Precompact Carathéodory ODE Families

• Precompact families of Carathéodory ODEs are sets of nonautonomous differential equations characterized by parametric b-measures and L^p-type topologies that ensure compact closure.
• They employ functional-analytic techniques such as equicontinuity, bounded variation, and the Arzelà–Ascoli theorem to guarantee solution stability and convergence.
• Applications include slow–fast systems and skew-product semiflows, providing averaged dynamics and robust frameworks for analyzing nonautonomous behavior.

Precompact families of Carathéodory ODEs constitute a central class in the analysis of nonautonomous differential equations, functional-analytic compactness properties, and modern dynamical systems frameworks. Recent advances characterize these families via parametric b-measures, $L^p_{\mathrm{loc}}$-type topologies, and generalized skew-product semiflows, enabling new approaches to stability, averaging, and the asymptotics of slow–fast systems (Novo et al., 16 Jan 2026, Longo et al., 2017).

1. Functional-Analytic Structures: Carathéodory Spaces and Parametric b-Measures

Carathéodory ODEs classically involve vector fields $$F \colon \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}^N$$ that are measurable in $t$ and continuous in $x$ for almost every $t$. These are generalized by parametric b-measures, defined as maps $\nu \colon \mathbb{R}^N \to M_c(B)$, where $M_c(B)$ denotes finitely additive set functions on the ring $B$ of bounded Borel subsets of $\mathbb{R}$ satisfying

• $\mu(\varnothing)=0$,
• finitely additive on disjoint unions,
• $\mu(\{t\})=0$ for every $t$.

The total variation $|\mu|$ is given by the Jordan decomposition. Each parametric b-measure is controlled by nondecreasing moduli of continuity $\{\omega_j\}$, with $m$-bounds and $l$-bounds ($m_j$, $l_j \in M_c(B)^+$) satisfying

• $|\nu_y|(A) \le m_j(A)$,
• $$|\nu_{y_1}-\nu_{y_2}|(A) \le l_j(A)\omega_j(|y_1-y_2|)$$.

These constructions situate Carathéodory ODEs in vector spaces $M_p$ of parametric b-measures, enabling integration along curves via limits of Riemann sums, and providing topologies generated by seminorms evaluating variations over tagged partitions or curves of prescribed modulus.

2. Topological Frameworks: $L^p_{\mathrm{loc}}$-type and Moduli-Based Topologies

Three principal Carathéodory function spaces underlie the analytic approach:

• $LC(\mathbb{R}^M)$ ("Lipschitz-Carathéodory"): functions are Borel in $(t,x)$, locally bounded by $m$-bounds, and locally Lipschitz in $x$ (controlled by $l$-bounds).
• $SC(\mathbb{R}^M)$ ("Strong-Carathéodory"): as above, with $x \mapsto f(t,x)$ continuous a.e. $t$.
• $TC_{(\Theta)}(\mathbb{R}^M)$ ("$\Theta$-Carathéodory"): continuity in $x$ in $L^p$-sense, quantified by suitable moduli $\Theta$.

Locally convex, metric $L^p_{\mathrm{loc}}$-type topologies are constructed via countable seminorm families:

• On $TC_{(\Theta)}$, $$p_{I,j}(f) = \sup_{x(\cdot) \in K_j^I} (\int_I |f(t,x(t))|^p dt)^{1/p}$$.
• On $SC$, topologies $T_B$ (over all trajectories), $T_D$ (countable dense points), and $T_\Theta$ ($\Theta$-based moduli).

A hierarchy exists: $T_D \le T_\Theta \le T_B$; all are metrizable, facilitating compactness criteria in these spaces (Longo et al., 2017).

3. Compactness and Precompactness Criteria

The central result for parametric b-measures is: if a family $E \subset M_p$ possesses equicontinuous $m$-bounds and bounded $l$-bounds, then its closure is compact in either topology ($\sigma_D$ or $\sigma_\Theta$) [(Novo et al., 16 Jan 2026), Theorem 3.13]. Compactness follows by:

• Showing boundedness of $l$-bounds equates the topologies on $E$,
• Associating to each $\nu \in E$ a map $F_\nu(t,y) = \int_0^t d\nu_y$, shown to be uniformly equicontinuous,
• Employing Arzelà–Ascoli and bounded variation arguments for subsequential convergence.

In classical $LC$ spaces, precompactness is characterized via the Fréchet–Kolmogorov–Riesz theorem: $E \subset LC$ is relatively compact iff each "section" $\{ t \mapsto f(t,x) : f \in E \}$ is relatively compact in $L^p_{\mathrm{loc}}(\mathbb{R})$ for all $x$ in a countable dense set.

Hull compactness is similarly characterized by compactness and uniform continuity of time translations within $L^p_{\mathrm{loc}}$–valued functions.

4. Skew-Product Semiflows and Nonautonomous Dynamical Systems

For $\nu \in M_p$ with equicontinuous $m$-bounds, translations $\nu \cdot t$ admit a compact hull $\mathrm{Hull}_{(M_p, T)}(\nu)$, equipped with a continuous $\mathbb{R}$–flow $\varphi(t, \mu) = \mu \cdot t$ [(Novo et al., 16 Jan 2026), Theorem 4.1]. Generalized ODEs driven by $\nu$, $y'(t)=\nu_{y(t)}$, are well-posed and exhibit continuous dependence on both hull element and initial data; the resulting skew–product flow on $\mathrm{Hull}(\nu_0) \times \mathbb{R}^N$ remains uniformly continuous.

Any Carathéodory ODE $y' = g(y,t)$ with equicontinuous $m$- and bounded $l$-bounds can be reframed in differential measure form. The family of time-translates is precompact in $M_p$, and classical solutions coincide with generalized ones, allowing representation of the skew–product flow via the hull [(Novo et al., 16 Jan 2026), Proposition 5.3].

Local and global skew-product semiflows in $LC$, $TC_{(\Theta)}$ are obtained when underlying vector fields satisfy the necessary bounds. Linearized skew-product semiflows extend this framework under additional differentiability, with stability and variation-of-constants identities preserved (even for limiting $G$ outside classical smoothness classes) [(Longo et al., 2017), Theorem 6.2].

5. Applications in Slow–Fast Systems and Averaging Behaviour

In slow–fast systems, the dynamics are written in slow time as

$$(\mathrm{S}) \left\{\begin{array}{l} \dot{x} = \epsilon f(x,y,\epsilon), \ \dot{y} = g(x,y), \end{array}\right.$$

or in fast time $\tau = t/\epsilon$ as

$$(\mathrm{F}) \left\{\begin{array}{l} x' = f(x,y,\epsilon), \ y' = g(x,y,\tau). \end{array}\right.$$

Assuming $g(x,y,\tau)$ is Carathéodory in $\tau$, continuous in $(x,y)$, and possesses locally integrable, equicontinuous $m$-bounds and bounded $l$-bounds across balls $B_j^{n+m}$, one constructs parametric b-measures $d(\nu_g)_y^x = g(x,y,\tau) \, d\tau$.

The hull $\mathcal{H} = \mathrm{Hull}(\nu_g)$ is compact, and for each fixed $x$ yields a continuous layer skew–product semiflow with pullback and global attractors $A^x \subset \mathcal{H} \times \mathbb{R}^m$ [(Novo et al., 16 Jan 2026), Proposition 6.3; Theorem 6.4].

As $\epsilon \to 0$, the main limit theorem demonstrates:

• Any vanishing sequence $\epsilon_j \to 0$ has a subsequence with $x_{\epsilon_k} \to x_0$ uniformly, with $x_0$ solving a differential inclusion representing the averaged dynamics over invariant measures,
• The fast variable $y_{\epsilon_k}(\tau)$ "tracks" the nonautonomous attractor, remaining within a prescribed $\delta$-inflated fiber for large times [(Novo et al., 16 Jan 2026), Theorem 6.6].

This establishes a rigorous averaged description for slow variables influenced by complex fast subsystems governed by Carathéodory vector fields.

6. Role and Implications of Compactness

Compactness and precompactness in spaces of Carathéodory ODEs (and their generalizations via parametric b-measures) underpin existence, uniqueness, and stability results. Equicontinuity and boundedness of $m$- and $l$-bounds are vital in ensuring closure properties and compact hulls in appropriate topologies.

Uniform continuity of solutions with respect to the hull element and initial data is a direct consequence, which is essential for nonautonomous dynamics and limit behaviour in perturbed and slow–fast systems.

A plausible implication is that the parametric b-measure perspective can extend to broader classes of nonautonomous equations, providing structure for general dynamical systems that depart from strict regularity assumptions and facilitating deeper analyses in mathematical and applied contexts (Novo et al., 16 Jan 2026, Longo et al., 2017).