Preamble

This work, following the foundational methods established in [1–4] and [7], addresses the qualitative analysis of differential equations in functional spaces. Building upon the results of [5, 6], we examine the properties of solutions within the framework of topological structures. Specifically, we investigate the behavior of solutions to equation (13) as discussed in [8] and [9]. The stability and asymptotic behavior of these solutions are analyzed under various conditions, particularly focusing on the properties of the shift operator and the compactness of trajectories in the function space $\mathcal{F}$, as defined in [15].

We consider a mapping $f$ from $T$ into the space $(X; Y)$, where $t \to f_t$ represents a continuous dependence as outlined in [16]. Let $X$ be a compact space and $Y$ a metric space. We define the space of continuous mappings $(T; Y)$ and $(X; Y)$ with the topology of uniform convergence. Following the definitions in [15], the distance between two functions $\phi$ and $\psi$ is given by:
$$\rho(\phi, \psi) = \sup \min { \rho(\phi(t), \psi(t)), 1 }$$
This metric ensures that the space $(T; Y)$ is complete if $Y$ is complete.

2.1 Stability and Convergence

In the context of the work by M. V. Belyaev [5, 6], we consider the conditions under which a solution $\phi$ is stable. Specifically, for any $\epsilon > 0$, there exists a $\delta > 0$ such that if $\rho(\phi, \psi) < \delta$, then the supremum of the distance between the trajectories remains bounded by $\epsilon$ for all $t$. This leads to the definition of almost-periodic functions in the sense of Bohr and Levitan, as discussed in [17].

We define the operator $G(t, x)$ acting on the domain $D \subset E$. The convergence of the sequence of functions ${h_n}$ to $f$ in the space $(T; (D; B))$ implies that the corresponding solutions $\phi_n$ of the differential equations:
$$x' = h_n(t)x$$
converge to the solution of the limit equation $x' = f(t)x$. As demonstrated in [15, p. 46], if $f$ and $g$ are continuous mappings, the distance between their images under the shift operator satisfies:
$$\rho(f^, g^) = \sup_{|t| < l} \rho(f_t, g_t)$$
This relationship is critical for establishing the existence of almost-periodic solutions for the system (13).

2.2 Asymptotic Properties

Consider the linear system $x' = A(t)x + a(t)$. According to the results in [19] and [20], if the coefficient matrix $A(t)$ and the vector function $a(t)$ satisfy certain regularity conditions, the existence of a bounded solution on the interval $[t_0, +\infty)$ implies the existence of an almost-periodic solution. This is further supported by the fixed-point theorems applied to the operator $T$ in the space of continuous functions.

The analysis of the equation $x' = (x-1)x$ for $0 < x < 1$ serves as a representative example of the behavior described in Theorem 3. While the conditions for stability are met, the global behavior of the trajectory depends heavily on the initial values and the compactness of the set $D = \phi_0(J)$.

References

1. Bohr, H. Fastperiodische Funktionen. Berlin, 1932.
2. Levitan, B. M. Almost Periodic Functions. Moscow, 1953.
3. Amerio, L. Sull'equazione differenziale non lineare di Riccati. Acta Math., 1950.
4. Gottschalk, W. H., and Hedlund, G. A. Topological Dynamics. Amer. Math. Soc. Colloquium Publications, 1955.
5. Belyaev, M. V. On the stability of motion. Prikl. Mat. Mekh., 1940.
6. Belyaev, M. V. Almost periodic solutions of differential equations. 1955.
7. Miller, R. K. Almost periodic differential equations. Jour. of Diff. Equations, 1, No. 3, 1965.
8. Arens, R. F. Topologies for spaces of transformations. Ann. of Math., 47, No. 3, 1946.
9. Bourbaki, N. Topologie générale, Chapter 10: Function Spaces. Hermann, Paris, 1953.
10. Shcherbakov, B. A. On the classification of motions in topological dynamics. Dokl. Akad. Nauk SSSR, 1966.
11. Bohr, H., and Neugebauer, O. Über lineare Differentialgleichungen mit konstanten Koeffizienten und fastperiodischer rechter Seite. Nachr. Ges. Wiss. Göttingen, 1926.