\documentstyle{SibMatJ} % \TestXML \topmatter \Author Levenshtam \Initial V. \Initial B. \Gender he \ORCID 0000-0003-2438-5307 \Email vlevenshtam\@yandex.ru \AffilRef 1 \AffilRef 2 \endAuthor \Affil 1 \Organization Steklov Mathematical Institute \City Moscow \Country Russia \endAffil \Affil 2 \Organization Southern Mathematical Institute \City Vladikavkaz \Country Russia \endAffil \Origin \Journal \VMZh \Year 2024 \CopyrightYear 2024 \Volume 26 \Issue 4 \Pages 95--104 \DOI 10.46698/y0708-2078-6879-i \endOrigin \datesubmitted May 3, 2024\enddatesubmitted \dateaccepted April 25, 2026\enddateaccepted %\? \UDclass 517.955.8+517.986.7 %34B10, 35K50, 34C29 \endUDclass \thanks The author was supported by the Russian Science Foundation (project 20--11--20141, \Url*{https://rscf.ru/project/23-11-45003/}). \endthanks \title Averaging of Abstract Parabolic Equations with Multipoint Integral Boundary Conditions \endtitle \abstract A multipoint boundary value problem for an~abstract parabolic equation with a~rapidly time-oscillating nonlinear part is considered on a~time interval. The operator $-A$, where $A$~is the senior stationary linear operator of the equation, is positive. The hypotheses are stated in terms of the semigroup theory and fractional powers of~$-A$. Multipoint boundary conditions on a~time interval contain integral terms. For this problem, which depends on a large parameter (high oscillation frequency), a~limit (averaged) multipoint boundary value problem is constructed and passing to the limit in the space of continuous vector-functions on a~time interval is justified. Thus, the Krylov--Bogolyubov averaging method is justified for abstract parabolic equations with multipoint boundary conditions. The results obtained are applicable to parabolic equations in a~bounded spatial domain with multipoint boundary conditions on a~time interval and some other problems of mathematical physics. \endabstract \keywords abstract parabolic equations, multipoint boundary conditions, averaging method \endkeywords \endtopmatter \head 1. Introduction \endhead The averaging method in time [1--3], which is associated with the names of Krylov and Bogolyubov, is one of the well-known asymptotic methods of the theory of differential equations. At present, it has been developed in great detail for ordinary differential equations, partial differential equations, and others (see, for example, [1--16]). However, for boundary value problems with respect to time (especially multipoint, i.e., for more than two points), it has not yet been sufficiently developed. The articles [12--16] dealing with~ODE systems belong to this area. Averaging of multipoint boundary value problems for evolutionary~PDEs (in particular, for problems in the form of abstract parabolic equations), to the best of the author's knowledge, has not been studied before. Our goal is to show that the averaging method is also applicable to some abstract parabolic equations with multipoint boundary conditions. We examine the following abstract parabolic equations with multipoint boundary conditions including integral terms on the interval $t \in [0; T]$, $T > 0$: \iftex $$ \align & \frac{dx}{dt} = Ax + f(x, t, \omega t), \tag1 \\ & \sum\limits_{k=1}^{m} B_k x(t_k) + \sum\limits_{k=1}^{n-1} \int\limits_{s_{k}}^{s_{k+1}} C_k(s) x(s)\, ds = a(\omega), \tag2 \endalign $$ \else $$ \frac{dx}{dt} = Ax + f(x, t, \omega t), \tag1 $$ $$ \sum\limits_{k=1}^{m} B_k x(t_k) + \sum\limits_{k=1}^{n-1} \int\limits_{s_{k}}^{s_{k+1}} C_k(s) x(s)\, ds = a(\omega), \tag2 $$ \fi where $m$ and $n$ are positive integers, $ n \geqslant 2$, and $\omega$ is a large parameter. The constraints on $A$, $B_k$, and~$C_k$, the nonlinear mapping~$f$, and the vector $a$ are stated below. The main part of the article is exposed in terms of the semigroup theory of and fractional powers of operators (see, for example, [17, Sections 13 and 14]). We use the methodology developed and applied in the article~[7] (see also the monograph~[8, Chapter~1]) to justify the averaging method for abstract parabolic equations in the case of a~problem with an~initial condition (a ``one-point'' problem). The results of the theorem established in this article (see \Sec*{Section~2}) can be applied to parabolic and other problems in accord with the same scheme as that in [8, Chapter~1]. However, the class of parabolic problems in our case is significantly narrower than that in the case of a~``one-point'' problem [8, Chapter~1]. Let us explain this fact. In the articles [12--16] devoted to the justification of the averaging method for ODE systems with multipoint boundary conditions, the most important specific requirement for the perturbed problem is that the matrix corresponding to the corresponding linear operator (the linearized right-hand side of the averaged equation) and the boundary conditions is nondegenerate, i.e., its determinant does not vanish. The transition from the finite-dimensional case [12--16] to the infinite-dimensional one, when the corresponding linear operator is bounded, does not cause any fundamental difficulties, in this case the condition on the determinant can be naturally replaced with the condition of invertibility of the corresponding operator. In the case of an~unbounded linear operator with the domain not coinciding with the entire phase (Banach) space, the problem becomes significantly more complicated, which leads to some additional strict constraints. \Par{C7}{Conditions~7} and~\Par{C9}{9} below are among them. The first requirement of \Par{C7}{condition~7} is a~natural generalization of the finite-dimensional condition on the nondegeneracy of the above matrix, and the remaining requirements of \Par{C7}{conditions~7} and~\Par{C9}{9} are related to the unboundedness of $A$ and the natural use of fractional powers of the positive operator $ - A$ when dealing with nonlinear abstract parabolic equations. In view of the above-mentioned, it is important to indicate the meaningful class of parabolic problems for which all conditions of the theorem of \Sec*{Section~2} are satisfied, and, hence, the averaging is justified. The first thing that comes to mind into this regard is to consider multipoint boundary value problems, which are close, in a~certain sense, to the ``single-point'' problems already studied in~[7]. The closeness here is reduced to the invertibility of the operator $B_1$ which is the first coefficient in the multipoint boundary conditions and to the smallness, in a~certain sense, of the remaining coefficients, i.e., the operators~$B_k$, $k=\overline{2,m}$, and $C_k$, $k=\overline{1,n-1}$. Finally, we describe the simplest problems, restricting ourselves to second-order parabolic equations and, even more, to the case of $A = \Delta$ with spatially homogeneous Dirichlet conditions. \head %\Label{Sec2} 2. The Main Result and Its Proof \endhead ${\bold 1^\circ}$. \Label{1c} In this section we state the main result of the article. 1. Let $A$ be a~linear operator acting in a~complex Banach space $B$ and generating an~analytic semigroup $e^{tA}$, with $-A$~a positive operator. Denote by $B^\gamma$, with $\gamma \geqslant 0$, the Banach space agreeing with the domain of $(-A)^\gamma$ and endowed with the norm $\| x \|_{B^\gamma} = \| (-A)^\gamma x \|_B$. 2. Assume that $0 < \delta$, $\alpha < 1$, $m$ and~$n$ are natural numbers, $0 = t_1 < t_2 < \dots < t_m = T$, $0 < s_1 < s_2 < \dots < s_n = T$, and~$f$ is a~continuous mapping of the set $B^\alpha \times [0; T] \times [0; \infty)$ into the space~$B^\delta$. 3. Assume also that the mapping $f(x, t, \tau)$ is Fr\`echet differentiable with respect to the first variable and the corresponding differential $(Df)(x, t, \tau)$ realizes a~continuous mapping of $B^\alpha \times [0; T] \times [0; \infty)$ in the space $\operatorname{Hom}(B^\alpha, B^\delta)$ (the latter symbol stands for the space of linear bounded operators from~$B^\alpha$ in~$B^\delta$ endowed with the conventional operator norm). 4. \Label{I4} There exists a~mapping $F: B^\alpha \times [0; T] \to B^\delta$ which together with its Fr\`echet derivative $DF: B^\alpha \times [0; T] \to \operatorname{Hom}(B^\alpha, B^\delta)$ is continuous with respect to the first variable for all $(x, t) \in B^\alpha \times [0; T]$, and the limit equalities $$ \lim\limits_{N \to \infty} \frac{1}{N} \int\limits_{0}^{N} f(x, t, \tau) \, d\tau = F(x, t), \quad \lim\limits_{N \to \infty} \frac{1}{N} \int\limits_{0}^{N} (Df)(x, t, \tau) \, d\tau = (DF)(x, t) $$ are valid in $B^\delta$ and $\operatorname{Hom}(B^\alpha, B^\delta)$, respectively. 5. For every set $K$ bounded in $B^\alpha$, the mappings $f(x, t, \tau)$ and $(Df)(x, t, \tau)$, $(x, t, \tau) \in K \times [0; T] \times [0; \infty)$, are uniformly bounded, equicontinuous in $\tau$, continuous in $(x, t)$, and the limits in \Par{I4}{Item~4} are uniform with respect to $(x, t) \in K \times [0; T]$. 6. The vectors $a(\omega)$ and $\omega \in (0; \infty)$ belong to the space $B^1$ and there exists a~vector $a_0$ such that $$\lim_{\omega \to \infty} a(\omega) = a_0. $$ Together with perturbed problem \Tag(1), \Tag(2), we consider the averaged problem \iftex $$ \align &\frac{dy}{dt} = Ay + F(y, t), \tag3 \\ &\sum\limits_{k=1}^{m} B_k y(t_k) + \sum\limits_{k=1}^{n-1} \int\limits_{s_k}^{s_{k+1}} C_k(s) y(s) \, ds = a_0. \tag4 \endalign $$ \else $$ \frac{dy}{dt} = Ay + F(y, t), \tag3 $$ $$ \sum\limits_{k=1}^{m} B_k y(t_k) + \sum\limits_{k=1}^{n-1} \int\limits_{s_k}^{s_{k+1}} C_k(s) y(s) \, ds = a_0. \tag4 $$ \fi 7. \Label{C7} Assume that the operator $$ P = \sum\limits_{k=1}^{m} B_k e^{t_k A} + \sum\limits_{k=1}^{n-1} \int\limits_{s_k}^{s_{k+1}} C_k(s) e^{sA} \, ds $$ in $B$ is invertible, the operators $P^{-1}B_k$, $P^{-1}C_k(s)$, and $P^{-1}$ belong to $\operatorname{Hom}(B^1, B^\alpha)$, and the norms of the operators of the above family depending on $s$ are uniformly bounded with respect to $s \in [s_1; T]$. Next, let the symbol $C([0; T], B^\alpha)$ designate the Banach space of continuous vector-functions $u: [0; T] \to B^\alpha$ endowed with the norm $$ \| x \|_{C([0; T], B^\alpha)} = \max\limits_{t \in [0; T]} \| x(t) \|_{B^\alpha}. $$ 8. Let problem \Tag(3), \Tag(4) have a~solution $\overset{\circ}\to{y}(t) \in C([0; T], B^\alpha)$. The Banach space $C_\mu^\gamma(B^\alpha)$, with $\gamma, \mu \in (0; 1)$, introduced in [7] plays an~important role in this article. We are interested in the case of $\gamma=\mu$ and, hence, we employ the space of mappings $x: [0; T] \to B^\alpha$ endowed with the norm $$ \| x \|_{C_{\mu}^{\mu}([0; T], B^\alpha)} = \sup\limits_{\varepsilon \in (0; T]} \varepsilon^{\mu} \Bigl\{\,\max\limits_{t\in [\varepsilon; T]} \| x(t) \|_{B^\alpha} + \sup\limits_{t_1, t_2\in [\varepsilon; T]} \frac{\| x(t_2) - x(t_1) \|_{B^\alpha}}{|t_2 - t_1|^\mu} \Bigr\} < \infty. $$ Following [7], denote by $C$ the space $$ C = C_{\mu}^{\mu}([0; T], B^\alpha) \cap C([0; T], B^\alpha). $$ 9. \Label{C9} Assume also that the operator $$ \align I &+ \int\limits_{0}^{t} e^{(t-\tau)A} (DF)(\overset{\circ}\to{y}(\tau), \tau) \, d\tau \\ & + e^{tA} P^{-1} \Biggl\{ \sum\limits_{i=1}^{m} B_i \int\limits_{0}^{t_i} e^{(t_i-\tau)A} (DF)(\overset{\circ}\to{y}(\tau), \tau)\, d\tau + \sum\limits_{i=1}^{n} \int\limits_{s_i}^{s_{i+1}} C_i(s) \Biggl[\,\int\limits_{0}^{s} e^{(s-\tau)A} (DF)(\overset{\circ}\to{y}(\tau), \tau)\, d\tau \Biggr] ds \Biggr\} \endalign $$ is invertible in $C$. \proclaim{Theorem} There exists a~number $\omega_0 > 0$ such that, for every $\omega > \omega_0$, problem \Tag(1), \Tag(2) in some $C$-neighborhood about the vector-function $\overset{\circ}\to{y}$ has a~unique solution $x_\omega(t)$ satisfying the limit equality $$ \lim\limits_{\omega \to \infty} \| x_\omega - \overset{\circ}\to{y} \|_{C([0; T], B^\alpha)} = 0. $$ \endproclaim ${\bold 2^\circ}$. This section is devoted to the proof of the theorem. A solution to equation \Tag(1) with the initial condition $x(0) = x_0 \in B^\alpha$, if exists, as is known, satisfies (see [17, Lemma~23.1]) to the equation $$ x(t) = e^{tA} x_0 + \int\limits_{0}^{t} e^{(t-\tau)A} f[x(\tau), \tau, \omega\tau] \, d\tau. \tag5 $$ Inserting \Tag(5) in \Tag(2), we obtain $$ \align &\Bigg[ \sum\limits_{i=1}^{m} B_i e^{t_iA} + \sum\limits_{i=1}^{n-1} \int\limits_{s_i}^{s_{i+1}} C_i(s) e^{sA}\, ds \Bigg] x_0 \\ &\qquad= a(\omega) - \sum\limits_{i=1}^{m} B_i \int\limits_{0}^{t_i} e^{(t_i-\tau)A} f[x(\tau), \tau, \omega\tau] \, d\tau \\ &\qquad\qquad\ - \sum\limits_{i=1}^{n-1} \int\limits_{s_i}^{s_{i+1}} C_i(s) \int\limits_{0}^{s} e^{(s-\tau)A} f[x(\tau), \tau, \omega\tau] \, d\tau ds. \endalign $$ Expressing $x_0$ and inserting the result in \Tag(5), we infer $$ \aligned x(t) &= e^{tA} P^{-1} \Biggl\{ a(\omega) - \sum\limits_{i=1}^{m} B_i \int\limits_{0}^{t_i} e^{(t_i-\tau)A} f[x(\tau), \tau, \omega\tau] \, d\tau \\ &\qquad - \sum\limits_{i=1}^{n-1} \int\limits_{s_i}^{s_{i+1}} \Bigg[ C_i(s) \int\limits_{0}^{s} e^{(s-\tau)A} f[x(\tau), \tau, \omega\tau] \, d\tau \Bigg] \, ds \Biggr\} \\ &\qquad + \int\limits_{0}^{t} e^{(t-\tau)A} f[x(\tau), \tau, \omega\tau] \, d\tau \equiv K_\omega(x)(t). \endaligned \tag6 $$ Similarly, a~solution to the averaged problem is representable in integral form $$ \aligned y(t) &\equiv e^{tA} P^{-1} \Biggl\{ a_0 - \sum\limits_{i=1}^{m} B_i \int\limits_{0}^{t_i} e^{(t_i-\tau)A} F[y(\tau), \tau] \, d\tau \\ &\qquad - \sum\limits_{i=1}^{n-1} \int\limits_{s_i}^{s_{i+1}} C_i(s) \Bigg[\,\int\limits_{0}^{s} e^{(s-\tau)A} F[y(\tau), \tau] \, d\tau \Bigg] ds \Biggr\} \\ &\qquad + \int\limits_{0}^{t} e^{(t-\tau)A} F[y(\tau), \tau] \, d\tau \equiv [ K_\infty(y) ] (t). \endgathered \tag7 $$ Introduce the operator $N(x, \omega)$ acting from $C \times [1; \infty]$ into $C$ by the rule $$ [ N(x, \omega) ] (t) = \cases x - [ K_\omega(x) ] (t), & \omega < \infty, \\ x - [ K_\infty(x) ] (t), & \omega = \infty. \endcases \tag8 $$ In the proof of the theorem, in particular, justifying well-posedness of the above-introduced operator $N(x, \omega)$, the following two estimates of the semigroup theory (see [17, Theorem~14.11; 8, Lemma~3.1]) play a~crucial role: $$ \| (-A)^\tau e^{tA} \|_B \leqslant \frac{c(\tau)}{t^\tau},\quad \text{ where } \tau > 0, \ t \in (0; T], \tag9 $$ $c(\tau)$~is a~constant depending on $\tau$; $$ \frac{\| e^{t_2A} - e^{t_1A} \|_{\operatorname{Hom}(B^\eta, B^\gamma)}}{(t_2 - t_1)^\nu} \leqslant \frac{c(\eta, \gamma, \nu)}{t_1^{\nu+\gamma-\eta}}, \tag10 $$ where $0 < t_1 < t_2 \leqslant T$, $0 < \nu \leqslant 1$, $\gamma \geqslant 0$, and $\nu + \gamma > \eta \geqslant 0$. Prove well-posedness of the above-introduced operator $N(x, \omega)$ (see \Tag(8)). To this end, we need to demonstrate that this operator takes $C \times [1; \infty]$ into $C$. We restrict ourselves to the case of $\omega = \infty$, since the case of $\omega < \infty$ is treated by analogy. So, let $y \in C$. Since $C \subset C([0; T], B^\alpha)$, conditions of Item~\Par{1c}{$1^\circ$} %\?\bold 1 yield $$ {F[ y(t), t] \in C([0; T], B^\delta)}. $$ Estimate~\Tag(9) implies that $$ \int\limits_{0}^{t_i} e^{(t_i-\tau)A} F[y(\tau), \tau] \, d\tau \in B^{1}, \quad \int\limits_{0}^{s} e^{(s-\tau)A} F[y(\tau), \tau] \, d\tau \in B^{1}. $$ Denoting the vector in the brackets of \Tag(7) by~$d$, in view of \Par{C7}{condition~7} of Item~\Par{1c}{$1^\circ$} and inequality \Tag(10), we obtain $$ \| e^{tA} P^{-1} d \|_{C_\mu^{\mu}(B^\alpha)} = \sup\limits_{\varepsilon \in (0; T]} \varepsilon^{\mu} \Bigl\{ \max\limits_{t \in [\varepsilon; T]} \| e^{tA} P^{-1} d \|_{B^\alpha} + \sup\limits_{\varepsilon \leqslant t_1 < t_2 \leqslant T} \frac{\| (e^{t_2A} - e^{t_1A}) P^{-1} d \|}{(t_2 - t_1)^\mu} B^\alpha \Bigr\} < \infty. $$ Note that \Tag(9) and \Tag(10) insure the membership $$\int\limits_{0}^{t} e^{(t-\tau)A} F[y(\tau), \tau] \, d\tau \in C_\mu^0([0; T], B^\alpha). $$ Well-posedness of the definition of~\Tag(8) of the operator $N$ is proved. The theorem follows from the classical implicit function theorem in a~Banach space and the following lemma. \proclaim{Lemma} The mapping $N$ is continuous and continuously differentiable with respect to~$x$ at~$(\overset{\circ}\to{y}, \infty)$. \endproclaim It is not difficult to prove this lemma applying the scheme of the proof of Theorem~1 in~[7]. Our \Par{C7}{conditions~7} and~\Par{C9}{9}, whose analogs are absent in~[7], fit naturally into this scheme. So, the proof of this lemma is excessive here. \head 3. An Application of the Theorem of \Sec*{Section~2} to a~Parabolic Problem \endhead Assume that $\Omega$~is a~bounded domain in $R^N$, with $C^2$-smooth boundary $\partial \Omega$, $N$~is a~positive integer, and $T > 0$. In the cylinder $(x, t) \in \overline{\Omega} \times [0; T]$, where $x$ designates an~independent space variable, we consider the following problem depending on a~large parameter $\omega$: $$ \aligned \frac{\partial u}{\partial t} &= \Delta u + \varphi \Bigl(x, u, \frac{\partial u}{\partial x_1}, \dots, \frac{\partial u}{\partial x_N}, t, \omega t \Bigr), \\ u \big|_{\partial \Omega} &= 0, \quad u(x, t_1) + \sum\limits_{k=2}^{m} B_k(x) u(x, t_k) + \sum\limits_{k=1}^{n-1} C_k(x) \int\limits_{s_k}^{s_{k+1}} u(x, s) \, ds = a_\omega(x). \endaligned \tag11 $$ Here $t_k$ and $s_k$ are those in Item~\Par{1c}{$1^\circ$} of \Sec*{Section~2}, and $\varphi(x, u_0, u_1, \dots, u_N, t, \tau)$~is a~continuous real function defined on the set $\overline{\Omega} \times R^{N+1} \times [0; T] \times [0; \infty)$ and satisfying the following conditions: 1. The function $\varphi$ is continuous and continuously differentiable with respect to $u_0$, $u_1,\dots,u_N$ and together with the derivatives satisfies the Lipschitz condition in the variables $(x, u_0, u_1,\dots, u_N, t)$ uniformly in $\tau$ provided that the vector $(u_0, u_1,\dots, u_N)$ belongs to a~bounded subset of the space $R^{N+1}$. 2. The function $\varphi$ is $2\pi$-periodic in $\tau$; the symbol $\Phi$ stands for its average in~$\tau$ over the period. 3. The functions $B_k(x)$ and $C_k(x)$ and the vector-function $a_\omega(x)$ as well are real, twice continuously differentiable with respect to $x$, and compactly supported. 4. There exists a~function $a_0(x)$ such that $$ \lim\limits_{\omega \to \infty} \| a_\omega(x) - a_0(x) \|_{C^2(\bar{\Omega})} = 0. $$ 5. Let $\delta > 0$ be a~number such that, for $k=2,\dots, m$ and $l=1, 2,\dots, n-1$, we have $$ \| B_k \|_{C^2(\bar{\Omega})} \leqslant \delta, \quad \| C_l \|_{C^2(\bar{\Omega})} \leqslant \delta. $$ Examine the averaged problem $$ \aligned \frac{\partial v}{\partial t} &= \Delta v + \Phi \Bigl( x, v, \frac{\partial v}{\partial x_1}, \dots, \frac{\partial v}{\partial x_N}, t \Bigr), \quad (x, t) \in \overline{\Omega} \times [0; T], \\ v\big|_{\partial \Omega} &= 0, \quad v(x, t_1) + \sum\limits_{k=2}^{m} B_k(x) v(x, t_k) + \sum\limits_{k=1}^{n-1} C_k(x) \int\limits_{s_k}^{s_{k+1}} v(x, s) \, ds = a_0. \endaligned \tag12 $$ 6. Let \Tag(12) have a~solution $\overset{\circ}\to{v}(x, t)$. The statement below follows from the theorem of \Sec*{Section~2}. \proclaim{Corollary} There exist positive numbers $\delta_0$ and $\omega_0$ such that, for $0 < \delta < \delta_0$ and $\omega > \omega_0$, problem~\Tag(11) has a~unique solution $u_\omega$ in the $C$-neighborhood about the function $\overset{\circ}\to{v}$ and $$ \lim\limits_{\omega \to \infty} \| u_\omega(x, t) - \overset{\circ}\to{v}(x, t) \|_{C([0; T], C^1(\overline{\Omega}))} = 0. $$ \endproclaim \demo{Proof} We need to prove that the conditions of this section ensure that of Item~\Par{1c}{$1^\circ$} of \Sec*{Section~2} stated for problem~\Tag(11). This proof ir realized mainly in accord with the scheme of [8, Section 4] in the derivation of Theorem~2.2 (for parabolic initial-boundary problems) and the proof of Theorem~2.1 (for abstract parabolic problems). In this regard, we confine the exposition to deriving from the conditions of the section only \Par{C7}{conditions~7} and~\Par{C9}{9} of Item~\Par{1c}{$1^\circ$} of \Sec*{Section~2}, which do not arise in [7; 8, Chapter~1]. First of all, we observe that, in view of [7; 8, Chapter~1], we need to take the space $L_p(\Omega)$, $p > N$, as the Banach space of \Sec*{Section~2}. Take~1 rather than~$\alpha$. Then $B^\alpha = B^1$~is a~closed subspace of the space~$W^2_p(\Omega)$ comprising the functions of this space vanishing on the boundary of the domain. The operator $P$ defined in \Sec*{Section~2} can be now written as $$ P = I + \sum\limits_{k=2}^{m} B_k e^{t_k \Delta} + \sum\limits_{k=1}^{n-1} \int\limits_{s_k}^{s_{k+1}} C_k e^{s \Delta} \, ds. $$ Show that, for $\delta$ sufficiently small, there exists the inverse operator $$ P^{-1} = I + \sum\limits_{r=1}^{\infty} (-1)^r \Bigg[ \sum\limits_{k=2}^{m} B_k e^{t_k \Delta} + \sum\limits_{k=1}^{n-1} \int\limits_{s_k}^{s_{k+1}} C_k e^{s \Delta} \, ds \Bigg]^{r} \equiv I + S \tag13 $$ from the space $\operatorname{Hom}(B^1, B^1)$. Take an~arbitrary function $\varphi \in C(\overline{\Omega})$ and consider the functions $B_k e^{t_k \Delta} \varphi$ and $$C_k \int\limits_{s_k}^{s_{k+1}} e^{s \Delta}\varphi\,ds. $$ These functions and their derivatives with respect to $x$ up to the second order are bounded by $\delta_1 \| \varphi \|_{C(\bar{\Omega})}$, with $\delta_1 \to 0$ as $\delta \to 0$. As is easily seen, there exists a~constant $K_0$ independent of $r$ such that $r$th summand of the series $S$ in bounded in the norm of the space $B^1$ by the number $K_0(n+m-1)^{r} \delta_1^r \| \varphi \|_B$ and, thereby, $S \in \operatorname{Hom}(B, B^1)$ and $P^{-1} \in \operatorname{Hom}(B^1, B^1)$ for $\delta$ sufficiently small. Hence, $$ P^{-1} B_k, P^{-1} C_k \in \operatorname{Hom}(B^1, B^1). $$ It remains to establish that, under the conditions of this section, \Par{C9}{condition~9} of Item~\Par{1c}{$1^\circ$} in \Sec*{Section~2} is valid, i.e., for $\delta$ sufficiently small, the operator $$ \align D_1 &+ e^{t \Delta} P^{-1} \Biggl\{ \sum\limits_{i=2}^{m} B_i \int\limits_{0}^{t_i} e^{(t_i - \tau) \Delta} (DF) (\overset{\circ}\to{y}(\tau), \tau) \, d\tau \\ & + \sum\limits_{i=1}^{n-1} \int\limits_{s_i}^{s_{i+1}} C_i(s) \Biggl[ \int\limits_{0}^{s} e^{(s-\tau) \Delta} (DF) (\overset{\circ}\to{y}(\tau), \tau) \, d\tau \Biggr] ds \Biggr\} \equiv D_1 + D_2, \endalign $$ where $$ D_1 = I + \int\limits_{0}^{t} e^{(t-\tau) \Delta} (DF)(\overset{\circ}\to{y}(\tau), \tau) \, d\tau $$ is invertible in $C$. The last claim follows %\? from the fact that $D_1$ is invertible in $C$ (it is proved in~[8]) and the operator $D_2$ also acting in $C$ is small (it follows %\? from the above arguments). 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