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Development of the new approach for existence of bounded solutions for point-type functional differential equations L. A. Beklaryanab a Central Economics and Mathematics Institute of the Russian Academy of Sciences, Moscow b Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Region
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    IntroductionSoliton solutions have been the object of intensive study in recent years [3]–[25]. Some early works establish relations between the theories of soliton solutions with those of solutions to functional differential equations [18]–[25]. For an account of the theory of functional differential equations, we refer the reader to [7], [26]–[29]. In [7]–[9], a formalism for soliton wedges was proposed, within which there is a correspondence between the soliton solutions of infinite-dimensional dynamical systems and the solutions of the corresponding point-type functional differential equations from the function–operator dual pair induced by the soliton wedge. This dualism is based on the group property of a soliton wedge, which secures the existence of a representative family of soliton solutions. A theorem for existence and uniqueness of a soliton solution and a solution of the corresponding functional differential equation for an arbitrary soliton wedge was established in the most general case. In the particular case where, for a canonical soliton wedge, the deviations of the argument in the functional differential equation are given by elements of a cyclic shift group, a proof can be found in the book [7]. § 1. Some preliminary concepts and resultsFollowing [8] and [9], we recall some definitions and results. In these works, a soliton wedge is defined in terms of the well-known Carathéodory conditions; there also appear the Carathéodory condition in the strong sense, which are required for the purposes of differential and integral calculus for infinite-dimensional vector functions. Definition 1. A set of constructions $(\Upsilon,d,s,\eta, G_\Gamma\,|\, Q,g)$, which is called a soliton wedge, is defined by a tuple $\Gamma=(\Upsilon,d,s, \eta,Q,g) $, where: 1) $\Upsilon$ is the finitely generated group with generators $\{\check{\gamma_1},\dots,\check{\gamma_d}\}$ and selected elements $\{\gamma_1,\dots,\gamma_s\}$, and
$$
\begin{equation*}
\mathcal K^n_\Upsilon=\overline{\prod}_{\gamma\in \Upsilon} R_\gamma,\qquad R_\gamma= \mathbb R^n,
\end{equation*}
\notag
$$
is the corresponding space of infinite sequences $\varkappa=\{x_\gamma\}_{\gamma\in \Upsilon}$, $x_\gamma\in \mathbb R^n$, $x_\gamma=(x_{\gamma 1},\dots, x_{\gamma n})'$, equipped with the standard topology of complete direct product; 2) $\mathbb T_\Upsilon=\{T_{\widehat \gamma}\colon \widehat\gamma \in \Upsilon\}$ is the finitely generated shift group acting in the space $\mathcal K^n_\Upsilon$ by
$$
\begin{equation*}
T_{\widehat{\gamma}} \{x_\gamma\}_{\gamma \in \Upsilon }=\{x_{\gamma\widehat{\gamma}}\}_{\gamma \in \Upsilon},\qquad \widehat{\gamma}\in \Upsilon,\quad \{x_\gamma\}_{\gamma \in \Upsilon}\in \mathcal K^n_\Upsilon,\quad T_{\widehat{\gamma}} \in \mathbb T_\Upsilon;
\end{equation*}
\notag
$$
3) $\eta\colon \Upsilon \to Q$ is an epimorphism, where $Q$ is the group of orientation preserving diffeomorphisms of the real line, that is, $Q\subset \mathrm{Diff}^{(1)}_+(\mathbb R)$, and, respectively, $Q$ is the finitely generated group with generators $\check{q_j}=\eta(\check{\gamma_j})$, $j=1,\dots,d$, and selected elements $q_j=\eta({\gamma_j})$, $j=1,\dots,s$; 4) $g\colon\mathbb R\times \mathbb R^{ns} \to \mathbb R^{n}$ satisfies the Carathéodory conditions; 5) the operator
$$
\begin{equation*}
G_{\Gamma}\colon \mathbb R\times \mathcal K^n_\Upsilon \to \mathcal K^n_\Upsilon,\qquad \Gamma=(\Upsilon,d,s,\eta,Q,g),
\end{equation*}
\notag
$$
is such that the coordinate $(G_{\Gamma}(t,\varkappa))_e$ of the infinite-dimensional vector function $G_{\Gamma}(t,\varkappa)$ corresponding to the unit element $e$ of the group $\Upsilon$ depends only on the finite number of coordinates and is equal to
$$
\begin{equation*}
\bigl(G_{\Gamma}(t,\varkappa)\bigr)_e=g(t,x_{{\gamma}_1},\dots,x_{{\gamma}_s});
\end{equation*}
\notag
$$
6) for almost each $t \in \mathbb R $ , the “almost commutative” relations
$$
\begin{equation*}
T_{\widehat\gamma} G_{\Gamma}(t,\varkappa)=\frac{d}{dt}\, \eta(\widehat\gamma)(t)\cdot G_{\Gamma}\bigl(\eta(\widehat\gamma)(t),T_{\widehat\gamma}\varkappa\bigr) \quad \forall\, \varkappa \in \mathcal K^n_\Upsilon,\quad \forall\, \widehat\gamma\in \Upsilon,
\end{equation*}
\notag
$$
hold. For any soliton wedge $(\Upsilon,d,s,\eta, G_\Gamma\,|\, Q,g)$ with $\Gamma=(\Upsilon,d,s, \eta,Q,g) $, consider the system
$$
\begin{equation}
\dot{\varkappa}(t)=G_\Gamma(t,\varkappa)\quad \text {for a.a. }\ t\in \mathbb R,
\end{equation}
\tag{1}
$$
$$
\begin{equation}
\varkappa(\eta(\widehat{\gamma})(t))=T_{\widehat{\gamma}}\varkappa(t)\quad \forall\, t, \widehat{\gamma},\quad t\in \mathbb R, \quad \widehat{\gamma}\in \Upsilon,
\end{equation}
\tag{2}
$$
in the phase space ${\mathcal K}^n_\Upsilon$ with phase variable $\varkappa \in {\mathcal K}^n_\Upsilon$. The derivative in the infinite-dimensional differential equation (1) is understood in the Gâteaux sense. A vector function $\varkappa\colon[a,b]\to \mathcal K^n_\Upsilon$, $\varkappa(\,{\cdot}\,) =\{x_\gamma(\,{\cdot}\,)\}_{\gamma \in \Upsilon}$, is called weakly differentiable at $t\in [a,b]$ if, for each, $\gamma \in \Upsilon$, the function $x_\gamma(\,{\cdot}\,)$ is differentiable at $t$. It is clear that such a function is weakly differentiable at $t\in [a,b]$ if and only if it is Gâteaux differentiable. A vector function $\varkappa\colon [a,b]\to \mathcal K^n_\Upsilon$, $\varkappa(\,{\cdot}\,)=\{x_\gamma(\,{\cdot}\,)\}_{\gamma \in \Upsilon} $, is weakly absolutely continuous (weakly measurable) if, for each, $\gamma \in \Upsilon$, the function $x_\gamma(\,{\cdot}\,)$ is absolutely continuous (measurable) on $[a,b]$. A solution of system (1), (2) is a weakly absolutely continuous vector function $\varkappa\colon\mathbb R \to \mathcal K^n_\Upsilon$, $\varkappa(\,{\cdot}\,)=\{x_\gamma(\,{\cdot}\,)\}_{\gamma \in \Upsilon} $, which satisfies (1) almost everywhere and obeys constraint (2) for each $t\in \mathbb R$. The non-local constraints (2) mean that the space shift of any solution of the system is equal to its time shift. Any solution of this system is called a travelling wave type solution (a soliton solution), and the group $Q$ is the characteristic of the travelling wave. For a soliton wedge $(\Upsilon,d,s,\eta, G_\Gamma\,|\, Q,g)$ with $\Gamma=(\Upsilon,d,s, \eta,Q,g) $, in parallel to system (1), (2), one considers the induced point-type functional differential equation
$$
\begin{equation}
\dot{x}(t)=g\bigl(t,x(q_1(t)),\dots,x(q_s(t))\bigr),\qquad t\in \mathbb R.
\end{equation}
\tag{3}
$$
Any soliton wedge $(\Upsilon,d,s,\eta, G_\Gamma\,|\, Q,g)$ with tuple $\Gamma=(\Upsilon,d,s, \eta,Q,g) $ induces the function–operator dual pair $(G_\Gamma\,|\, Q,g)$ and also the canonical soliton wedge $(Q,d,s, \mathcal I,G_\Gamma\,|\,Q,g)$, $\Gamma=(Q,d,s,\mathcal I,Q,g)$, with the canonical dual pair $(G_\Gamma\,|\,Q,g)$, where $\mathcal I$ is the identity automorphism of the group $Q$. Given $Q$, $g$ (a given functional differential equation with right-hand side $g(\,{\cdot}\,)$ and deviations of the argument $q_1(\,{\cdot}\,),\dots,q_s(\,{\cdot}\,)\in Q$), the canonical soliton wedge features the most simple structure of $\Gamma=(Q,d,s, \mathcal I,Q,g)$ and, respectively, of the operator $G_\Gamma$. For a soliton wedge $(\Upsilon,d,s,\eta, G_\Gamma\,|\, Q,g)$ with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, to each solution $x(t)$, $t\in \mathbb R$, of the point-type functional differential equation (3) there corresponds a solution ${\varkappa}(t)=\{x_{{\gamma}}(t)\}_{{\gamma}\in \Upsilon}$, $t \in \mathbb R$, of the infinite-dimensional system (1), (2) (a soliton solution), and vice versa. These solutions are related as follows:
$$
\begin{equation*}
x_{\gamma}(t)=x_e(\eta(\gamma)(t)),\quad x_e(t)=x(t)\qquad \forall\, t,\gamma, \quad t\in \mathbb R,\quad \gamma\in \Upsilon.
\end{equation*}
\notag
$$
The space $\mathcal K^n_\Upsilon$ is only metrizable, and so the study of the space soliton solutions (the space of solutions of system (1), (2)) is a quite involved problem. A more advanced consideration requires a localization of the space of soliton solutions based on certain facts, and, in particular, on the consideration of the spatial asymptotics of the solutions, which are considered as paths in the infinite-dimensional sequence space $\mathcal K^n_\Upsilon$. In this way, we define the family of Banach subspaces $\mathcal K^n_{\Upsilon \infty\mu \check{h}}$, $\mu \in (0,1]$, $\check{h}\in\,[0,+\infty)$, as follows:
$$
\begin{equation*}
\begin{gathered} \, \mathcal K^n_{\Upsilon \infty\mu \check{h}}=\Bigl\{\varkappa\colon \varkappa\in \mathcal K^n_{\Upsilon};\ \sup_{\gamma\in \Upsilon}\|x_\gamma\|_{\mathbb R^n}\mu^{l(\gamma)\check{h}}<+\infty\Bigr\}, \\ \|\varkappa\|_{\Upsilon \infty\mu \check{h}}=\sup_{\gamma\in \Upsilon} \|x_\gamma\|_{\mathbb R^n} \mu^{l(\gamma)\check{h}}, \end{gathered}
\end{equation*}
\notag
$$
where $l(\gamma)$ is the length of a word $\gamma \in \Upsilon$ in the finitely generated group $\Upsilon$ (see [30]). The parameter $\check{h}$ in these Banach spaces is defined in terms of the group $Q$ from the soliton wedge under consideration as follows:
$$
\begin{equation*}
\check{h}=\max_{j\in \{1,\dots,d\}}\,h_{\check{q}_j},\quad h_{\check{q}_j}=\sup_{t\in \mathbb R}|t-\check{q}_j(t)|,\qquad j=1,\dots,d,
\end{equation*}
\notag
$$
where $\check{q}_j$, $j=1,\dots,d$, are the generators of the group $Q$. Let us formulate additional conditions on the function $g$ in terms of the growth rate and a condition for the group of diffeomorphisms $Q$ from the soliton wedge under consideration: ($\mathrm a^*$) the function $g\colon\mathbb R\times \mathbb R^{ns}\to \mathbb R^n$ satisfies the Carathéodory conditions; ($\mathrm b^*$) for all $z_j$, $\overline{z}_j$, $j=1,\dots,s$, and almost all $t$, the growth rate condition
$$
\begin{equation*}
\| g(t,z_1,\dots,z_s)\|_{\mathbb R^n}\leqslant M_0(t)+M_1\sum_{j=1}^s \|z_j\|_{\mathbb R^n}
\end{equation*}
\notag
$$
and the Lipschitz condition
$$
\begin{equation*}
\begin{gathered} \, \| g(t,z_1,\dots,z_s)-g(t,\overline z_1,\dots,\overline z_s)\|_{\mathbb R^n}\leqslant M_2\sum_{j=1}^s \|z_j-\overline z_j\|_{\mathbb R^n}, \\ M_0(t)=\|g(t,0,\dots,0)\|_{\mathbb R^n},\quad t\in \mathbb R, \qquad M_1=M_2, \end{gathered}
\end{equation*}
\notag
$$
hold; ($\mathrm c^*$) there exists $\mu^*\in (0,1]$ such that
$$
\begin{equation*}
\begin{gathered} \, M_0(\,{\cdot}\,)\in \mathcal L^1_{\mu^\ast}L_\infty(\mathbb{R}), \\ \mathcal L^1_\mu L_\infty(\mathbb R)=\bigl\{x(\,{\cdot}\,)\colon x(t)\mu^{|t|}=y(t),\, t\in \mathbb R;\, y(\,{\cdot}\,) \in L_\infty(\mathbb{R},\mathbb{R}) \bigr\}; \end{gathered}
\end{equation*}
\notag
$$
($\mathrm d^*$) there is a group of diffeomorphisms $Q$ with system of generators $\check{q_1},\dots,\check{q_d}$ and selected elements $q_1,\dots,q_s\in Q$ such that, for each $q\in Q$, the quantities
$$
\begin{equation*}
h_{q}=\sup_{t\in \mathbb R}|t-q(t)|,\qquad h_{\dot{q}}=\sup_{t\in \mathbb R} \dot{q}(t)
\end{equation*}
\notag
$$
are all finite, the family $h_{\dot{q}}$, $q\in Q$, is uniformly bounded, and hence the quantities
$$
\begin{equation*}
\check{h}=\max_{j\in \{1,\dots,d\}}\, \sup_{t\in \mathbb R}|t-\check{q}_j(t)|, \qquad h_{\dot{Q}}=\sup_{q\in Q} h_{\dot{q}}
\end{equation*}
\notag
$$
are finite. All the above conditions for the function $g$, which is the right-hand side of the differential equation, are standard in the theory of differential equations. The right-hand side $g(\,{\cdot}\,)$ of the point-type functional differential equation (3) is an element of the Banach function space $V_{\mu^*}(\mathbb R\times \mathbb R^{ns},\mathbb R^n)$ with the Lipschitz norm
$$
\begin{equation*}
\begin{gathered} \, V_{\mu^*}(\mathbb R\times \mathbb R^{ns},\mathbb R^n)=\{ g(\,{\cdot}\,)\colon g(\,{\cdot}\,) \text{ satisfies conditions $(\mathrm a^*){-}(\mathrm c^*)$}\}, \\ \begin{aligned} \, &\|g(\,{\cdot}\,)\|_{\mathrm{Lip}}=\sup_{t\in \mathbb R}\| g(t, 0,\dots,0)(\mu^*)^{|t|}\|_{\mathbb R^n} \\ &\qquad+\sup_{(t,z_1,\dots,z_s,\overline z_1,\dots,\overline z_s)\in \mathbb R^{1+2ns}}\frac{\| g(t,z_1,\dots,z_s)-g(t,\overline z_1,\dots,\overline z_s)\|_{\mathbb R^n}}{\sum_{j=1}^s \| z_j-\overline z_j\|_{\mathbb R^n}}, \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
where the parameter $\mu^*\in \mathbb R_+$ coincides with the corresponding constant from condition $(\mathrm c^*)$. In the framework of the dualism between the soliton solutions and the solutions of the corresponding point-type functional differential equation (3), we will use the solutions of the latter equations; we will also find characteristics of the equation in parametric families of Banach subspaces (weighted function spaces)
$$
\begin{equation*}
\begin{aligned} \, {\mathcal L}_{\mu}^nC^{(0)}(\mathbb R) &=\Bigl\{ x(\,{\cdot}\,)\colon x(\,{\cdot}\,)\in C^{(0)}(\mathbb R, \mathbb R^n),\, \sup_{t\in \mathbb R} \|x(t)\mu^{|t|}\|_{\mathbb R^n} < +\infty\Bigr\}, \\ \mathcal L^n_\mu L_\infty(\mathbb R) &=\Bigl\{x(\,{\cdot}\,)\colon x(t)\mu^{|t|}=y(t),\, t\in \mathbb R;\, y(\,{\cdot}\,) \in L_\infty(\mathbb{R},\mathbb{R}^n) \Bigr\}, \end{aligned}
\end{equation*}
\notag
$$
with parameter $\mu \in (0,1]$ and the norms
$$
\begin{equation*}
\| x(\,{\cdot}\,)\|_{\mu}^{(0)}=\sup_{t\in \mathbb R} \| x(t)\mu^{|t|}\|_{\mathbb R^n},\qquad \|x(\,{\cdot}\,)\|_\mu=\operatorname*{vrai\,sup}_{t\in \mathbb R}\|x(t)\mu^{|t|}\|_{\mathbb R^n}.
\end{equation*}
\notag
$$
We will prove the growth estimate and the Lipschitz property
$$
\begin{equation*}
\begin{gathered} \, \|G_{\Gamma \infty\mu \check{h}}(t,\varkappa )\|_{\Upsilon \infty\mu \check{h}}\leqslant \mathcal M_{0\Upsilon \infty\mu \check{h}}(t)+\mathcal M_{1\Upsilon \infty\mu \check{h}}\|\varkappa\|_{\Upsilon \infty\mu \check{h}}, \\ \|G_{\Gamma \infty\mu \check{h}}(t,\varkappa )-G_{\Gamma \infty\mu \check{h}}(t,\overline\varkappa )\|_{\Upsilon \infty\mu \check{h}}\leqslant \mathcal M_{2\Upsilon \infty\mu \check{h}}\|\varkappa -\overline\varkappa\|_{\Upsilon \infty\mu \check{h}} \end{gathered}
\end{equation*}
\notag
$$
for the operator $G_{\Gamma \infty\mu h}=G_\Gamma|_{\mathbb R\times\mathcal K^n_{\Upsilon \infty\mu h}}$ on the space $\mathcal K^n_{\Upsilon \infty\mu \check{h}}$, where
$$
\begin{equation*}
\begin{gathered} \, \mathcal M_{0\Upsilon \infty\mu \check{h}}(t)=h_{\dot{Q}}\sup_{\gamma\in \Upsilon} M_0(\eta(\gamma)(t))\mu^{l(\gamma)\check{h}},\qquad \mathcal M_{0\Upsilon \infty\mu \check{h}}(\,{\cdot}\,) \in \mathcal L^1_{\mu^*}L_\infty(\mathbb{R}), \\ \mathcal P_{\Upsilon \mu \check{h}}=\sum_{j=1}^s \mu^{-l(\gamma_j)\check{h}},\qquad \mathcal M_{1\Upsilon \infty\mu \check{h}}= \mathcal M_{2\Upsilon \infty\mu \check{h}}=h_{\dot{Q}}M_2\mathcal P_{\Upsilon \mu \check{h}}. \end{gathered}
\end{equation*}
\notag
$$
Given $\mu\in (0,1]$, $\check{h}\in [0,+\infty)$, in parallel with system (1), (2) in the original phase space $\mathcal K^n_\Upsilon$, we consider the system
$$
\begin{equation}
\frac{d\varkappa(t)}{dt}=G_{\Gamma \infty \mu \check{h}}(t,\varkappa)\quad \text {for a.a. }\ t\in \mathbb R,
\end{equation}
\tag{4}
$$
$$
\begin{equation}
\varkappa(\eta(\widehat{\gamma})(t))=T_{\widehat{\gamma}}\varkappa(t)\quad \forall\, t, \widehat{\gamma}, \quad t\in \mathbb R, \quad \widehat{\gamma}\in \Upsilon,
\end{equation}
\tag{5}
$$
in the space $\mathcal K^n_{\Upsilon \infty\mu \check{h}}$. This system is considered as the differential equation (4) with non-local constraints (5). Here, the differential equation (4) is understood in the strong sense (with the Fréchet derivative), rather in the weak sense (the Gâteaux derivative), as in (1). According to Lemma 3 in [8], the operator $G_{\Gamma \infty \mu \check{h}}$ is strongly measurable. Hence, for each strongly measurable (and, a fortiori, continuous) vector function $\varkappa\colon\mathbb R \to \mathcal K^n_{\Upsilon \infty\mu \check{h}}$, the superposition $G_{\Gamma \infty \mu \check{h}}(t,\varkappa(t))$, $t\in \mathbb R$, is a strongly measurable Bochner integrable vector function. By a solution of this infinite-dimensional differential equation with phase space $\mathcal K^n_{\Upsilon \infty\mu \check{h}}$, one means any almost everywhere strongly absolutely continuous vector function $\varkappa(t)$, $t\in \mathbb R$ (a continuous vector function which is Fréchet differentiable almost everywhere with Bochner integrable derivative $ \dot{\varkappa}(\,{\cdot}\,)$; the vector function $\varkappa(\,{\cdot}\,)$ is uniquely recovered from the derivative $\dot{\varkappa}(\,{\cdot}\,)$), satisfying this equation. For this infinite-dimensional differential equation (4), there is a theorem on existence and uniqueness of a solution of the Cauchy problem, and the spatial asymptotics of the solutions depends on the parameter $\mu$. An important problem here is to also find the time asymptotics of the solution of system (4), (5) (a soliton solution). In this way, Proposition 1 in [8] puts forward, for a solution $\varkappa(\,{\cdot}\,)$ of the infinite-dimensional differential equation (4), the time asymptotics defined by the embedding
$$
\begin{equation*}
\rho(\,{\cdot}\,) \in \mathcal L^1_{\widehat\mu} C^{(0)}(R),\qquad \rho(t)=\|\varkappa(t)\|_{\Upsilon \infty\mu \check{h}},\quad t \in \mathbb R, \quad \widehat{\mu}= \min \bigl\{\mu^*,\,e^{-\mathcal M_{2\Upsilon p\mu \check{h}}}\bigr\}.
\end{equation*}
\notag
$$
Theorem 1 in [8] establishes a one-one correspondence between the entire space of soliton solutions ${\varkappa}(\,{\cdot}\,)\in C^{(0)}( \mathbb R,\mathcal K^n_{\Upsilon })$ (the solutions of system (1), (2)) and the corresponding solutions $x(\,{\cdot}\,)\in C^{(0)}(\mathbb R,\mathbb R^n)$ of the point-type functional differential equation (3) in the form of a dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$, where $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $x_{\gamma}(t)=x_e(\eta(\gamma)(t))$, $x_e(t)=x(t)$, $\gamma\in \Upsilon$, $t\in \mathbb R$. For a subfamily of such dual pairs of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$, that is, $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $x_{\gamma}(t)=x_e(\eta(\gamma)(t))$, $x_e(t)=x(t)$, $\gamma\in \Upsilon$, $t\in \mathbb R$, consisting of the solutions of the point-type functional differential equation (3) and the solutions of system (4), (5) (the soliton solutions), Theorem 2 in [8] refines the time asymptotics for the former solutions and gives a spatial and time asymptotics for the latter solutions. For each solution,
$$
\begin{equation}
x(\,{\cdot}\,)\in \mathcal L^n_{\overline \mu} C^{(0)}(R),\qquad \overline\mu\in (0,1],
\end{equation}
\tag{6}
$$
of the point-type functional differential equation (3) in the dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$, that is, $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $x_{\gamma}(t)=x_e(\eta(\gamma)(t))$, $x_e(t)=x(t)$, $\gamma\in \Upsilon$, $t\in \mathbb R$, the vector function $\varkappa(\,{\cdot}\,)$ for each $\mu\leqslant \min\{\mu^*,\,\overline{\mu}\}$ is a soliton solution $\varkappa(t)\in \mathcal K^n_{\Upsilon \infty\mu \check{h}}$, $t \in \mathbb R$, with time asymptotics
$$
\begin{equation}
\rho(\,{\cdot}\,) \in \mathcal L^1_{\overline\mu} C^{(0)}(R),\qquad \rho(t)=\|\varkappa(t)\|_{\Upsilon \infty\mu \check{h}},\quad t \in \mathbb R.
\end{equation}
\tag{7}
$$
Conversely, for each soliton solution ${\varkappa}(t)\in \mathcal K^n_{\Upsilon \infty\mu \check{h}}$, $t \in \mathbb R$, $\mu\in (0,\mu^*]$, with time asymptotics
$$
\begin{equation*}
\rho(\,{\cdot}\,) \in \mathcal L^1_{\overline\mu} C^{(0)}(R),\qquad \rho(t)=\|\varkappa(t)\|_{\Upsilon \infty\mu \check{h}},\quad t \in \mathbb R,\quad \overline\mu \in (0,1],
\end{equation*}
\notag
$$
in the dual pair $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$ of solutions, that is, $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $x_{\gamma}(t)=x_e(\eta(\gamma)(t))$, $x_e(t)=x(t)$, $\gamma\in \Upsilon$, $t\in \mathbb R$, the vector function $x(\,{\cdot}\,)$ is a solution of the point-type functional differential equation (3) with time asymptotics $x(\,{\cdot}\,)\in \mathcal L^n_{ \overline\mu} C^{(0)}(R)$. In this regard, one of the most principal problems here is to refine the range of $\overline \mu$ as a function of $\mu$. On the other hand, in Theorem 4 of [8] it was verified that each solution of system (4), (5) (a soliton solution), for each fixed initial value $\overline{t}\in \mathbb R$, is a solution of the infinite-dimensional boundary-value problem
$$
\begin{equation}
\dot{\varkappa}(t)= G_{\Gamma \infty \mu \check{h}}(t,\varkappa)\quad \text {for a.a. }\ t\in \mathbb R,
\end{equation}
\tag{8}
$$
$$
\begin{equation}
\varkappa(\eta(\check{\gamma}_k)(\overline{t}))=T_{\check{\gamma}_k}\varkappa(\overline{t}),\qquad k=1,\dots,d,\quad \overline{t}\in \mathbb R,
\end{equation}
\tag{9}
$$
and vice versa. So, the study of the soliton solutions for system (4), (5) is equivalent to that of the involved but quite transparent boundary-value problem (8), (9). 1.1. The principal infinite-dimensional initial-boundary value problem and infinite-dimensional integral equationFollowing [9], we give some constructions and results. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, where the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$ and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*$). In the phase space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$, $\mu \in (0,\mu^*]$, $\check{h}\in [0,+\infty)$, for a given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, and a coordinate index $ \overline \gamma\in \Upsilon$, consider the principal infinite-dimensional initial-boundary value problem
$$
\begin{equation}
\dot{\varkappa}(t)= G_{\Gamma\infty \mu \check{h}}(t,\varkappa) \quad \text {for a.a. }\ t\in \mathbb R,
\end{equation}
\tag{10}
$$
$$
\begin{equation}
\varkappa(\eta(\check\gamma_k)(\overline{t}))=T_{\check\gamma_k}\varkappa(\overline{t}),\qquad k=1,\dots,d,
\end{equation}
\tag{11}
$$
$$
\begin{equation}
(\varkappa(\overline{t}))_{\overline{\gamma}}=\overline x,\qquad \overline{t}\in \mathbb R,\quad \overline{x} \in \mathbb R^n, \quad \overline{\gamma} \in \Upsilon.
\end{equation}
\tag{12}
$$
A solution for equation (10) was defined in the previous section, where it was also noted that, in the phase space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$, the soliton solutions are precisely the solutions of the infinite-dimensional boundary value problem (10), (11), and, therefore, are solutions of the principal infinite-dimensional initial-boundary value problem (10)–(12) for various initial values $ \overline x\in \mathbb R^n$. For a given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, and a coordinate index $ \overline \gamma\in \Upsilon$, consider the corresponding principal infinite-dimensional integral equation
$$
\begin{equation}
\begin{aligned} \, \varkappa(t) &=\biggl[\widetilde{S}_{\Upsilon\infty \mu \check{h}}\overline x - \sum_{\gamma \in \Upsilon} P_\gamma\int^{\eta(\gamma^{-1}\overline \gamma)(\overline{t})}_{\overline{t}} G_{\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau \biggr] \nonumber \\ &\qquad+\int^t_{\overline{t}} G_{\Gamma \infty \mu \check{h}}(\tau,\varkappa(\tau))\,d\tau,\qquad t\in \mathbb R, \end{aligned}
\end{equation}
\tag{13}
$$
where $ P_\gamma$ is the projection in the Banach space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$, for which $\operatorname{Im} P_\gamma$ coincides with the subspace generated by the coordinate $x_\gamma$, and $\widetilde{S}_{\Upsilon\infty \mu \check{h}}\colon \mathbb R^n \to \mathcal K^n_{\Upsilon \infty \mu \check{h}}$ is the operator defined by
$$
\begin{equation*}
(\widetilde{S}_{\Upsilon\infty \mu \check{h}}\overline x)_\gamma=\overline x\quad \forall\, \overline x,\gamma,\quad \overline x\in \mathbb R^n,\quad \gamma\in \Upsilon.
\end{equation*}
\notag
$$
Definition 2. For a given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, and a coordinate index $ \overline \gamma\in \Upsilon$, by a solution of the principal infinite-dimensional integral equation (13) we will mean any continuous vector function $\varkappa:\mathbb R \to \mathcal K^n_{\Upsilon \infty \mu \check{h}}$ satisfying this equation. There exist several canonical transformations for the space of soliton wedges. Let us defined one of such transformations. Definition 3. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, where the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$, а group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*$). For a shift-time diffeomorphism $\theta(t)=t+\xi$, $t\in \mathbb R$, $\theta(\,{\cdot}\,)\in \mathrm{Diff}_+^{(1)}(\mathbb R)$, consider the soliton wedge $(\Upsilon, d,s,\eta_\theta, G_{\Gamma_\theta}\,|\, Q_\theta,g_\theta)$ with $\Gamma_\theta=(\Upsilon,d,s,\eta_\theta,Q_\theta,g_\theta)$, where Along with the infinite-dimensional initial-boundary value problem (10)–(12), we will consider the initial-boundary value problem
$$
\begin{equation}
\dot{\varkappa}(t)= G_{\Gamma_\theta \infty \mu_\theta \check{h}_\theta}(t,\varkappa)\quad \text{for a.a. } \ t\in \mathbb R,
\end{equation}
\tag{14}
$$
$$
\begin{equation}
\varkappa(\eta_\theta(\check\gamma_k)(\overline{t}_\theta))=T_{\check\gamma_k}\varkappa(\overline{t}_\theta),\qquad k\in \{1,\dots,d\},
\end{equation}
\tag{15}
$$
$$
\begin{equation}
(\varkappa(\overline{t}_\theta))_{\overline{\gamma}}=\overline x,\qquad \overline{t}_\theta \in \mathbb R,\quad \overline{x} \in \mathbb R^n, \quad \overline{\gamma} \in \Upsilon.
\end{equation}
\tag{16}
$$
Lemma 1. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma\,{=}\,(\Upsilon,d,s,\eta,Q,g)$, where the function $g\colon\mathbb R \,{\times}\, \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$ and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*$). Then, for the time shift $\theta(t)= t+\xi$, the soliton wedge $(\Upsilon, d,s,\eta_\theta, G_{\Gamma_\theta}\,|\, Q_\theta,g_\theta)$ with $\Gamma_\theta=(\Upsilon,d,s,\eta_\theta,Q_\theta,g_\theta)$ satisfies In view of Lemma 1 it can be assumed without loss of generality that in the principal infinite-dimensional initial-boundary value problem (10)–(12) the initial time is zero, that is, $\overline t=0$. In the space of continuous vector functions $C^{(0)}(\mathbb R,\mathcal K^n_{\Upsilon \infty \mu \check{h}})$, consider the subspace of vector functions
$$
\begin{equation*}
C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{\Upsilon \infty \mu \check{h}})=\Bigl\{\varkappa(\,{\cdot}\,)\colon \varkappa(\,{\cdot}\,)\in C^{(0)}(\mathbb R,\mathcal K^n_{\Upsilon \infty \mu \check{h}}), \, \sup_{t\in \mathbb R} \|\varkappa(t)\|_{\Upsilon \infty \mu \check{h}} \vartheta^{|t|}<+\infty\Bigr\}
\end{equation*}
\notag
$$
equipped with the norm
$$
\begin{equation*}
\|\varkappa(\,{\cdot}\,)\|_{\Upsilon \infty \mu \check{h} C_{\vartheta}^0}=\sup_{t\in \mathbb R} \|\varkappa(t)\|_{\Upsilon \infty \mu \check{h}} \vartheta^{|t|}.
\end{equation*}
\notag
$$
We are interested with continuous vector functions $\varkappa\colon \mathbb R \to \mathcal K^n_{\Upsilon \infty p \check{h}}$ satisfying the principal infinite-dimensional integral equation (13). It is clear that each such a solution satisfies the infinite-dimensional differential equation(10). As noted in the previous section, the solution $\varkappa(\,{\cdot}\,)$ of this infinite-dimensional differential equation has the time asymptotics
$$
\begin{equation}
\rho(\,{\cdot}\,) \in \mathcal L^1_{\widehat\mu} C^{(0)}(R),\qquad \rho(t)=\|\varkappa(t)\|_{\Upsilon \infty \mu \check{h}},\quad t \in \mathbb R,\quad \widehat{\mu}=\min \bigl\{\mu^*,\,e^{-\mathcal M_{2\Upsilon \infty \mu \check{h}}}\bigr\}
\end{equation}
\tag{17}
$$
(an embedding); that is, this solution lies in the Banach space of continuous vector functions $C^{(0)}_{\widehat{\mu}}(\mathbb R,\mathcal K^n_{\Upsilon \infty \mu \check{h}})$. In what follow, we will need another transformation of a soliton wedge based on the following renormalization of the phase space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$: for each $r=0,1,\dots$, by $\mathcal K^n_{r\Upsilon \infty \mu \check{h}}$ we denote the space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$ equipped with the norm
$$
\begin{equation*}
\|\varkappa\|_{r\Upsilon \infty\mu \check{h}}=\sup_{\gamma \in \Upsilon} \|x_\gamma\|_{R^n}\mu^{l(\gamma)\check{h}}\mu^{r\check{h}}.
\end{equation*}
\notag
$$
There is a natural isomorphism $\mathcal K^n_{r\Upsilon \infty \mu \check{h}} \cong \mathcal K^n_{\Upsilon \infty \mu \check{h}}$ between these spaces; the norm of $C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r\Upsilon \infty \mu \check{h}})$ is denoted by $\|\varkappa(\,{\cdot}\,)\|_{r\Upsilon \infty\mu \check{h} C^{(0)}_{\nu}}$. Definition 4. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, where the function $g\colon \mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$ and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*$). For $r\,{=}\,0,1,\dots$, by $(\Upsilon, d,s,\eta, G_{r\Gamma}\,|\, Q,g)_r$, where $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, we denote the soliton wedge $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ with the phase space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$ replaced by the renormed phase space $\mathcal K^n_{r\Upsilon \infty \mu \check{h}}$. For a given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, a coordinate index $ \overline \gamma\in \Upsilon$, and $r\in \{0,1,\dots\}$, the corresponding principal infinite-dimensional initial-boundary value problem reads as
$$
\begin{equation}
\dot{\varkappa}(t)= G_{r\Gamma\infty \mu \check{h}}(t,\varkappa)\quad \text {for a.a. }\ t\in \mathbb R,
\end{equation}
\tag{18}
$$
$$
\begin{equation}
\varkappa(\eta(\check\gamma_k)(\overline{t}))=T_{\check\gamma_k}\varkappa(\overline{t}),\qquad k=1,\dots,d,
\end{equation}
\tag{19}
$$
$$
\begin{equation}
(\varkappa(\overline{t}))_{\overline{\gamma}}=\overline x,\qquad \overline{t}\in \mathbb R,\quad \overline{x} \in \mathbb R^n, \quad \overline{\gamma} \in \Upsilon,\quad r\in \{0,1,\dots\}.
\end{equation}
\tag{20}
$$
The index $r$ means that the phase space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}$ is replaced by the renormed phase space $\mathcal K^n_{r\Upsilon \infty \mu \check{h}}$. It is clear that, for $r=0$, we have $\mathcal K^n_{0\Upsilon \infty \mu \check{h}}=\mathcal K^n_{\Upsilon \infty \mu \check{h}}$, $G_{0\Gamma\infty \mu \check{h}}=G_{\Gamma\infty \mu \check{h}}$, and we get the principal initial-boundary value problem (10)–(12). For each $r\in \{0,1,\dots\}$, all the parameters of the soliton wedge $(\Upsilon, d,s,\eta, G_{r\Gamma}\,|\, Q,g)_r$ coincide with those of the soliton wedge $(\Upsilon, d,s,\eta, G_{\Gamma}\,|\, Q,g)$, except for the parameter $\mathcal M_{r0\Upsilon \infty\mu \check{h}}(\,{\cdot}\,)$, which satisfies
$$
\begin{equation*}
\mathcal M_{r0\Upsilon \infty\mu \check{h}}(\tau)=\mathcal M_{0\Upsilon \infty\mu \check{h}}(\tau)\mu^{rh}.
\end{equation*}
\notag
$$
Clearly, each solution of the principal initial-boundary value problem (10)–(12) is a solution of the principal initial-boundary value problem (18)–(20), and vice versa. For a given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, a coordinate index $\overline \gamma\in \Upsilon$, and $r\in \{0,1,\dots \}$, consider the corresponding principal infinite-dimensional integral equation
$$
\begin{equation}
\begin{aligned} \, \varkappa(t) &=\biggl[\widetilde{S}_{r\Upsilon\infty \mu h}\overline x - \sum_{\gamma \in \Upsilon} P_\gamma\int^{\eta(\gamma^{-1}\overline \gamma)(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu h} (\tau,\varkappa(\tau)) \,d\tau\biggr] \nonumber \\ &\qquad+\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau,\qquad t\in \mathbb R, \end{aligned}
\end{equation}
\tag{21}
$$
where $\widetilde{S}_{r\Upsilon\infty \mu \check{h}}\colon \mathbb R^n \to \mathcal K^n_{r\Upsilon \infty \mu \check{h}}$ is the operator defined by
$$
\begin{equation*}
(\widetilde{S}_{r\Upsilon\infty \mu \check{h}}\overline x)_\gamma=\overline x\quad \forall\, \overline x,\gamma,\quad \overline x\in \mathbb R^n,\quad \gamma\in \Upsilon.
\end{equation*}
\notag
$$
Consider the operators
$$
\begin{equation*}
\begin{aligned} \, A_r\colon \varkappa(\,{\cdot}\,) &\to\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau, \qquad t \in \mathbb R, \\ \mathcal A_r\colon \varkappa(\,{\cdot}\,) &\to \biggl[\widetilde{S}_{r\Upsilon\infty \mu h}\overline x - \sum_{\gamma \in \Upsilon} P_\gamma \bigl[(A_r[\varkappa(\,{\cdot}\,),\overline{x}]) (\eta(\gamma^{-1}\overline{\gamma})(\overline t))\bigr] \biggr] \\ &\qquad+(A_r[\varkappa(\,{\cdot}\,),\overline{x}])(t),\qquad t \in \mathbb R, \end{aligned}
\end{equation*}
\notag
$$
on $C^{(0)}(\mathbb R,\mathcal K^n_{r\Upsilon \infty\mu \check{h}})$. The operator $ A_r$, $r\in \{0,1,\dots \}$, is obtained from the operator $ A_0$, where the phase space $\mathcal K^n_{\Upsilon \infty \mu \check{h}}=\mathcal K^n_{0\Upsilon \infty \mu \check{h}}$ is replaced by the renormed phase space $\mathcal K^n_{r\Upsilon \infty \mu \check{h}}$. For $r=0$, the principal infinite-dimensional integral equation (21) coincides with the principal infinite-dimensional integral equation (13). Now the principal infinite-dimensional integral equation (21) is solvable for some $r\in \{0,1,\dots\}$ and, in particular, for $r=0$, if and only if it is solvable for each $r\in \{0,1,\dots\}$. We show that if, for all given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, and a coordinate index $ \overline \gamma\in \Upsilon$, the principal infinite-dimensional integral equation (13) has a unique solution, then there exists a one-one correspondence between the solutions of the principal infinite-dimensional initial-boundary value problem (10)–(12) and those of the principal infinite-dimensional integral equation (13). Let us give one principal result and provide its equivalent reformulation, which guarantee the existence and uniqueness of a solution of the principal infinite-dimensional integral equation (13). Lemma 2. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma=(\Upsilon,d, s,\eta,Q,g)$, where the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$ and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. Then, for a parameter $\check{h}\in [0,+\infty)$, the parameter $\mu \in (0,\mu^*]$ satisfies Under the conditions of Lemma 2, in view of embeddings (17), (24) and estimates (23), each solution ${\varkappa}(t)\in \mathcal K^n_{\Upsilon \infty\mu \check{h}}$, $t \in \mathbb R$, $\mu\in (0,\mu^*]$, of the infinite-dimensional differential equation (10) satisfies the time asymptotics
$$
\begin{equation*}
\rho(\,{\cdot}\,) \in \mathcal L^1_{\mu} C^{(0)}(R),\qquad \rho(t)=\|\varkappa(t)\|_{\Upsilon \infty\mu \check{h}},\quad t \in \mathbb R,
\end{equation*}
\notag
$$
which, in turn, inverts (6) and (7). Let us describe properties of the operators $\mathcal A_r$, $r\in \{0,1,\dots \}$, depending on the first relation in (23) of Lemma 2 and also on the first and second relations together. Lemma 3. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)_r$ be a soliton wedge with $\Gamma=(\Upsilon,d, s,\eta,Q,g)$ and $r\in \{0,1,\dots\}$, where the phase space $\mathcal K^n_{\Upsilon \infty \mu h}$ is replaced by the renormed phase space $\mathcal K^n_{r\Upsilon \infty \mu \check{h}}$, the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$, and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. If, for a given parameter $\check{h}\in [0,+\infty)$, the parameters $\mu \in (0,\mu^*]$, $\vartheta \in (0,1]$ satisfy then, for a given initial moment $\overline{t}\in \mathbb R$, an initial value $\overline x\in \mathbb R^n$, a coordinate index $\overline \gamma\in \Upsilon$, and $r\in \{0,1,\dots \}$, the operator $\widehat{\mathcal A_r}$ is continuous on $ C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r\Upsilon \infty \mu \check{h}})$. In addition, if then the operator $\mathcal A_r$, which is continuous on $ C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r\Upsilon \infty \mu \check{h}})$, is contracting for sufficiently large $r$. In addition, for each $r\in \{0,1,\dots \}$, the operator $ A_r$ is contracting. For each $r\in \{0,1,\dots \}$, the principal infinite-dimensional integral equation (21) can be written in the operator form
$$
\begin{equation}
\varkappa(t)\equiv({\mathcal A_r}[\varkappa(\,{\cdot}\,)])(t),\qquad t \in \mathbb R.
\end{equation}
\tag{26}
$$
Under the conditions of Lemma 3, which involve conditions (25) or the equivalent conditions (23), the operator ${\mathcal A_r}$ is contracting, and, for the principal infinite-dimensional integral equation in the operator form (26), for each initial moment $\overline{t}\in \mathbb R$, an initial value $\overline{x} \in \mathbb R$, a coordinate index $\overline \gamma \in \Upsilon$, and, for sufficiently large $r\in \{0,1,\dots\}$, there is a theorem on existence and uniqueness of a solution of the principal infinite-dimensional integral equation (21), which implies a theorem of existence and uniqueness of the solution for the principal infinite-dimensional integral equation (13). This result is formulated as follows. Theorem 1. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, where the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$, and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. Assume that, for a given parameter $\check{h}\in [0,+\infty)$, The bounded soliton solutions constitute an important class of soliton solutions. Similarly, the bounded solutions constitute an important class of solutions of point-type functional differential equations. In the framework of the dualism for the canonical soliton wedge, it was verified, in particular, that there is a correspondence between the bounded solutions of a principal initial-boundary value problem and of the corresponding functional differential equation from the “function–operator” dual pair. Theorem 2. Let $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a soliton wedge with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, where the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$, and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. Assume that an absolutely continuous function $x(\,{\cdot}\,)\in \mathcal L^n_1C^{(0)}(R)$ is a solution (a bounded solution) of the functional differential equation (3). Then, in the dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$, that is, $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $x_{\gamma}(t)=x_e(\eta(\gamma)(t))$, $x_e(t)=x(t)$, $\gamma\in \Upsilon$, $t\in \mathbb R$, for an arbitrary $\mu \in(0,\mu^*]$, the vector function $\varkappa(\,{\cdot}\,)$ satisfies $\varkappa(t)\in \mathcal K^n_{\Upsilon p \mu \check{h}}$, $t \in \mathbb R$, and is such that: – $\varkappa(\,{\cdot}\,)$ is strongly absolutely continuous; – $\varkappa(\,{\cdot}\,)$ is a solution of the infinite-dimensional ordinary differential equation (10) and satisfies the non-local constraints (11) (the soliton condition of the solution); – the function $\rho(t)=\|\varkappa(t)\|_{\Upsilon p\mu \check{h}}$, $t \in \mathbb R$, satisfies $\rho(\,{\cdot}\,) \in \mathcal L^1_1 C^{(0)}(R)$ (a bounded solution). If, for $\mu \in(0,\mu^*]$, the strongly absolutely continuous vector function $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $\varkappa(t)\in \mathcal K^n_{\Upsilon p \mu \check{h}}$, $t \in \mathbb R$, is a solution of the infinite-dimensional system (10), (11) (a soliton solution) and the function $\rho(t)=\|\varkappa(t)\|_{\Upsilon p\mu \check{h}}$, $t \in \mathbb R$, satisfies $\rho(\,{\cdot}\,) \in \mathcal L^1_1 C^{(0)}(R)$ (a bounded solution), then, in the dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$, that is, $\varkappa(t)=\{x_{\gamma}(t)\}_{\gamma\in \Upsilon}$, $x_{\gamma}(t)=x_e(\eta(\gamma)(t))$, $x_e(t)=x(t)$, $\gamma\in \Upsilon$, $t\in \mathbb R$, to $\rho(t)$ there corresponds a solution $x(t)$, $t\in \mathbb R$, of the functional differential equation (3), and $x(\,{\cdot}\,)\in\mathcal L^n_1 C^{(0)}(R)$ (a bounded solution). For a given initial moment $\overline{t}\in \mathbb R$ and an initial value $ \overline x\in \mathbb R^n$, we pose the Cauchy problem for equation (3):
$$
\begin{equation}
\dot{x}(t)=g\bigl(t,x(q_1(t)),\dots,x(q_s(t))\bigr),\qquad t\in \mathbb R,
\end{equation}
\tag{27}
$$
$$
\begin{equation}
x(\widehat{t})=\overline{x},\qquad \overline{x}\in \mathbb R^n,\quad \widehat t\in \mathbb R.
\end{equation}
\tag{28}
$$
In the actual fact, the correspondences described in Theorem 2 of [8] between the solutions of the principal infinite-dimensional boundary-value problem (10), (11) (the soliton solutions) and the solutions of the functional differential equation (27), can be extended to correspondences between the solutions of the principal infinite-dimensional initial-boundary value problem (10)–(12) and the solution of the Cauchy problem (27), (28). This correspondence is most informative for the Cauchy problem (27), (28) in the case a canonical soliton wedge. To this end, for a soliton wedge $(\Upsilon, d,s,\eta, G_\Gamma\,|\, Q,g)$ with $\Gamma=(\Upsilon,d,s,\eta,Q,g)$, we transform to the canonical soliton wedge $(Q, d,s,\mathcal I, G_\Gamma\,|\, Q,g)$ with $\Gamma=(Q,d,s,\mathcal I,Q,g)$. Using the dualism and, respectively, Theorem 2 of [8] on interactions, Theorem 1 for a canonical soliton wedge was reformulated in terms applicable to the point-type functional differential equation (27). Theorem 3. Let the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfy conditions $(\mathrm a^*)$–$(\mathrm c^*)$ and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. Assume that § 2. Existence of bounded soliton solutions and of bounded solutions for a functional differential equation from the “function–operator” dual pairConditions for the existence of a bounded solution for ordinary differential equation based on properties of the shift operator along solutions have been obtained in many papers, see, in particular, [31], [32]. In these studies, for ordinary differential equations such conditions, which are formulated in terms of a property of the right-hand side of the equation on some sphere (as a geometric property of the phase portrait), assume the form
$$
\begin{equation}
(\overline{x},g(t,\overline{x}))\leqslant 0, \qquad t\in \mathbb R,\quad \|\overline{x}\|_{\mathbb R^n}= \varrho,
\end{equation}
\tag{30}
$$
with some fixed $\varrho>0$. The meaning of constraints (30) is that the image of the phase curve $x(t)$ should not leave the ball of radius $\varrho$ in the phase space $\mathbb R^n$. It is quite important that the conditions we provide are free of (quite involved) spectral properties of the equation in variations. For the equation with time delay a condition similar to (30) assumes the form
$$
\begin{equation*}
(\overline{x},g(t,\overline{x}, \overline{y}))\leqslant 0, \qquad t\in \mathbb R,\quad \|\overline{x}\|_{\mathbb R^n}= \varrho,\quad \overline{y}\in \mathbb R^n,
\end{equation*}
\notag
$$
which strongly narrows the class of equations with this property. The principal drawback of this approach for functional differential equations is the point localization of the conditions which guarantee the boundedness of the image of the phase curve $x(\,{\cdot}\,)$. The papers [1] and [2] give a new type of conditions to guarantee the boundedness of the image of the phase curve $x(\,{\cdot}\,)$ for functional differential equations with deviating arguments defined on a cyclic shift group. In these studies, an approach was used based on consideration of asymptotic properties of solutions of point-type functional differential equations, which can be identified on shifts of solutions and can be formulated in terms of mean values over some fixed period related to a fixed sphere in the phase space. For general type point-type functional differential equations (with deviations of the argument and generated by an arbitrary finitely generated group of diffeomorphisms of $\mathbb R$), by advancing this approach one can also obtain solutions for existence of bounded solutions. This approach is realized in the framework of the dualism with the theory of soliton solutions and solutions of point-type functional differential equations. Let a function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfy conditions $(\mathrm a^*)$–$(\mathrm c^*)$ and a group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. Consider the point-type functional differential equation (3)
$$
\begin{equation}
\dot x(t)=g\bigl(t,x(q_1(t)),\dots,x(q_s(t))\bigr)\quad \text{for a.a. }\ t\in \mathbb R.
\end{equation}
\tag{31}
$$
For a solution $x(\,{\cdot}\,)$ of equation (31), an initial state $\overline{x}$, and a real number $\omega>0$, we set
$$
\begin{equation*}
\begin{aligned} \, y_{t\omega g}(x(\,{\cdot}\,)) &=\int_t^{t+\omega} g\bigl(\tau,x(q_1(\tau)),\dots,x(q_s(\tau))\bigr) \, d\tau, \\ y_{t\omega g}(\overline{x}) &=\int_t^{t+\omega} g(\tau,\overline{x},\dots,\overline{x}) \, d\tau. \end{aligned}
\end{equation*}
\notag
$$
Using the duality of the theory of soliton solutions and that of solutions of point-type functional differential equations, we will establish a sufficient condition for the existence of a bounded solution for the point-type functional differential equation (31). This result is as follows: for some fixed $\omega> 0$, $\varrho>0$, for each $t\in \mathbb R$, any solution $x(\,{\cdot}\,)$ of equation (31) such that $x(t)=\overline{x}$, $\|\overline{x}\|_{\mathbb R^n}= \varrho$, satisfies
$$
\begin{equation}
\|\overline{x}+y_{t\omega g}(x(\,{\cdot}\,))\|_{\mathbb R^n}\leqslant \varrho.
\end{equation}
\tag{32}
$$
Squaring both sides of inequality (32) and writing the left-hand side as a squared inner product, we can formulated condition (32) in the following equivalent form: for some fixed $\omega> 0$, $\varrho>0$, for all $t\in \mathbb R$, any solution $x(\,{\cdot}\,)$ of equation (31) such that $ x(t)=\overline{x}$, $\|\overline{x}\|_{\mathbb R^n}= \varrho $, should satisfy
$$
\begin{equation}
(\overline{x},{y}_{t\omega g}(\overline{x}))\leqslant - \bigl(\overline{x},\,y_{t\omega g}(x(\,{\cdot}\,))-{y}_{t\omega g}(\overline{x})\bigr)-\frac{1}{2}\|y_{t\omega g}(x(\,{\cdot}\,))\|^2_{\mathbb R^n}.
\end{equation}
\tag{33}
$$
It is easily seen that condition (33) holds if for some fixed $\omega> 0$, $r>0$, for each $t\in \mathbb R$, any solution $x(\,{\cdot}\,)$ of equation (31) such that $ x(t)=\overline{x}$, $\|\overline{x}\|_{\mathbb R^n}= \varrho$, satisfies
$$
\begin{equation}
\begin{aligned} \, &\biggl(\frac{\overline{x}}{\|\overline{x}\|_{\mathbb R^n}},\,y_{t\omega g}(\overline{x})\biggr)\leqslant -\|y_{t\omega g}(x(\,{\cdot}\,))-y_{t\omega g}(\overline{x})\|_{\mathbb R^n} \nonumber \\ &\qquad-\frac{1}{2\varrho}\bigl(\|y_{t\omega g}(x(\,{\cdot}\,))-y_{t\omega g}(\overline{x})\|_{\mathbb R^n} +\|y_{t\omega g}(\overline{x}) \|_{\mathbb R^n} \bigr)^2. \end{aligned}
\end{equation}
\tag{34}
$$
We will next give an estimate of the right-hand side of inequality (34) in terms of the exponent of the majorizing exponential function in a localization of the space of solutions of the functional differential equation (31), the Lipschitz constants of the right-hand side of this equation, and deviations of the argument. Inequality (34) will be ensured by existence of a bounded solution for this equation. To this end, we will use Theorem 2 on a correspondence between the soliton solutions and the solutions of the corresponding point-type functional differential equation in the case of a canonical soliton wedge. Consider a canonical soliton wedge $(Q, d,s,\mathcal I, G_\Gamma\,|\, Q,g)$ with $\Gamma\,{=}\,(Q,d, s,\mathcal I,Q,g)$, where the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$, and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. By $(Q, d,s,\eta, G_{r\Gamma}\,|\, Q,g)_r$, where $r=0,1,\dots$ and $\Gamma=(Q,d,s,\eta,Q,g)$, we denote the soliton wedge $(Q, d,s,\eta, G_\Gamma\,|\, Q,g)$, for which the phase space $\mathcal K^n_{Q \infty \mu \check{h}}$ is replaced by the renormed phase space $\mathcal K^n_{rQ \infty \mu \check{h}}$ and the corresponding principal infinite-dimensional initial-boundary value problem
$$
\begin{equation}
\dot{\varkappa}(t)= G_{r\Gamma \infty \mu \check{h}}(t,\varkappa)\quad \text{for a.a. }\ t\in \mathbb R,
\end{equation}
\tag{35}
$$
$$
\begin{equation}
\varkappa(\check q_k(\overline{t}))=T_{\check q_k}\varkappa(\overline{t}),\qquad k=1,\dots,d,
\end{equation}
\tag{36}
$$
$$
\begin{equation}
(\varkappa(\overline{t}))_{\overline{q}}=\overline x,\qquad \overline{t}\in \mathbb R,\quad \overline{x} \in \mathbb R^n, \quad \overline{q} \in Q,
\end{equation}
\tag{37}
$$
with a given initial moment $\overline{t}\in \mathbb R$, an initial value $ \overline x\in \mathbb R^n$, an index $ \overline q\in Q$, and $r\in \{0,1,\dots\}$, in the phase space $\mathcal K^n_{r Q \infty \mu \check{h}}$. We will also consider the corresponding principal infinite-dimensional integral equation
$$
\begin{equation}
\begin{aligned} \, \varkappa(t) &=\biggl[\widetilde{S}_{rQ\infty \mu \check{h}}\overline x - \sum_{q \in Q} P_q\int^{(q^{-1}\overline q)(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau\biggr] \nonumber \\ &\qquad+\int^t_{\overline{t}} G_{r\Gamma \infty \mu \check{h}}(\tau,\varkappa(\tau))\,d\tau, \qquad t\in \mathbb R, \end{aligned}
\end{equation}
\tag{38}
$$
where $P_q$ is the projection in the Banach space $\mathcal K^n_{r Q \infty \mu \check{h}}$ such that $\operatorname{Im} P_q$ coincides with the subspace generated by the coordinate $x_q$, and $\widetilde{S}_{rQ\infty \mu \check{h}}\colon \mathbb R^n \to \mathcal K^n_{r Q \infty \mu \check{h}}$ is the operator defined by
$$
\begin{equation*}
(\widetilde{S}_{rQ\infty \mu \check{h}}\overline x)_q=\overline x\quad \forall\, \overline x,q,\quad \overline x\in \mathbb R^n,\quad q\in Q.
\end{equation*}
\notag
$$
As noted above, the duality will enable us, in the finite-dimensional phase space $\mathbb R^n$, to check condition (34) for a solution of the functional differential equation (31), while the terms on the right-hand side of inequality (34) will be estimated via properties of the dual soliton solution of the principal infinite-dimensional initial-boundary value problem (35)–(37) (the principal infinite-dimensional integral equation (38)) with the infinite-dimensional phase space $\mathcal K^n_{r Q \infty \mu \check{h}}$. If the principal inequality (29) holds and if a coordinate index $\overline q\in Q$ is fixed, then, due to the existence of an isomorphism $ R^n \simeq \operatorname{Im}\widetilde S_{rQ\infty\mu \check{h}}$ and since the integral equation (38) has a unique solution with fixed initial moment $\overline t\in \mathbb R$ and initial value $\overline{x}$ (Theorem 1), there exists a natural homeomorphism
$$
\begin{equation*}
\Psi_{\overline q}\colon\operatorname{Im}\widetilde S_{rQ\infty\mu \check{h}}\to \mathcal X
\end{equation*}
\notag
$$
between the space $\operatorname{Im}\widetilde S_{rQ\infty\mu \check{h}}$ and the space $\mathcal X \subset C^{(0)}(\mathbb R,K^n_{r Q\infty\mu \check{h}})$ of solutions $\varkappa(\,{\cdot}\,)$ of the integral equation (38). It is clear that
$$
\begin{equation*}
\bigl(\Psi_{\overline q}(\widetilde S_{rQ\infty\mu \check{h}}\overline{x})(\overline{t})\bigr)_{\overline q}=\overline{x}\quad \forall\, \overline{x}\in \mathbb R^n.
\end{equation*}
\notag
$$
Given $\vartheta \in (0,1]$, $\mu\in (0,\mu^*]$, $\mu\leqslant \vartheta$, $\vartheta< \widehat{\mu}$, $\varrho\geqslant 0$, and for sufficiently large $r\in \{0,1,\dots\}$, we set
$$
\begin{equation*}
\begin{aligned} \, \Delta_{r\vartheta\mu}(\varrho) &=\bigl\{1- [\ln{\vartheta}^{-1}]^{-1}\mathcal M_{2Q \infty\mu \check{h}} \bigl[1+\vartheta^{-l(\overline q){\check{h}}}\mu^{r\check{h}}\operatorname{sign} \check{h}\bigr]\bigr\}^{-1} \\ &\qquad\times \bigl\{[\ln{\vartheta}^{-1}]^{-1} \|\mathcal M_{r0Q \infty\mu \check{h}}(\,{\cdot}\,)\|_{\vartheta}^{(0)} \bigl[1+\vartheta^{-l(\overline q){\check{h}}} \mu^{r\check{h}}\operatorname{sign} \check{h}\bigr]+\varrho \mu^{r\check{h}} \bigr\}. \end{aligned}
\end{equation*}
\notag
$$
Some estimates were proved in [9]. In particular, for $\mu\in (0,\mu^*]$, $\vartheta\in (0,1]$, $\mu\leqslant \vartheta$, $\vartheta< \widehat{\mu}$, $\varkappa(\,{\cdot}\,),\widetilde{\varkappa}(\,{\cdot}\,) \in C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{rQ \infty \mu \check{h}})$ and $\overline{t}=0$,
$$
\begin{equation}
\biggl\|\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau \biggr\|_{rQ \infty \mu \check{h}} \leqslant \int^t_{\overline{t}} \|G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,\|_{rQ \infty \mu \check{h}}\, d\tau \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\leqslant [\vartheta^{-|t|}-1][\ln{\vartheta}^{-1}]^{-1}\bigl[\|\mathcal M_{r0Q \infty\mu \check{h}}(\,{\cdot}\,)\|_{\vartheta}^{(0)}+\mathcal M_{2Q \infty\mu \check{h}} \|\varkappa(\,{\cdot}\,)\|_{rQ\infty\mu \check{h} C_{\vartheta}^0}\bigr],
\end{equation}
\tag{39}
$$
$$
\begin{equation}
\biggl\|\int^t_{\overline{t}} [G_{r\Gamma \infty \mu \check{h}}(\tau,\varkappa(\tau))- G_{r\Gamma \infty \mu \check{h}}(\tau,\widetilde{\varkappa}(\tau))]\,d\tau\biggr\|_{rQ\infty\mu \check{h}} \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\leqslant\int^t_{\overline{t}} \|G_{r\Gamma \infty \mu \check{h}}(\tau,\varkappa(\tau))- G_{r\Gamma \infty \mu \check{h}}(\tau,\widetilde{\varkappa}(\tau))\|_{rQ\infty\mu \check{h}}\,d\tau \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\qquad\leqslant [\vartheta^{-|t|}-1][\ln{\vartheta}^{-1}]^{-1}\mathcal M_{2Q \infty\mu \check{h}}\|\varkappa(\,{\cdot}\,)-\widetilde{\varkappa}(\,{\cdot}\,)\|_{rQ\infty\mu \check{h} C_{\vartheta}^0},
\end{equation}
\tag{40}
$$
$$
\begin{equation}
\biggl\|\sum_{q \in Q} P_q\int^{q^{-1}\overline q(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau\biggr\|_{rQ \infty\mu \check{h}} \nonumber
\end{equation}
\notag
$$
$$
\begin{equation}
\quad\leqslant [\ln{\vartheta}^{-1}]^{-1} \bigl[\|\mathcal M_{r0 Q \infty\mu \check{h}}(\,{\cdot}\,)\|_{\vartheta}^{(0)} + \mathcal M_{2Q \infty\mu \check{h}}\|\varkappa(\,{\cdot}\,)\|_{rQ\infty\mu \check{h} C_{\vartheta}^0}\bigr] \vartheta^{-l(\overline q){\check{h}}}\mu^{r\check{h}}\operatorname{sign} \check{h}.
\end{equation}
\tag{41}
$$
Lemma 4. Let $(Q, d,s,\eta,G_\Gamma\,|\, Q,g)_r$ be a canonical soliton wedge with tuple $\Gamma = (Q,d,s,\eta,Q,g)$, where $r \in \{0,1,\dots\}$ is sufficiently large and for which the phase space $\mathcal K^n_{Q \infty \mu h}$ is replaced by the renormed phase space $\mathcal K^n_{r Q \infty \mu \check{h}}$, let a function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfy conditions $(\mathrm a^*)$–$(\mathrm c^*)$, and let a group of diffeomorphisms $Q$ obey conditions $(\mathrm d^*)$. Assume that, for a given parameter $\check{h}\in [0,+\infty)$, Proof. Let us verify (43). From the integral equation (38) it follows that its solution ${\varkappa}(\,{\cdot}\,)\in C^{(0)}_\vartheta(\mathbb R,K^n_{r Q\infty\mu \check{h}})$ satisfies, for each $t\in \mathbb R$,
$$
\begin{equation*}
\begin{aligned} \, \|\varkappa(t)\|_{rQ \infty\mu \check{h}} &\leqslant \|\overline{x}\|_{\mathbb R^n}\,\mu^{r\check{h}}+ \biggl\|\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau \biggr\|_{rQ \infty \mu \check{h}} \\ &\qquad+ \biggl\|\sum_{q \in Q} P_q\int^{q^{-1}\overline q(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau\biggr\|_{rQ \infty\mu \check{h}}. \end{aligned}
\end{equation*}
\notag
$$
From the last estimate we have
$$
\begin{equation*}
\begin{aligned} \, &\|{\varkappa}(\,{\cdot}\,)\|_{rQ \infty\mu \check{h} C^{(0)}_{\vartheta}} =\sup_{t\in \mathbb R} \vartheta^{|t|} \|{\varkappa}(t)\|_{rQ \infty\mu \check{h}} \\ &\qquad\leqslant \sup_{t\in \mathbb R} \vartheta^{|t|} \|\overline{x}\|_{\mathbb R^n} \mu^{r\check{h}} +\sup_{t\in \mathbb R} \vartheta^{|t|} \biggl\|\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau \biggr\|_{rQ \infty \mu \check{h}} \\ &\qquad\qquad+ \sup_{t\in \mathbb R} \vartheta^t \biggl\|\sum_{q \in Q} P_q\int^{q^{-1}\overline q(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau \biggr\|_{rQ \infty\mu \check{h}}. \end{aligned}
\end{equation*}
\notag
$$
Now estimate (43) is obtained if we use estimates (39), (41) and collect all the factors multiplying $\|{\varkappa}(\,{\cdot}\,)\|_{rQ \infty\mu \check{h} C^{(0)}_{\vartheta}}$ in the left-hand side.
Let us verify (44). From the integral equation (38) it follows that its solution ${\varkappa}(\,{\cdot}\,)\in C^{(0)}_\vartheta(\mathbb R,K^n_{r Q\infty\mu \check{h}})$ also satisfies the estimate
$$
\begin{equation*}
\begin{aligned} \, \|\varkappa(t)- \widetilde S_{rQ\infty\mu\check{h}}\overline{x}\|_{rQ \infty\mu \check{h}} &\leqslant \biggl\|\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau \biggr\|_{rQ \infty \mu \check{h}} \\ &\qquad+ \biggl\|\sum_{q \in Q} P_q\int^{q^{-1}\overline q(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau \biggr\|_{rQ \infty\mu \check{h}} \end{aligned}
\end{equation*}
\notag
$$
for each $t\in \mathbb R$. As a result,
$$
\begin{equation*}
\begin{aligned} \, &\|\varkappa(\,{\cdot}\,)-\widetilde S_{rQ\infty\mu\check{h}}\overline{x}\|_{rQ \infty\mu \check{h} C^{(0)}_{\vartheta}} =\sup_{t\in \mathbb R} \vartheta^{|t|} \|\varkappa(t)-\widetilde S_{rQ\infty\mu\check{h}}\overline{x}\|_{rQ \infty\mu \check{h}} \\ &\qquad\leqslant\sup_{t\in \mathbb R} \vartheta^{|t|} \biggl\|\int^t_{\overline{t}} G_{r\Gamma \infty \mu h}(\tau,\varkappa(\tau))\,d\tau \biggr\|_{rQ \infty \mu \check{h}} \\ &\qquad\qquad+ \sup_{t\in \mathbb R} \vartheta^{|t|} \biggl\|\sum_{q \in Q} P_q\int^{q^{-1}\overline q(\overline{t})}_{\overline{t}} G_{r\Gamma \infty \mu \check{h}} (\tau,\varkappa(\tau)) \,d\tau \biggr\|_{rQ \infty\mu \check{h}}. \end{aligned}
\end{equation*}
\notag
$$
Now from (39) and (40) we have
$$
\begin{equation*}
\begin{aligned} \, &\|{\varkappa}(\,{\cdot}\,)-\widetilde S_{rQ\infty\mu\check{h}}\overline{x}\|_{rQ \infty\mu \check{h} C^{(0)}_{\vartheta}} \\ &\qquad\leqslant [\ln{\vartheta}^{-1}]^{-1}\bigl[\|\mathcal M_{r0 Q \infty\mu \check{h}}(\,{\cdot}\,)\|_{\vartheta}^{(0)}+ \mathcal M_{2Q \infty\mu \check{h}} \|{\varkappa}(\,{\cdot}\,)\|_{rQ \infty\mu \check{h} C^{(0)}_{\vartheta}}\bigr] \\ &\qquad\qquad\times\bigl[1+\vartheta^{-l(\overline q){\check{h}}} \mu^{r\check{h}}\operatorname{sign} \check{h} \bigr]. \end{aligned}
\end{equation*}
\notag
$$
Now estimate (44) follows by applying estimate (43) in the right-hand side of the resulting relation for $\|{\varkappa}(\,{\cdot}\,)\|_{rQ \infty\mu \check{h} C^{(0)}_{\vartheta}}$. This proves the lemma. For a uniformly bounded function $g(t,0,\dots,0)$, we set
$$
\begin{equation}
\overline{\mathcal M}_{0Q}=h_{\dot{Q}}\sup_{t\in \mathbb R} \|g(t,0,\dots,0)\|.
\end{equation}
\tag{45}
$$
It is clear that
$$
\begin{equation}
\sup_{t\in \mathbb R}{\mathcal M}_{0Q \infty\mu \check{h}}(t)\leqslant \overline{\mathcal M}_{0Q}.
\end{equation}
\tag{46}
$$
We set
$$
\begin{equation}
\overline{\Delta}_{\vartheta\mu}(\rho)= \bigl\{1- [\ln{\vartheta}^{-1}]^{-1} \mathcal M_{2Q \infty\mu \check{h}} \bigr\}^{-1} \bigl\{[\ln{\vartheta}^{-1}]^{-1} \overline{\mathcal M}_{0Q} + \rho\bigr\}.
\end{equation}
\tag{47}
$$
Corollary 1. Under the conditions of Lemma 4, let the function $g(t,0,\dots,0)$ be uniformly bounded on $\mathbb R$, that is, $\mu^*=1$. Then Proof. The left- and right-hand sides of (43) and (44) are homogeneous in $\mu^{r\check{h}}$ with degree of homogeneity 1, and since these relations also hold for any sufficiently large $r\in \{0,1,\dots\}$, we get the first part of the claim. The second part follows from the first one and the fact that $(\mu,\mu)$ is also a dual pair. Lemma 5. Let $(Q, d,s,\eta, G_\Gamma\,|\, Q,g)$ be a canonical soliton wedge with tuple $\Gamma=(Q,d,s,\eta,Q,g)$, let $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfy conditions $(\mathrm a^*)$–$(\mathrm c^*)$, let $g(t,0,\dots,0)$ be uniformly bounded on $\mathbb R$, that is, $\mu^*=1$, and let the group of diffeomorphisms $Q$ satisfy conditions $(\mathrm d^*)$. Assume that, for a given parameter $\check{h}\in [0,+\infty)$, Proof. From (6) and (7) it follows that, for the solution $x(\,{\cdot}\,)\in \mathcal L^n_{\mu} C^{(0)}(R)$, $\mu\in (0,\mu^*]$, of the point-type functional differential equation (3), the vector function $\varkappa(\,{\cdot}\,)$ in the dual pair $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$ is a soliton solution with spatial asymptotics ${\varkappa}(t)\in \mathcal K^n_{Q \infty\mu \check{h}}$, $t \in \mathbb R$ and time asymptotics
Let us prove (53). Consider the canonical soliton wedge $(Q, d,s,\eta, G_\Gamma\,|\, Q,g)_r$ with $\Gamma=(Q,d,s,\eta,Q,g)$, and $ r\in \{0,1,\dots\}$ is sufficiently large. We have
$$
\begin{equation}
\begin{aligned} \, &\|y_{\overline t\omega g}(x(\,{\cdot}\,))-{y}_{\overline t\omega g}(\overline{x})\|_{\mathbb R^n} \nonumber \\ &\qquad\leqslant \biggl\|\int_{\overline t}^{\overline t+\omega} \bigl[g\bigl(\tau,x(q_1(\tau)),\dots,x(q_s(\tau))\bigr) -g(\tau,\overline x,\dots,\overline x)\bigr] \, d\tau \biggr\|_{\mathbb R^n} \nonumber \\ &\qquad\leqslant \int_{\overline t}^{\overline t+\omega} \bigl\| g\bigl(\tau,x(q_1(\tau)),\dots,x(q_s(\tau))\bigr) -g(\tau,\overline x,\dots,\overline x)\bigr\|_{\mathbb R^n} \, d\tau \nonumber \\ &\qquad\leqslant \mu^{-r\check{h}} \int_{\overline t}^{\overline t+\omega} \bigl\| G(\tau,\varkappa(\tau)) -G(\tau,\widetilde{S}_{rQ\infty \mu \check{h}}\overline x)\bigr\|_{rQ \infty \mu \check{h}} \, d\tau \nonumber \\ &\qquad\leqslant \mu^{-r\check{h}} [\vartheta^{-\omega}-1] [\ln{\vartheta}^{-1}]^{-1} \mathcal M_{2Q \infty\mu \check{h}} \bigl\|\varkappa(\,{\cdot}\,)- \widetilde{S}_{rQ\infty \mu \check{h}}\overline x \bigr\|_{r Q \infty \mu \check{h}C^{(0)}_{\vartheta}} \nonumber \\ &\qquad\leqslant \mu^{-r\check{h}} [\vartheta^{-\omega}-1] [\ln{\vartheta}^{-1}]^{-1} \mathcal M_{2Q \infty\mu \check{h}} \nonumber \\ &\qquad\qquad\times [\ln{\vartheta}^{-1}]^{-1} \bigl[\|\mathcal M_{r0 Q \infty\mu \check{h}}(\,{\cdot}\,) \|_{\vartheta}^{(0)} + \mathcal M_{2Q \infty\mu \check{h}} \Delta_{r\vartheta\mu}(\|\overline{x}\|) \bigr] \nonumber \\ &\qquad\qquad\times\bigl[1+\vartheta^{-l(\overline q){\check{h}}} \mu^{r\check{h}}\operatorname{sign} \check{h}\bigr]. \end{aligned}
\end{equation}
\tag{55}
$$
From (46) we have
$$
\begin{equation*}
\|\mathcal M_{r0 Q \infty\mu \check{h}}(\,{\cdot}\,)\|_{\vartheta}^{(0)}=\sup_{t\in \mathbb R} \vartheta^{|t|} \|\mathcal M_{r0 Q \infty\mu \check{h}}(t)\|_{\mathbb R}\leqslant \overline{\mathcal M}_{0Q} \mu^{r\check{h}}.
\end{equation*}
\notag
$$
Now inequality (55) can be written as
$$
\begin{equation}
\begin{aligned} \, \|y_{\overline t\omega g}(x(\,{\cdot}\,))-y_{\overline t\omega g}(\overline{x})\|_{\mathbb R^n} &\leqslant \mu^{-r\check{h}}[\vartheta^{-{\omega}}-1] [\ln{\vartheta}^{-1}]^{-1} \mathcal M_{2Q \infty\mu \check{h}} \nonumber \\ &\qquad\times [\ln{\vartheta}^{-1}]^{-1} \bigl[\, \overline{\mathcal M}_{0Q} \mu^{r\check{h}} + \mathcal M_{2Q \infty\mu \check{h}} \Delta_{r\vartheta \mu}(\|\overline{x}\|)\bigr] \nonumber \\ &\qquad\times\bigl[1+\vartheta^{-l(\overline q){\check{h}}} \mu^{r\check{h}}\operatorname{sign} \check{h} \bigr]. \end{aligned}
\end{equation}
\tag{56}
$$
The second line in (56) is homogeneous in $\mu^{r\check{h}}$ with degree of homogeneity 1. This verifies (53), because (56) holds for arbitrarily large $r\in \{0,1,\dots\}$.
Let us verify (54). Using (39) and (47), we have
$$
\begin{equation*}
\begin{aligned} \, &\|y_{\overline t\omega g}(\overline{x})\|_{\mathbb R^n}\leqslant \biggl\|\int_{\overline t}^{\overline t+\omega} g(\tau,\overline x,\dots,\overline x) \, d\tau \biggr\|_{\mathbb R^n}\leqslant \int_{\overline t}^{\overline t+\omega} \| g(\tau,\overline x,\dots,\overline x)\|_{\mathbb R^n} \, d\tau \\ &\ \leqslant \mu^{-r\check{h}} \int_{\overline t}^{\overline t+\omega} \bigl\| G(\tau,\widetilde{S}_{rQ\infty \mu \check{h}}\overline x) \bigr\|_{r Q \infty \mu \check{h}} \, d\tau \\ &\ \leqslant \mu^{-r\check{h}} [\vartheta^{-\omega}-1] [\ln{\vartheta}^{-1}]^{-1} \bigl[\|\mathcal M_{r0Q \infty\mu \check{h}}(\,{\cdot}\,)\|_{\vartheta}^{(0)}+\mathcal M_{2Q \infty\mu \check{h}} \|\widetilde{S}_{rQ\infty \mu \check{h}}\overline x\|_{rQ\infty\mu \check{h} C_{\vartheta}^0} \bigr] \\ &\ \leqslant \mu^{-r\check{h}} [\vartheta^{-\omega}-1] [\ln{\vartheta}^{-1}]^{-1} \bigl[\, \overline{\mathcal M}_{0Q} \mu^{r\check{h}} +\mathcal M_{2Q \infty\mu \check{h}} \|\overline x\|_{\mathbb R^n} \mu^{r\check{h}}\bigr] \\ &\ \leqslant [\vartheta^{-\omega}-1] [\ln{\vartheta}^{-1}]^{-1} \bigl[\, \overline{\mathcal M}_{0Q} +\mathcal M_{2Q \infty\mu \check{h}} \|\overline x\|_{\mathbb R^n} \bigr]. \end{aligned}
\end{equation*}
\notag
$$
This proves the lemma. Corollary 2. Under the conditions of Lemma 4, Proof. The first two inequalities in the corollary can be verified directly using (53) and (54). Inequalities (57) and (58) follow from the first two inequalities and the fact that $(\mu,\mu)$ is also a dual pair. This proves the corollary. For each $\vartheta,\mu\in (0,1]$, $\check{h}\in [0,+\infty)$, $\omega>0$, $r\in \{0,1,\dots\}$, consider the shift operators
$$
\begin{equation*}
\begin{aligned} \, &\mathcal T_\omega\colon C^{(0)}_\vartheta(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}})\to C^{(0)}_\vartheta(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}}),\qquad \mathcal T_\omega[\varkappa(\,{\cdot}\,)](t)=\varkappa(t+\omega)\quad \forall\, t\in \mathbb R, \\ &T_\omega\colon \mathcal L^n_{\mu} C^{(0)}(R) \to \mathcal L^n_{\mu} C^{(0)}(R),\qquad T_\omega[x(\,{\cdot}\,)](t)=x(t+\omega)\quad \forall \, t\in \mathbb R \end{aligned}
\end{equation*}
\notag
$$
in the spaces $ C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}})$ and $\mathcal L^n_{\mu} C^{(0)}(R)$. Let us establish a criterion for boundedness of vector functions from $ C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}}) $ and $\mathcal L^n_{\mu} C^{(0)}(R)$ in terms of orbits of the shift operators $\mathcal T_\omega$ and $T_\omega$. Lemma 6. Let $\varkappa(\,{\cdot}\,) \in C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}})$, $r\in \{0,1,\dots\}$, and $x(\,{\cdot}\,) \in \mathcal L^n_{\mu} C^{(0)}(R)$. Then a necessary and sufficient condition that the vector functions ${\varkappa}(\,{\cdot}\,)$ and $ x(\,{\cdot}\,)$ be bounded, that is, ${\varkappa}(\,{\cdot}\,)\in C^{(0)}_1(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}})$ and $x(\,{\cdot}\,)\in \mathcal L^n_1 C^{(0)}(R)$, is that the orbits $ \{\mathcal T_\omega^k [\varkappa(\,{\cdot}\,)]\}_{k=-\infty}^{+\infty}$ and $\{ T_\omega^k [x(\,{\cdot}\,)]\}_{k=-\infty}^{+\infty}$ be bounded in the spaces $C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}})$ and $ \mathcal L^n_{\mu} C^{(0)}(R)$, respectively. Proof. The necessity part is clear. Assume that the orbits $ \{\mathcal T_\omega^k [\varkappa(\,{\cdot}\,)]\}_{k=-\infty}^{+\infty}$ and $ \{ T_\omega^k [x(\,{\cdot}\,)]\}_{k=-\infty}^{+\infty}$ are bounded in the spaces $C^{(0)}_{\vartheta}(\mathbb R,\mathcal K^n_{r Q \infty \mu \check{h}})$ and $\mathcal L^n_{\mu} C^{(0)}(R)$, respectively. In this case, in particular, the images of the family of vector functions $\{\varkappa(t+k\omega)\}_{k=-\infty}^{+\infty}$, $\{x(t+k\omega)\}_{k=-\infty}^{+\infty}$, $t\in [0,\omega]$, are uniformly bounded in the phase spaces $K^n_{r Q \infty \mu \check{h}}$ and $\mathbb R^n$, respectively. Hence ${\varkappa}(\,{\cdot}\,)$ and $x(\,{\cdot}\,)$ are bounded. This proves the lemma. Let us prove the existence of bounded solutions as a dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$. Theorem 4. Assume that a canonical soliton wedge $(Q, d,s,\mathcal I, G_\Gamma\,|\, Q,g)$, where $\Gamma=(Q,d,s,\eta,Q,g)$, is such that the function $g\colon\mathbb R \times \mathbb R^{ns} \to \mathbb R^n $ satisfies conditions $(\mathrm a^*)$–$(\mathrm c^*)$, the function $g(t,0,\dots,0)$ is uniformly bounded on $\mathbb R$, that is, $\mu^*=1$, and the group of diffeomorphisms $Q$ obeys conditions $(\mathrm d^*)$. We also assume that In the dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,x(\,{\cdot}\,))$, that is, $\varkappa(t)\,{=}\,\{x_q(t)\}_{q\in Q}$, $x_q(t) = x_e(q(t))$, $x_e(t)=x(t)$, $q\in Q$, $t\in \mathbb R$, the vector function $\varkappa(\,{\cdot}\,)$ satisfies $\varkappa(t)\in \mathcal K^n_{Q \infty \mu \check{h}}$, $t \in \mathbb R$, and also satisfies system (35), (36) (a soliton solution), is a bounded soliton solution, that is, $\varkappa(\,{\cdot}\,)\in C^{(0)}_1(\mathbb R,\mathcal K^n_{ Q \infty \mu \check{h}})$, and lies in a ball of the phase space $ \mathcal K^n_{ Q \infty \mu \check{h}}$ of radius $\mathcal R$. Proof. By condition (59) and Theorem 3, the Cauchy problem (27), (28) has a solution $\widehat{x}(\,{\cdot}\,)\in \mathcal L^n_{\mu} C^{(0)}(R)$. This solution is unique and depends continuously on all parameters. In (6) and (7), it was noted that, for the solution $\widehat{x}(\,{\cdot}\,)\in \mathcal L^n_{ \mu} C^{(0)}(R)$ of the point-type functional differential equation (27) in the dual pair of vector functions $(\widehat{\varkappa}(\,{\cdot}\,)\,|\,\widehat{x}(\,{\cdot}\,))$, that is, $\widehat{\varkappa}(t)=\{\widehat{x}_q(t)\}_{q\in Q}$, $\widehat{x}_q(t)=\widehat{x}_e(q(t))$, $\widehat{x}_e(t)=\widehat{x}(t)$, $q \in Q$, $t\in \mathbb R$, the vector function $\widehat{\varkappa}(t)\in \mathcal K^n_{Q \infty\mu \check{h}}$, $t \in \mathbb R$, is a soliton solution with time asymptotics
$$
\begin{equation*}
\rho(\,{\cdot}\,) \in \mathcal L^1_{\mu} C^{(0)}(R),\qquad \rho(t)=\|\widehat{\varkappa}(t)\|_{Q \infty\mu \check{h}},\quad t \in \mathbb R.
\end{equation*}
\notag
$$
Let $\widehat{t}$ be an initial time and $\overline x$ be an initial value such that $\|\overline x\|=\varrho$. For these initial data, the Cauchy problem (27), (28) has a solution $\widehat{x}(\,{\cdot}\,)$, and $(\widehat{\varkappa}(\,{\cdot}\,)\,|\,\widehat{x}(\,{\cdot}\,))$ is a dual pair of functions. Consider the time shift $\theta(t)=t-\widehat{t}$. By Lemma 1, under this time shift, the soliton wedge is transformed to a new soliton wedge, for which all the parameters in Theorem 4 remain the same. The solution $\widehat x_\theta(\,{\cdot}\,)$ of the new functional differential equation is a shift of the solution $\widehat x(\,{\cdot}\,)$ of the original equation with the initial time $\widehat{t}_\theta=0$ and the initial value $ \|\overline x\|=\varrho$. In addition, the images of the original solution $\widehat x(\,{\cdot}\,)$ and of its shift $\widehat x_\theta(\,{\cdot}\,)$ are equal. For the solution $\widehat x_\theta(\,{\cdot}\,)$, by $(\widehat \varkappa_\theta(\,{\cdot}\,)|\widehat x_\theta(\,{\cdot}\,))$ we denote the corresponding dual pair of vector functions.
From Lemma 2 and inequality (59) it follows that $(\mu,\mu)$ is also a dual pair. Hence by (50) in Corollary 1, for the vector function $\widehat\varkappa_\theta(\,{\cdot}\,)$ from the dual pair of vector functions $(\widehat\varkappa_\theta(\,{\cdot}\,)|\widehat x_\theta(\,{\cdot}\,))$, we have the estimate
$$
\begin{equation}
\|\widehat{\varkappa}_\theta(\,{\cdot}\,)\|_{Q \infty\mu \check{h} C^{(0)}_{\mu}}\leqslant \overline{\Delta}_{\mu\mu}(\varrho).
\end{equation}
\tag{61}
$$
On the other hand,
$$
\begin{equation}
\sup_{t\in \mathbb R} \|\widehat x_\theta(t)\|_{\mathbb R^n}\mu^{|t|}\leqslant \|\widehat{\varkappa}_\theta(\,{\cdot}\,)\|_{Q \infty\mu \check{h} C^{(0)}_{\mu}}.
\end{equation}
\tag{62}
$$
Now from estimates (61) and (62) we have the estimate
$$
\begin{equation*}
\sup_{t\in [0,\omega]} \| \widehat x_\theta(t)\|_{\mathbb R^n}\leqslant \mu^{-\omega}\overline{\Delta}_{\mu\mu}(\varrho).
\end{equation*}
\notag
$$
From condition (60) and relations (32)–(34), (57), (58) it follows that this solution $\widehat x_\theta(\,{\cdot}\,)$ of the Cauchy problem satisfies $\|\widehat x_\theta(\omega)\|_{\mathbb R^n}\leqslant \varrho$. If $\|\widehat x_\theta(\widetilde{t})\|_{\mathbb R^n}\leqslant \varrho$ for each $t>\omega$, then we have shown that the function $\widehat x_\theta(\,{\cdot}\,)$ satisfies $\|\widehat x_\theta({t})\|_{\mathbb R^n}\leqslant \mu^{-\omega}\overline{\Delta}_{\mu\mu}(\varrho)$ on $[0,+\infty)$ and the length of maximal open connected time intervals on $[0, +\infty)$ for which the image of the curve $ \widehat x_\theta(\,{\cdot}\,)$ does not lies in a ball in $\mathbb R^n$ of radius $\varrho$ is at most $\omega$. This means that the original solution $\widehat x(\,{\cdot}\,)$ satisfies the estimate
$$
\begin{equation}
\|\widehat x(t)\|_{\mathbb R^n}\leqslant \mu^{-\omega}\overline{\Delta}_{\mu\mu}(\varrho),\qquad t\in [\widehat{t}, +\infty),
\end{equation}
\tag{63}
$$
and the length of maximal open connected time intervals for which the image of the original curve $ \widehat x(\,{\cdot}\,)$ does not lie in any ball in $\mathbb R^n$ of radius $\varrho$ is at most $\omega$.
Assume that this is not so. Let $\widetilde{t}\geqslant \omega$ be the first point at which $\|\widehat x_\theta(\widetilde{t})\|_{\mathbb R^n}=\varrho$, after which the curve leaves the ball of radius $\varrho$. In this case, we consider the time shift $\theta(t)=t-\widetilde{t}$ and proceed as in the previous step. Iterating, we can show that the original solution $\widehat x(\,{\cdot}\,)$ obeys estimate (63) and the length of maximal open connected time intervals of $[\widehat{t}, +\infty)$ for which the image of the original curve $\widehat x(\,{\cdot}\,)$ does not lie in a ball in $\mathbb R^n$ of radius $\varrho$ is at most $\omega$. Let us now proceed with the study of the family of solutions of the Cauchy problem. Let $\widehat{t}_i$, $i=1,2,\dots$, $\lim_{i\to +\infty} \widehat{t}_i=-\infty $, be initial times, $ \overline{x}\in \mathbb R^n$, $\|\overline{x}\|_{\mathbb R^n}=\varrho$, be an initial value, and let $\widehat x_i(\,{\cdot}\,)$, $i=1,2,\dots$, be the corresponding solutions of the Cauchy problem, where $\widehat x_i(\widehat t_i)=\overline{x}$, $i=1,2,\dots$ . According to the above, for each $ i=1,2,\dots$, the solutions $\widehat x_i(\,{\cdot}\,)$ satisfy
$$
\begin{equation}
\|\widehat x_i({t})\|_{\mathbb R^n}\leqslant \mu^{-\omega}\overline{\Delta}_{\mu\mu}(\varrho),\qquad t\in [\widehat{t}_i, +\infty),
\end{equation}
\tag{64}
$$
and the length of maximal open connected time intervals of $[\widehat{t}_i, +\infty)$ for which the image of the curve $\widehat x_i(\,{\cdot}\,)$ does not lie in a ball in $\mathbb R^n$ of radius $\varrho$ is at most $\omega$. Consider the sequence $\widehat x_i(0)$, $i=1,2,\dots$ . In view of (64), this sequence is bounded, and hence it has some subsequence (which we can identify with the original sequence) converging to a point $\check{x}\in \mathbb R^n$. By the existence and uniqueness theorem and in view of continuous dependence on parameters, there exists a solution ${x}(\,{\cdot}\,)$ satisfying the initial condition $x(0)=\check{x}$. By continuous dependence of the solution on initial conditions, this solution satisfies
$$
\begin{equation*}
\|{x}({t})\|_{\mathbb R^n}\leqslant \mu^{-\omega}\overline{\Delta}_{\mu\mu}(\varrho),\qquad t\in \mathbb R,
\end{equation*}
\notag
$$
and, in addition, the length of maximal open connected time intervals of $\mathbb R$ for which the image of the curve ${x}(\,{\cdot}\,)$ does not lie in a ball in $\mathbb R^n$ of radius $\varrho$ is at most $\omega$. This proves the first part of the theorem.
Let us proceed with the proof of the second part of the theorem. For a solution $x(\,{\cdot}\,)$ of the functional differential equation, consider the dual pair of vector functions $(\varkappa(\,{\cdot}\,)\,|\,{x}(\,{\cdot}\,))$, that is, $\varkappa(t)=\{x_q(t)\}_{q\in Q}$, $x_q(t)={x}_e(q(t))$, $x_e(t)=x(t)$, $q\in Q$, $t\in \mathbb R$. Let $\widehat{t}\in \mathbb R \setminus \mathbb P$, that is, $\|x(\widehat{t})\|_{\mathbb R^n}\leqslant \varrho$. According to the first part of the proof, this vector function $\varkappa(\,{\cdot}\,)$ satisfies
$$
\begin{equation*}
\|\varkappa(\,{\cdot}\,)\|_{Q \infty\mu \check{h} C^{(0)}_{\mu}}=\sup_{t\in \mathbb R} \|\varkappa(\,{\cdot}\,)\|_{Q \infty\mu \check{h}} \mu^{|t|} \leqslant \overline{\Delta}_{\mu\mu}(\varrho),
\end{equation*}
\notag
$$
whence The length of maximal open connected intervals of the open set $\mathbb P$ is at most $\omega$, and so we have This proves the theorem. § 3. ExampleLet us construct a representative class of point-type functional differential equations admitting a bounded solution. Consider the equation
$$
\begin{equation}
\begin{gathered} \, \dot{x}(t)=\sum_{i=1}^{p}\sum_{j=1}^{s_i}\alpha_{ij} x(t+k_{ij}{h_i})+ \Phi(t,x(t)),\qquad t\in \mathbb R,\quad x \in \mathbb R, \\ p,s_i\in \mathbb Z_+,\quad\alpha_{ij}\in \mathbb R,\quad h_i\geqslant 0,\quad k_{ij}\in \mathbb Z,\quad i=1,\dots,p,\quad j=1,\dots,s_i, \end{gathered}
\end{equation}
\tag{65}
$$
where the continuous function $\Phi(t,x)$ is Lipschitz continuous in $x$ with Lipschitz constant $L_{\Phi}$. We assume that, for each $x\in \mathbb R$, there exists $t\in \mathbb R$ such that
$$
\begin{equation*}
\sum_{i=1}^{p}\sum_{j=1}^{s_i}\,\alpha_{ij}x+ \Phi(t,x) \neq 0.
\end{equation*}
\notag
$$
This shows that equation (65) has no stationary solutions. The function $\Phi(\,{\cdot}\,)$ satisfies for large $|x|$ and each $t\in \mathbb R$, and, for given $\omega$, $\varrho>0$, any small $\delta>0$ and each $t\in \mathbb R$, the function
$$
\begin{equation*}
g(t,x_{00},x_{11},\dots,x_{1s_1},\dots,x_{p1},\dots,x_{ps_p})= \sum_{i=1}^{p}\sum_{j=1}^{s_i} \alpha_{ij}x_{ij}+ \Phi(t,x_{00})
\end{equation*}
\notag
$$
satisfies
$$
\begin{equation}
\biggl(\frac{\overline{x}}{\|\overline{x}\|_{\mathbb R^n}},\int_t^{t+\omega} g(\tau,\overline{x},\dots,\overline{x}) \, d\tau\biggr)<-\delta,\qquad \varrho=\overline x,\quad \overline x>0.
\end{equation}
\tag{67}
$$
It is easily seen that, for this equation, the group of homeomorphisms $Q$ has the form $Q=\langle t+h_1,\dots,t+h_p\rangle$, and, correspondingly, $\check{h}=\max_{i\in \{1,\dots,p\}} h_i$. We fix $\mu\in (0,1)$. In this case,
$$
\begin{equation*}
\mathcal M_{2Q \infty\mu \check{h}}=M_2 \biggl[\sum_{i=1}^p\sum_{j=1}^{s_i}\mu^{-|k_{ij}|\check{h}}+1\biggr].
\end{equation*}
\notag
$$
We can find a sufficiently small constant $M_2$, that is, we can choose the coefficients in equation (65) and the Lipschitz constant $L_\Phi$ so as to satisfy inequality (59), which has the form
$$
\begin{equation}
M_2\biggl[\sum_{i=1}^p\sum_{j=1}^{s_i}\mu^{-|k_{ij}|\check{h}}+1\biggr]<\ln \mu^{-1},
\end{equation}
\tag{68}
$$
and inequality (67) remains valid for sufficiently large $\varrho= M_2^{-1}$. For this example, the above constants are such that
$$
\begin{equation*}
\begin{gathered} \, \mathbb{A}_{g\mu} = \frac{\mu^{-\omega}-1}{\ln \mu^{-1}},\qquad \mathbb{B}_{g\mu\mu} = \frac{M_2 \bigl[\sum_{i=1}^p\sum_{j=1}^{s_i}\mu^{-|k_{ij}|\check{h}}+1\bigr]}{\bigl(\ln \mu^{-1}-M_2 \bigl[\sum_{i=1}^p\sum_{j=1}^{s_i}\mu^{-|k_{ij}|\check{h}}+1\bigr]\bigr)}, \\ \mathbb{C}_{g\mu}(\varrho)= \overline{\mathcal M}_{0Q} + M_2 \biggl[\sum_{i=1}^p\sum_{j=1}^{s_i}\mu^{-|k_{ij}|\check{h}}+1\biggr]\varrho. \end{gathered}
\end{equation*}
\notag
$$
The main condition (60) for $|\overline x|=\varrho$ assumes the form
$$
\begin{equation}
\begin{aligned} \, &\biggl(\frac{\overline{x}}{\|\overline{x}\|_{\mathbb R^n}},\int_t^{t+\omega} g(\tau,\overline{x},\dots,\overline{x}) \, d\tau\biggr) \nonumber \\ &\qquad\leqslant -\mathbb{A}_{g\mu} \mathbb{B}_{g\mu\mu} \mathbb{C}_{g\mu}(\|\overline x\|_{\mathbb R^n})- \frac{1}{2\varrho}\bigl[\mathbb{A}_{g\mu} (\mathbb {B}_{g\mu\mu}+1) \mathbb{C}_{g\mu}(\|\overline x\|_{\mathbb R^n}) \bigr]^2. \end{aligned}
\end{equation}
\tag{69}
$$
Inequality (69) should be tested only for positive $\overline x=\varrho$. By condition (66), for negative $\overline x$, $|\overline x|=\varrho$, this inequality also holds. In the above constructions, we can make sure that $\varrho=M_2^{-1}$ and that, for each choice of parameters, the value $\overline{\mathcal M}_{0Q}$, which was defined in (45), is kept unchanged. As noted above, this will not violate inequality (67). Now, since $M_2$ is small (in this case, $\varrho=M_2^{-1}$ is large), we can make sure that the absolute values of the first and second terms on the right-hand side of inequality (69) are arbitrarily small. In this case, inequality (69) holds, since inequality (67) remains true. This equation satisfies the conditions of the existence theorem for a bounded solution (conditions (68) and (69)). Hence this equation has a bounded solution. It is clear that the equations thus constructed constitute a representative class of equations.
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\Bibitem{Bek25} |
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