {Int}\widetilde K_B$, given $0<\varepsilon <1$, there is a $\delta (\varepsilon )\geq 0$ such that if $\| y_s(\omega,\,\phi )-\psi \|_L<\delta$ for some $s\geq 0$ and some $\psi \in \operatorname {Int}\widetilde K_B$, then $p(y_s(\omega,\,\phi ),\,\psi ) <\varepsilon = \ln (e^{\varepsilon })$, that is, $e^{-\varepsilon } y_s(\omega,\,\phi )\leq _B\psi \leq _B\,{\rm e}^{\varepsilon } y_s(\omega,\,\phi )$, and again monotonicity and sublinearity yield

\[ e^{-\varepsilon} y_{s+t}(\omega, \phi)\leq_B y_t(\omega{\cdot} s,\psi)\leq_B e^{\varepsilon} y_{s+t}(\omega,\phi)\quad \text{for each } t\geq 0\,, \]

i.e., $p(y_t(\omega {\cdot }s,\,\psi ),\,y_{s+t}(\omega,\,\phi ))\leq \ln (e^{\varepsilon })=\varepsilon$ for $t\geq 0$, which together with (3.3) provides the uniform stability, as claimed.

Finally, the continuity of $\tau$ restricted to the compact invariant set $\mathcal {O}(\omega,\,\phi )\subset \Omega \times \operatorname {Int}\widetilde K_B$ follows from the continuity of $\tau$ in (2.2) and the equivalence of the topologies on $\Omega \times X$ and $\Omega \times X_L$ when restricted to $\mathcal {O}(\omega,\,\phi )$. The proof is finished.

Proof. Since the semitrajectory of $(\omega _0,\,\phi _0)$ is bounded and strongly apart from $0$, and $\phi _0\gg _B 0$, we can find a $\mu _0>1$ big enough so that ${\mu _0}^{-1}\phi _0\leq _B y_t(\omega _0,\,\phi _0)\leq _B \mu _0\,\phi _0$ for all $t\geq 0$. Now, given any $\omega \in \Omega$ we can take a pair $(\omega,\,\phi )\in \mathcal {O}(\omega _0,\,\phi _0)$ which also satisfies ${\mu _0}^{-1}\phi _0\leq _B \phi \leq _B \mu _0\,\phi _0$. Applying monotonicity, sublinearity and the fact that $\tau _t(\omega,\,\phi )$ remains in $\mathcal {O}(\omega _0,\,\phi _0)$ for all $t\geq 0$, we have that ${\mu _0}^{-1}y_t(\omega,\,\phi _0)\leq _B y_t(\omega,\,\phi )\leq _B \mu _0\,\phi _0$ and also ${\mu _0}^{-1}\phi _0\leq _B y_t(\omega,\,\phi )\leq _B \mu _0\,y_t(\omega,\,\phi _0)$ for all $t\geq 0$. As a consequence, the proof of (i) is finished by taking $\mu =\mu _0^2$.

Item (ii) follows straightforward by applying monotonicity, sublinearity and (i). The proof is finished.

Proof. First of all notice that $E\subset \Omega \times \operatorname {Int}\widetilde K_B\subset \Omega \times X_L$ by proposition 3.2 and $E$ is invariant, that is, $\tau _t(E)=E$ for all $t\geq 0$. To see that $E$ is locally compact, let us consider the family of open subsets in $E$ given by

(3.5)\begin{equation} V_j:=\left\{(\omega,\varphi)\in E \,\left|\; \frac{1}{j}\,\phi_0 \ll_B \varphi \ll_B j\,\phi_0 \right.\right\} \quad\text{for each}\;j\geq 1\,, \end{equation}

where $(\omega _0,\,\phi _0)$ is the pair in condition (F5). Note that, given any $(\widetilde \omega,\,\widetilde \varphi )\in E$, there is a sufficiently big $j$ such that $(\widetilde \omega,\,\widetilde \varphi )\in V_j$. Thus, if we prove that the sets $V_j$ are relatively compact in $\Omega \times X_L$, we are done.

With this purpose, fix a $j\geq 1$ and a sequence $\{(\omega _n,\,\varphi _n)\} \subset V_j$ and let us check that it has a convergent subsequence. Note that, for each $n\geq 1$, $(\omega _n,\,\varphi _n)\in \mathcal {O}(\widetilde \omega _n,\,\widetilde \varphi _n)$ for some $\widetilde \omega _n\in \Omega$ and $\widetilde \varphi _n\gg _B 0$. By proposition 3.3, the set $\mathcal {O}(\widetilde \omega _n,\,\widetilde \varphi _n)$ admits a flow extension, that is, there exists a unique backward orbit for each of its points. Then, there exists a unique $\psi _n\in X_L$ such that $\tau (r,\,\omega _n{\cdot }(-r),\, \psi _n)=(\omega _n,\,\varphi _n)$. If the sequence $\{\psi _n\}$ were relatively compact in $X$ for the sup-norm, then for a subsequence $\psi _{n_k}\to \psi _0\in X$ as $k\to \infty$. Since $\Omega$ is compact, we can assume without loss of generality (w.l.o.g.) that $\omega _{n_k}\to \omega \in \Omega$ as $k\to \infty$. Therefore, by the continuity of the map $\tau _r:\Omega \times X\to \Omega \times X_L$ [see (2.4)], $y_r(\omega _{n_k}{\cdot }(-r),\, \psi _{n_k})= \varphi _{n_k} \to y_r(\omega {\cdot }(-r),\, \psi _0)$ in $X_L$ as $k\to \infty$ and we would have found a convergent subsequence $\{(\omega _{n_k},\,\varphi _{n_k})\}$, as wanted.

Therefore, to finish the proof of (i) we need to see that $\{\psi _n\}$ is relatively compact in $X$ for the sup-norm. The main idea is to choose appropriate pairs $(\omega _n^*,\,\varphi _n^*)\in \Omega \times \operatorname {Int}\widetilde K_B$ in such a way that $(\omega _n,\,\varphi _n)\in \mathcal {O}(\omega _n^*,\,\varphi _n^*)$ for each $n\geq 1$ and the set

\[ H^*=\{y_{t+r}(\omega_n^*,\varphi_n^*)\mid n\geq 1,\,t\geq 0\}\subset X \]

is relatively compact. Note that then, also $(\omega _n{\cdot }(-r),\, \psi _n) \in \mathcal {O}(\omega _n^*,\,\varphi _n^*)$ for each $n\geq 1$, and thus $\{\psi _n\}\subset \operatorname {cls} H^*$, the closure of $H^*$ under the sup-norm, a compact set.

By the definition of $V_j$, for each $n\geq 1$, we have that $j^{-1}\phi _0 \ll _B \varphi _n \ll _B j\,\phi _0$ and, since $(\omega _n,\,\varphi _n)\in \mathcal {O}(\widetilde \omega _n,\,\widetilde \varphi _n)$, for each $n\geq 1$, there exists a time $s_n$ big enough so that $j^{-1}\phi _0 \ll _B y_{s_n}(\widetilde \omega _n,\,\widetilde \varphi _n) \ll _B j\,\phi _0$. Let $(\omega _n^*,\,\varphi _n^*)=\tau (s_n,\,\widetilde \omega _n,\,\widetilde \varphi _n)$. It is clear that $(\omega _n,\,\varphi _n)\in \mathcal {O}(\omega _n^*,\,\varphi _n^*)$ for each $n\geq 1$ and $\varphi _n^* \in C_j$. By lemma 3.4, $y_t(\omega _n^*,\,\varphi _n^*) \in C_{\mu j}$ for all $t\geq 0$ and $n\geq 1$, so that in particular $y_t(\omega _n^*,\,\varphi _n^*)$ is in the interval for the usual order in $X$, $I=[j^{-1}\mu ^{-1}\,\phi _0,\,j\,\mu \,\phi _0]$. Then, by (2.3), we can rewrite

\[ H^*=\{y_r(\omega_n^*{\cdot}t,y_t(\omega_n^*,\varphi_n^*))\mid n\geq 1,\,t\geq 0\}\subset y_r\left(\Omega\times I\right)\,, \]

which is relatively compact in $X$ because the map $y_r$ is compact. Therefore, $H^*$ is relatively compact too and the proof of (i) is finished.

Most of the arguments for the proof of (ii) are identical to the previous ones, so that we just give a sketch of the proof. To check the joint continuity of $\tau$ on $\mathbb {R}^+\times E$, we take $\{(t_n,\,\omega _n,\,\varphi _n)\}\subset \mathbb {R}^+\times E$ converging to $(\bar t,\,\bar \omega,\,\bar \varphi )\in \mathbb {R}^+\times E$ as $n\to \infty$ with the norm $\|\cdot \|_L$ in $X_L$. We already know that $\lim _{n\to \infty }\omega _n{\cdot }t_n=\bar \omega {\cdot }\bar t$. For the second component of $\tau$, note that, for each $n\geq 1$, $(\omega _n,\,\varphi _n)\in \mathcal {O}(\widetilde \omega _n,\,\widetilde \varphi _n)$ for some $\widetilde \omega _n\in \Omega$ and $\widetilde \varphi _n\gg _B 0$. Again thanks to (2.3) we can write

(3.6)\begin{equation} y_{t_n}(\omega_n,\varphi_n)= y_{t_n+r}(\omega_n{\cdot}({-}r), \psi_n)\,, \end{equation}

where $\psi _n\in X_L$ satisfies $y_r(\omega _n{\cdot }(-r),\, \psi _n)=\varphi _n$. If the sequence $\{\psi _n\}$ were relatively compact in $X$ for the sup-norm, then $\psi _n\to \psi _0\in X$ as $n\to \infty$, up to taking a subsequence. By the continuity of $\tau : [r,\,\infty )\times \Omega \times X\to \Omega \times X_L$, we would first have that $y_r(\omega _n{\cdot }(-r),\, \psi _n)=\varphi _n\to y_r(\bar \omega {\cdot }(-r),\, \psi _0)=\bar \varphi$ in $X_L$ as $n\to \infty$, and then, taking limits in (3.6), $y_{t_n}(\omega _n,\,\varphi _n)\to y_{\bar t+r}(\bar \omega {\cdot }(-r),\, \psi _0)=y_{\bar t}(\bar \omega,\, y_r(\bar \omega {\cdot }(-r),\, \psi _0))=y_{\bar t}(\bar \omega,\,\bar \varphi )$ also in $X_L$ as $n\to \infty$, and we would be done.

Therefore, it remains to check that $\{\psi _n\}$ is relatively compact in $X$, and this is done just as in the proof of (i). We just point out that this time we can take a big $j\geq 1$ and a ball $D$ centred at $\bar \varphi \gg _B 0$ so that ${j}^{-1}\phi _0\leq _B \phi \leq _B j\,\phi _0$ for all $\phi \in D$. Since $\varphi _n\to \bar \varphi$ as $n\to \infty$ and $(\omega _n,\,\varphi _n)\in \mathcal {O}(\widetilde \omega _n,\,\widetilde \varphi _n)$, for each $n\geq 1$ we can find a time $s_n$ big enough so that $y_{s_n}(\widetilde \omega _n,\,\widetilde \varphi _n)\in D$. From here the proof is finished just as before.

In the proof of (iii) we will use some properties of the Hausdorff semidistance (2.8). Namely, ${\rm dist}(Y_1,\,Y_2)={\rm dist}(Y_1,\,\overline {Y}_{\! 2})$; if ${\rm dist}(Y_1,\,Y_2)=0$, then $Y_1\subseteq \overline {Y}_{\! 2}$; if $Y_1\subset Z_1$, then ${\rm dist}(Y_1,\,Y_2)\leq {\rm dist}(Z_1,\,Y_2)$; and if $Z_2\subset Y_2$, then ${\rm dist}(Y_1,\,Y_2)\leq {\rm dist}(Y_1,\,Z_2)$, for subsets $Y_1,\, Y_2,\, Z_1,\, Z_2$ of a metric space $Y$, denoting by $\overline {Y}_{\! 2}$ the closure of the set, for the sake of notation. Finally, recall that when $Y_1$ is a singleton, the Hausdorff semidistance is just the usual distance between a point and a set, ${\rm d}(y_1,\,Y_2)$.

The proof is organized into a series of claims. Let us fix a $j\geq 1$ and let $\mu >1$ be the one in lemma 3.4.

Claim 1. Let $j_0\geq \mu \,j$ and consider the set $E_{j_0}:=\bigcup _{(\omega,\varphi )\in \Omega \times C_{j_0}}\mathcal {O}(\omega,\,\varphi )\subset E$. Then, $E_{j_0}$ is invariant and relatively compact in $\Omega \times X_L$, the section map $\omega \in \Omega \mapsto \overline E_{j_0}(\omega )$ is continuous at every point of $\Omega$, and for each fixed $\widetilde \omega \in \Omega$, $\lim _{t\to \infty }{\rm dist}(y_t(\widetilde \omega,\,C_j),\,E_{j_0}(\widetilde \omega {\cdot }t))=0$.

First of all, note that by taking $E_{j_0}$ we are reducing the zone of the set $E$ to which $y_t(\omega,\,C_j)$ is going to approach, with the aim that it is relatively compact and uniformly stable. Recall that if $\varphi \in C_j$, then $\mathcal {O}(\omega,\,\varphi )\subset \Omega \times C_{\mu j}$ for $\omega \in \Omega$, and $C_{\mu j}\subseteq C_{j_0}$. Iterating this fact, taking $j_1> \mu \,j_0$, $E_{j_0}\subset V_{j_1}$ for the relatively compact set $V_{j_1}$ defined in (3.5), so that $E_{j_0}$ is relatively compact too. Besides, arguing as in the proof of proposition 3.2, we get the uniform stability of this set, that is, given any $\varepsilon >0$, there is a $\delta >0$ such that if $(\omega,\,\phi )\in E_{j_0}$ and $\psi \in \operatorname {Int}\widetilde K_B$ satisfy $\|\phi -\psi \|_L<\delta$, then $\|y_t(\omega,\,\phi )-y_t(\omega,\,\psi )\|_L<\varepsilon$ for $t\geq 0$. Note also that $E_{j_0}$ admits a flow extension, by proposition 3.3. Then, we can apply [Reference Novo, Obaya and Sanz16, theorem 3.3] to the invariant compact set $\overline E_{j_0}\subset \Omega \times X_L$, whence the section map $\omega \in \Omega \mapsto \overline E_{j_0}(\omega )$ is continuous at every point of $\Omega$ for the Hausdorff metric, which implies its continuity also for the Hausdorff semidistance we are using.

At this point, fix an $\widetilde \omega \in \Omega$. To see that $\lim _{t\to \infty } {\rm dist}(y_t(\widetilde \omega,\,C_j),\,E_{j_0}(\widetilde \omega {\cdot }t))=0$, let us argue by contradiction and assume that there exists an $\varepsilon _0>0$ and there are sequences $\{t_n\}\uparrow \infty$ and $\{\varphi _n\}\subset C_j$ such that

(3.7)\begin{equation} {\rm d}(y_{t_n}(\widetilde\omega,\varphi_n), E_{j_0}(\widetilde\omega{\cdot}t_n))\geq \varepsilon_0 \quad\text{for}\ n\geq 1\,. \end{equation}

Now, as in the proof of theorem 2.8, the set $H=\operatorname {cls}\{y_r(\omega,\,\varphi )\mid (\omega,\,\varphi )\in \Omega \times C_{j_0}\}$ is compact in $X$ and $\widetilde H=\{y_r(\omega,\,\varphi )\mid (\omega,\,\varphi )\in \Omega \times H\}$ is compact in $X_L$. Besides, by using the semicocycle property (2.3),

(3.8)\begin{equation} y_t(\omega,\varphi)\in \widetilde H \quad \text{whenever }\;\omega\in \Omega\,,\;\varphi\in C_j\;\text{ and }t\geq 2\,r\,. \end{equation}

Then, supposing that $t_n\geq 2r$ for $n\geq 1$, we have that $y_{t_n}(\widetilde \omega,\,\varphi _n)\in \widetilde H$ for $n\geq 1$ and we can assume w.l.o.g. that $(\widetilde \omega {\cdot }t_n,\,y_{t_n}(\widetilde \omega,\,\varphi _n))\to (\omega ^*,\,\varphi ^*)$ as $n\to \infty$. Due to the continuity of the section map $\omega \in \Omega \mapsto \overline E_{j_0}(\omega )$ at $\omega ^*$, taking limits in (3.7) we conclude that ${\rm d}(\varphi ^*,\,\overline E_{j_0}(\omega ^*))\geq \varepsilon _0$.

On the other hand, since also $\{y_{2r}(\widetilde \omega,\,\varphi _n)\}\subset \widetilde H$, once more we can assume w.l.o.g. that $y_{2r}(\widetilde \omega,\,\varphi _n)\to \widetilde \varphi \in \widetilde H \cap C_{j_0}$ as $n\to \infty$. Arguing again as in the proof of proposition 3.2, we obtain the uniform stability of the compact set $\widetilde H\subset \operatorname {Int} \widetilde K_B$, that is, given any $\varepsilon >0$ there is a $\delta >0$ such that, if $\phi \in \widetilde H$ and $\psi \in \operatorname {Int}\widetilde K_B$ satisfy $\|\phi -\psi \|_L<\delta$, then $\|y_t(\omega,\,\phi )-y_t(\omega,\,\psi )\|_L<\varepsilon$ for all $\omega \in \Omega$ and $t\geq 0$. Then, given any $\varepsilon >0$ there is an $n_{\varepsilon }$ such that $\|y_{t_n-2r}(\widetilde \omega {\cdot }2r,\,y_{2r}(\widetilde \omega,\,\varphi _n))-y_{t_n-2r}(\widetilde \omega {\cdot }2r,\,\widetilde \varphi )\|_L<\varepsilon$ for $n\geq n_{\varepsilon }$, that is, $\|y_{t_n}(\widetilde \omega,\,\varphi _n)-y_{t_n-2r}(\widetilde \omega {\cdot }2r,\,\widetilde \varphi )\|_L<\varepsilon$ for $n\geq n_{\varepsilon }$. It follows that $\lim _{n\to \infty } \tau _{t_n-2r}(\widetilde \omega {\cdot }2r,\, \widetilde \varphi )=(\omega ^*,\,\varphi ^*)$, that is, $(\omega ^*,\,\varphi ^*)\in \mathcal {O}(\widetilde \omega {\cdot }2r,\,\widetilde \varphi )\subset E_{j_0}$ by the construction. Therefore, ${\rm d}(\varphi ^*,\,E_{j_0}(\omega ^*))=0$, a contradiction.

Claim 2. The omega-limit set of $\Omega \times C_j$, $\mathcal {O}_j:=\{(\omega,\,\varphi )\in \Omega \times X_L \mid (\omega,\,\varphi )=\lim _{n\to \infty } (\omega _n{\cdot }t_n,\,y_{t_n}(\omega _n,\,\varphi _n))$ for some $\{\omega _n\}\subset \Omega$, $\{\varphi _n\}\subset C_j$, $\{t_n\}\uparrow \infty \}$, is an invariant compact set and the section map $\omega \mapsto \mathcal {O}_j(\omega )$ is continuous at every $\omega \in \Omega$.

By the construction, $\mathcal {O}_j$ is a closed set. It is positively invariant because of the semiflow property and the continuity of the map $\tau _t:\Omega \times X_L\to \Omega \times X_L$ for each fixed $t\geq 0$. Besides, according to (3.8), $\mathcal {O}_j\subset \Omega \times \widetilde H$, so that also $\mathcal {O}_j$ is compact and uniformly stable. Last but not least, there are backward extensions inside $\mathcal {O}_j$. With all these properties, we can apply [Reference Novo, Obaya and Sanz16, theorem 3.4] to get the continuity of the section map.

Claim 3. $\lim _{t\to \infty }{\rm dist}(y_t(\omega,\,C_j),\,\mathcal {O}_j(\omega {\cdot }t))=0$ uniformly for $\omega \in \Omega$.

To see it, argue by contradiction and assume that there exist an $\varepsilon _0>0$ and sequences $\{t_n\}\uparrow \infty$, $\{\omega _n\}\subset \Omega$ and $\{\varphi _n\}\subset C_j$ such that ${\rm d}(y_{t_n}(\omega _n,\,\varphi _n),\, \mathcal {O}_j(\omega _n{\cdot }t_n))\geq \varepsilon _0$ for $n\geq 1$. Once more, by (3.8) we can assume w.l.o.g. that $(\omega _n{\cdot }t_n,\,y_{t_n}(\omega _n,\,\varphi _n))\to (\omega ^*,\,\varphi ^*)\in \mathcal {O}_j$. Thus, on the one hand ${\rm d}(\varphi ^*,\, \mathcal {O}_j(\omega ^*))=0$, but on the other, by the continuity of the section map, ${\rm d}(\varphi ^*,\, \mathcal {O}_j(\omega ^*))\geq \varepsilon _0$, which is a contradiction.

Claim 4. Let $j_0\geq \mu \,j$ and $j_1\geq \mu \,j_0$. Then, $\mathcal {O}_j(\omega )\subseteq \overline E_{j_1}(\omega )$ for all $\omega \in \Omega$.

Note that by its definition, $\mathcal {O}_j\subset \Omega \times C_{j_0}$. Since $\mathcal {O}_j$ is an invariant set, for a fixed $\widetilde \omega \in \Omega$, $\mathcal {O}_j(\widetilde \omega {\cdot }t) \subset y_t(\widetilde \omega,\,C_{j_0})$ for all $t\geq 0$. Therefore, ${\rm dist}(\mathcal {O}_j(\widetilde \omega {\cdot }t),\,E_{j_1}(\widetilde \omega {\cdot }t))\leq {\rm dist}(y_t(\widetilde \omega,\,C_{j_0}),\,E_{j_1}(\widetilde \omega {\cdot }t))\to 0$ as $t\to \infty$ by claim 1 applied to $\widetilde \omega$, $j_0$ and $j_1$.

From here, by the minimal character of $\Omega$, for each $\omega \in \Omega$ we can find a sequence $\{t_n\}\uparrow \infty$ such that $\widetilde \omega {\cdot }t_n\to \omega$. By the continuity of the section maps expressed in claims 1 and 2, ${\rm dist_{\mathcal {H}}}(\mathcal {O}_j(\omega ),\, \overline E_{j_1}(\omega ))=\lim _{n\to \infty }{\rm dist_{\mathcal {H}}}(\mathcal {O}_j(\widetilde \omega {\cdot }t_n),\,\overline E_{j_1}(\widetilde \omega {\cdot }t_n))=0$. As a consequence, $\mathcal {O}_j(\omega )\subseteq \overline E_{j_1}(\omega )$ for all $\omega \in \Omega$.

Claim 5. $\lim _{t\to \infty }{\rm dist}(y_t(\omega,\,C_j),\,E(\omega {\cdot }t))=0$ uniformly for $\omega \in \Omega$.

For each $\omega \in \Omega$, $E_{j_1}(\omega ) \subset E(\omega )$ and, by claim 4, $\mathcal {O}_j(\omega )\subset \overline E_{j_1}(\omega )$. Then, for all $t\!\geq\! 0$, ${\rm dist}(y_t(\omega,\,C_j),\,E(\omega {\cdot }t))\!\leq\! {\rm dist}(y_t( \omega,\,C_j),\,E_{j_1}(\omega {\cdot }t))\leq {\rm dist}(y_t(\omega,\,C_j),\,\mathcal {O}_j(\omega {\cdot }t))$, and this goes to $0$ as $t$ goes to $\infty$ uniformly for $\omega \in \Omega$, as stated in claim 3.

Finally, note that the second limit in (iii) follows straightforward from the first one. The proof is finished.

Proof. First notice that there is at least one strongly positive minimal set because, if we take the point $(\omega _0,\,\phi _0)$ in condition (F5), proposition 3.3 asserts that $M=\mathcal {O}(\omega _0,\,\phi _0)$ is a strongly positive minimal set, $M\gg _B 0$. Assume on the contrary that there are two distinct minimal sets $M_1\gg _B 0$ and $M_2\gg _B 0$, and consider $(\omega,\,\psi _1)\in M_1$ and $(\omega,\,\psi _2)\in M_2$. Let $\omega _1\in \Omega$ and $t_1>0$ be the point and time of condition (F6). We take a sequence $\{s_n\}\downarrow -\infty$ with $s_{n-1}-s_n>t_1$ for each $n\geq 2$, and such that

(3.10)\begin{equation} \lim_{n\to\infty}(\omega{\cdot}s_n, y_{s_n}(\omega,\psi_i))=(\omega_1,\psi^*_i)\in M_i \;\text{ for }\; i=1,2. \end{equation}

Note that in particular $\psi ^*_1\not =\psi ^*_2$, since distinct minimal sets have an empty intersection. Now, from (F2), (F4) and (F6), i.e., monotonicity, sublinearity and strong sublinearity at time $t_1$ for $\omega _1$, it is well-known (see, e.g., Chueshov [Reference Chueshov4, lemma 4.2.1]) that the part metric $p$ defined in (3.2) is decreasing along the trajectories, that is,

\[ p(\varphi_1,\varphi_2)\geq p(y_t(\omega,\varphi_1),y_t(\omega,\varphi_2)) \quad \text{whenever } \omega\in\Omega,\;\varphi_1,\varphi_2\in \operatorname{Int}\widetilde K_B \text{ and }t\geq 0\,, \]

and strictly decreasing at $t_1$ for the point $\omega _1$, that is,

\[ p(\varphi_1,\varphi_2)>p(y_{t_1}(\omega_1,\varphi_1),y_{t_1}(\omega_1,\varphi_2))\quad \text{whenever } \varphi_1,\varphi_2\in \operatorname{Int}\widetilde K_B,\;\varphi_1\not=\varphi_2\,. \]

This fact, together with (3.10), the continuity of the part metric, and the inequalities $s_{n-1}> t_1+s_n$ for $n\geq 2$, yield

\begin{align*} p(\psi^*_1,\psi^*_2)& >p(y_{t_1}(\omega_1,\psi^*_1),y_{t_1}(\omega_1,\psi^*_2)) =\lim_{n\to\infty} p(y_{t_1+s_n}(\omega,\psi_1),y_{t_1+s_n}(\omega,\psi_2))\\ & \geq \lim_{n\to\infty} p(y_{s_{n-1}}(\omega,\psi_1),y_{s_{n-1}}(\omega,\psi_2))= p(\psi^*_1,\psi^*_2), \end{align*}

a contradiction. Thus, there is a unique strongly positive minimal set $M\gg _B 0$ and now we check that it is a copy of the base. Once more we argue by contradiction and assume that for certain $\omega \in \Omega$ there are two distinct pairs $(\omega,\,\psi _1),\, (\omega,\,\psi _2)\in M$. Then, we get a contradiction arguing as before. Just note that this time we get that $\psi ^*_1\not =\psi ^*_2$ because $M$ has a fibre distal flow extension (see proposition 3.3). Consequently, $M$ is a copy of the base, $b$ is continuous and (3.9) follows from $\mathcal {O}(\omega,\,\phi )=M$ for all $\omega \in \Omega$ and $\phi \gg _B 0$. The proof is finished.

Proof. Take into account the sufficient condition (4.4) for (F2) and the fact that, for each $i\in \{1,\,\ldots,\,m\}$, by the hull construction, $-d_i(\omega )+\mu _i-{\rm e}^{\mu _ir_i-2}\beta _i(\omega )\geq -d_i^++\mu _i-{\rm e}^{\mu _ir_i-2} \beta _i^+$ for all $\omega \in \Omega$. Then, it suffices to do an analytical study of the map

\[ f_i(\mu)={-} d_i^+{+}\mu -\frac{1}{e^2}\,\beta_i^+\,{\rm e}^{\mu r_i},\quad \mu\geq 0, \]

to see that $f_i(\mu _i)\geq 0$ provided that (4.7) holds. More precisely, $\mu _i > 0$ is the point where the map $f_i$ reaches its maximum value on the positive semiaxis and condition (4.7) guarantees that this maximum value is $\geq$ $0$.

When (4.8) holds, then the maximum value of $f_i$ on the positive semiaxis is positive, the inequalities in (4.4) are strict, and thus condition (F3) holds. Now, take $\omega \in \Omega$ and $\phi \geq 0$. Then, for $t\geq r$, $y_t(\omega,\,\phi )$ is a smooth map, $y_t(\omega,\,\phi )\geq 0$ and

\[ y_i'(t\!+\!s,\omega,\phi)\geq{-}d_i(\omega{\cdot}(t\!+\!s))\,y_i(t\!+\!s,\omega,\phi)\geq{-}\mu_i\, y_i(t\!+\!s,\omega,\phi)\quad \text{for all } s\in [{-}r_i,0] \]

and $1\leq i\leq m$. Hence, $y_t(\omega,\,\phi )\geq _B 0$ for $t\geq r$. Finally, if also $\phi (0)\gg 0$, as we have already mentioned, then $y(t,\,\omega,\,\phi )\gg 0$ for all $t\geq 0$. To finish, we can argue exactly as in the proof of proposition 2.1(i).

Proof. Just note that inside a minimal set there are backward extensions of the semitrajectories. Then, when we move backwards we remain in the interior of the standard positive cone, and when we come back forwards we enter the interior of the exponential ordering cone, by theorem 4.2(ii.2).

Proof. Let us assume that $\tau$ is uniformly persistent in $\operatorname {Int}\widetilde K_B$ and let us see that it is also uniformly persistent in $\operatorname {Int} X_+$. Let $\psi \gg _B 0$ be the one in definition 2.6(ii) and take $\omega \in \Omega$ and $\varphi \gg 0$. By theorem 4.2(ii) we know that $y_{2r}(\omega,\,\varphi )\gg _B 0$, so that there exists a $t_0=t_0(\omega,\,\varphi,\,2r)$ such that $y_t(\omega {\cdot }2r,\, y_{2r}(\omega,\,\varphi ))=y_{t+2r}(\omega,\,\varphi )\geq _B \psi$ for all $t\geq t_0$. By the definition of $\leq _B$ in particular $y_{t+2r}(\omega,\,\varphi )\geq \psi \gg 0$, and we are done.

Proof. In order to apply theorem 3.7, we write the systems (4.3) in the general form $y'(t)=F(\omega {\cdot }t,\,y_t)$, $\omega \in \Omega$. Then, condition (F1) holds and, by theorem 4.2(ii), (F3) [and hence (F2)] follows from (4.8). It is also easy to see that the sublinearity condition (F4) actually holds for all $\psi \geq 0$, so that by proposition 3.1 the semiflow is sublinear for the exponential ordering.

As mentioned before, [Reference Obaya and Sanz21, theorem 3.4] says that definition 4.1 is equivalent to definition 2.6(i). Since the solutions are ultimately bounded, taking any pair $(\omega,\,\phi )\in \Omega \times \operatorname {Int} X_+$, its omega-limit set contains a minimal set which satisfies $M\gg 0$ by the uniform persistence property. But then, by corollary 4.3, $M\gg _B 0$ and all the semitrajectories inside $M$ satisfy condition (F5).

Finally, for condition (F6), let $\omega _1$ be the map in $\Omega$ determining the initial Nicholson system (4.1), which we write as $y_i'(t)=f_i(t,\,y_t)$, $1\leq i\leq m$, for the maps $f_i:\mathbb {R}\times X\to \mathbb {R}$ given in (4.2). By assumptions (a4)–(a5), for each fixed $t$, the real map $h(x)=\widetilde \beta _{i}(t)\,x\,{\rm e}^{-\widetilde c_i(t)\,x}$ is strictly sublinear for $x>0$. Thus, it is easy to check that, for each $\lambda \in (0,\,1)$ and $t\geq 0$,

(4.10)\begin{equation} \phi\geq 0\; \text{with}\;\,\phi_i({-}r_i)>0\,,\; 1\leq i\leq m \;\Rightarrow\; \lambda\, f(t,\phi)\ll f(t,\lambda\,\phi)\,. \end{equation}

We claim that (F6) holds for $\omega _1$ and every $t_1>r:=\max (r_1,\,\ldots,\,r_m)$. To check it, take $\psi \gg _B 0$ and $\lambda \in (0,\,1)$. Having in mind the expression of $\operatorname {Int}\widetilde K_B$, let us first check that $\lambda \, y_t(\omega _1,\,\psi )\ll y_t(\omega _1,\,\lambda \, \psi )$ for all $t>r$ by comparing the solutions. It suffices to see that $\lambda \, y(t,\,\omega _1,\,\psi )\ll y(t,\,\omega _1,\,\lambda \, \psi )$ for all $t>0$. Consider $\widetilde r:=\min (r_1,\,\ldots,\,r_m)>0$ and let $z(t):=\lambda \, y(t,\,\omega _1,\,\psi )$, $t\geq 0$. Then, since in particular $\psi \gg 0$, again by strict sublinearity, for $t\in (0,\,\widetilde r]$ and $1\leq i\leq m$ we have that

\begin{align*} z_i'(t)& ={-}\widetilde d_i(t)\,z_i(t) +\displaystyle \sum_{j=1}^m \widetilde a_{ij}(t)\,z_j(t) + \widetilde\beta_{i}(t)\,\lambda\,\psi_i(t-r_i)\,{\rm e}^{-\widetilde c_i(t)\,\psi_i(t-r_i)}\\ & <{-}\widetilde d_i(t)\,z_i(t) +\displaystyle \sum_{j=1}^m \widetilde a_{ij}(t)\,z_j(t) + \widetilde\beta_{i}(t)\,\lambda\,\psi_i(t-r_i)\,{\rm e}^{-\lambda\,\widetilde c_i(t)\,\psi_i(t-r_i)}\,. \end{align*}

A standard comparison theorem for cooperative ODEs [thanks to (a3)] implies that $z(t)\ll y(t,\,\omega _1,\,\lambda \, \psi )$ for $t\in (0,\,\widetilde r]$. Iterating the procedure interval by interval, we can assert that $\lambda \, y(t,\,\omega _1,\,\psi )\ll y(t,\,\omega _1,\,\lambda \,\psi )$ for all $t>0$, as we wanted.

Secondly, note that, since $\psi \gg 0$, we know that $y(t,\,\omega _1,\,\psi )\gg 0$ for all $t\geq 0$, so that $y_t(\omega _1,\,\psi )\gg 0$ for all $t\geq 0$. Then, for all $t>0$ and $1\leq i\leq m$ we write

\begin{align*} & y_i'(t,\omega_1,\lambda\,\psi) -\lambda \,y_i'(t,\omega_1,\psi)+\mu_i(y_i(t,\omega_1,\lambda\,\psi) -\lambda\, y_i(t,\omega_1,\psi)) \\ & \quad = f_i(t,y_t(\omega_1,\lambda\,\psi))- \lambda\, f_i(t,y_t(\omega_1,\psi))+\mu_i(y_i(t,\omega_1,\lambda\,\psi) -\lambda\, y_i(t,\omega_1,\psi))\\ & \quad > f_i(t,y_t(\omega_1,\lambda\,\psi))- f_i(t,\lambda\, y_t(\omega_1,\psi))+\mu_i(y_i(t,\omega_1,\lambda\,\psi) -\lambda\, y_i(t,\omega_1,\psi)) \geq 0, \end{align*}

where (4.10) applied to $y_t(\omega _1,\,\psi )\gg 0$ has been used in the first inequality, and the sublinearity of the semiflow for $\leq _B$ permits to apply (F2) to justify the second inequality. Then, by a continuity argument we deduce that $y_t(\omega _1,\,\lambda \,\psi )\gg _B \lambda \,y_t(\omega _1,\,\psi )$ for all $t>r$, whence (F6) holds for $\omega _1$ and every $t_1> r$, as claimed.

Therefore, we can apply theorem 3.7 saying that there exists a unique strongly positive minimal set $M\gg _B 0$ which is a copy of the base and attracts all the semiorbits starting in $\Omega \times \operatorname {Int} \widetilde K_B$ as $t\to \infty$ for the $\|\cdot \|_L$ norm. By corollary 4.3, $M$ is also the unique minimal set in $\operatorname {Int} X_+$. Finally, (4.9) follows from theorem 4.2(ii.2), since the orbit of each $(\omega,\,\phi )$ with $\phi \geq 0$ and $\phi (0)\gg 0$ enters $\Omega \times \operatorname {Int} \widetilde K_B$ after time $2r$ and it is then attracted to the minimal set forwards in time. Since the immersion $X_L\hookrightarrow X$ is continuous, the proof is finished.

Proof. One implication is corollary 4.4. Assume that $\tau$ is uniformly persistent in $\operatorname {Int} X_+$. Thanks to condition (4.8), theorem 4.5 applies so that the set $M=\{(\omega,\,b(\omega ))\mid \omega \in \Omega \}\gg _B 0$ is the attractor of positive solutions. In particular, there exists a $\psi \gg _B 0$ such that $b(\omega )\geq _B \psi$ for all $\in \Omega$. Then, taking any $\omega \in \Omega$ and $\varphi \gg _B 0$, the semiorbit $\{(\omega {\cdot }t,\,y_t(\omega,\,\varphi ))\mid t\geq 0\}$ approaches $M$ as $t\to \infty$ and therefore, there exists a $t_0>0$ such that $y_t(\omega,\,\varphi )\geq _B \psi /2$ for all $t\geq t_0$. The proof is finished.

Proof. Consider the family of systems (4.3) over the hull $\Omega$, which is minimal and uniquely ergodic because of the almost periodicity assumption (a1). Let $\nu$ be the unique ergodic measure on $\Omega$. Roughly speaking, the idea is to make a joint change of variables which for each $\omega \in \Omega$ takes the almost periodic coefficients $d_i(\omega {\cdot }t)$, $1\leq i\leq m$ into their respective mean values $d_{i0}$, or at least arbitrarily close to them. Recall that, by Birkhoff's ergodic theorem, $d_{i0}=\int _\Omega d_i\,d\nu$ for $1\leq i\leq m$. Consider the Banach space $C_0(\Omega )=\left \{a\in C(\Omega )\mid \int _\Omega a\,d\nu =0\right \}$ of continuous maps on $\Omega$ with null mean value, and its vector subspace $BP(\Omega )=\{ a\in C_0(\Omega ) \mid a \,\text { has a bounded primitive}\}$. In the case of periodic coefficients, $C_0(\Omega )=BP(\Omega )$, whereas, if $\Omega$ is aperiodic, then $BP(\Omega )\subsetneq C_0(\Omega )$ and it is a dense subset of first category. See Campos et al. [Reference Campos, Obaya and Tarallo1, lemma 5.1] for this result in a more general setting, but note that the almost periodic case was already proved by Johnson [Reference Johnson11].

We could skip the first case we are going to consider, but it helps to understand the condition required in (4.11). Let us first assume that the maps $\widehat d_i (\omega ) = d_i(\omega )-d_{i0}$, $\omega \in \Omega$ for $1\leq i\leq m$, which are in $C_0(\Omega )$, admit bounded primitives, that is, there exist maps $h_i\in C(\Omega )$, $1\leq i\leq m$, such that

(4.12)\begin{equation} \int_0^t \widehat d_i (\omega{\cdot}s)\,{\rm d}s = h_i(\omega{\cdot}t)-h_i(\omega) \quad\text{for all}\ \omega\in\Omega\,,\; t\in \mathbb{R}. \end{equation}

Then, in (4.3), for each $\omega \in \Omega$ we make the change of variables $z_i(t)={\rm e}^{h_i(\omega {\cdot }t)}\,y_i(t)$, $1\leq i\leq m$, which preserves the boundedness of the solutions and the property of uniform persistence. The transformed systems have a similar structure to the Nicholson systems, namely, for each $\omega \in \Omega$,

\[ z'_i(t)={-} d_{i0}\,z_i(t) +\displaystyle \sum_{j=1}^m a_{ij}^*(\omega{\cdot}t)\,z_j(t) + \beta_{i}^*(\omega{\cdot}t)\,z_i(t-r_i)\,{\rm e}^{{-}c_i^*(\omega{\cdot}t)\,z_i(t-r_i)} \]

for the new coefficients

\begin{align*} a_{ij}^*(\omega):& = a_{ij}(\omega)\,{\rm e}^{h_i(\omega)-h_j(\omega)}, \\ \beta_{i}^*(\omega):& = \beta_{i}(\omega)\,{\rm e}^{h_i(\omega)-h_i(\omega{\cdot}({-}r_i))}\,,\\ c_{i}^*(\omega):& = c_{i}(\omega)\,{\rm e}^{{-}h_i(\omega{\cdot}({-}r_i))}\,, \end{align*}

and it is easy to check that the transformed system of (4.1) also satisfies (a1)–(a5), although it might not satisfy assumption (a6). However, this is unimportant because, even if this condition has a biological meaning in the original system, its analytical implications—it is basically used to prove the ultimate boundedness of solutions—still hold, by the expression of the change of variables, the fact that $\Omega$ is compact, and $h_i$ are continuous maps.

Therefore, we can apply theorem 4.5 to the new systems, provided that the monotonicity condition for the exponential ordering (4.8) holds, but that is exactly condition (4.11) in the hypotheses. To check it, let $\omega _1\in \Omega$ be the one determining the initial Nicholson system (4.1) and note that now the parameters involved for each $1\leq i\leq m$ are $(d_i^*)^+= d_{i0}$ and

\[ (\beta_i^*)^+ := \sup\left\{ \beta_i^*(\omega_1{\cdot}t) \mid t\in\mathbb{R}\right\} =\sup\left\{ \beta_{i}(\omega_1{\cdot}t)\,{\rm e}^{h_i(\omega_1{\cdot}t)-h_i(\omega_1{\cdot}(t-r_i))}\,\big|\, t\in\mathbb{R}\right\}\,, \]

which can be rewritten as $(\beta _i^*)^+ =\sup \left \{\widetilde \beta _i(t)\,{\rm e}^{\int _{t-r_i}^t \widetilde d_i(s)\,{\rm d}s -d_{i0}r_i} \,\big |\, t\in \mathbb {R}\right \}$, by (4.12). As a consequence, for the transformed systems, there exists a unique minimal set $M^*=\{(\omega,\,b^*(\omega ))\mid \omega \in \Omega \}$ for a continuous map $b^*:\Omega \to \operatorname {Int} X_+$, which is an attractor of the semitrajectory of every pair $(\omega,\,\phi )$ provided that $\phi \geq 0$ and $\phi (0)\gg 0$. Reversing the change of variables, we see that the original systems (4.3) also have a unique minimal set $M=\{(\omega,\,b(\omega ))\mid \omega \in \Omega \}$ for a continuous map $b:\Omega \to \operatorname {Int} X_+$ which is also an attractor of the semitrajectory of every pair $(\omega,\,\phi )$ provided that $\phi \geq 0$ and $\phi (0)\gg 0$.

To finish the proof it remains to assume that some of the maps $\widehat d_i (\omega ) = d_i(\omega )-d_{i0}$, $\omega \in \Omega$, for $1\leq i\leq m$, which are in $C_0(\Omega )$, do not admit a bounded primitive. Then, since the subspace $BP(\Omega )$ is dense in $C_0(\Omega )$ and (4.11) holds, given any $\varepsilon >0$ with the following restrictions: