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     \begin{center}
     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
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     \noindent
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     \vspace{1mm}
     {\flushleft\it Том 16, стр. 144--144 (2019) \hspace{35mm}{\rm\small UDK 517.958.72+517.958:536.2 \udcs}} %Volume, pages in Russian,
                                                                                    %filled by Editorial board
     \newline
         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\begin{document}
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\title[Kernel determination problem ...]{Kernel determination problem for a
parabolic integro--differential equation with a variable thermal
conductivity}
\author{{D.K. Durdiev,  Zh.Z. Nuriddinov}}%
\address{Durdimurod Kalandarivich Durdiev
\newline\hphantom{iii} Senior Lecturer, Department of
Mathematics, Bukhara State University,
\newline\hphantom{iii} St. M.Ikbol, 11,
\newline\hphantom{iii} 705018, Bukhara,  Uzbekistan.}%
\email{durdiev65@mail.ru}%

\address{ Nuriddinov Zhavlon Zafarovich
\newline\hphantom{iii} Doctor of Physics and Mathematics, Professor, Department of
Mathematics, Bukhara State University,
\newline\hphantom{iii} St. M.Ikbol, 11,
\newline\hphantom{iii} 705018, Bukhara,  Uzbekistan.}%
\email{j.zafarovich@mail.ru}%
\thanks{\sc Durdiev D.K, Nuriddinov Zh.Z.
Kernel determination problem for a parabolic integro--differential
equation with a variable thermal conductivity}
\thanks{\copyright \ 2020 Durdiev D.K, Nuriddinov Zh.Z.}
\thanks{\rm The work of first author  was supported by the research
project  of the Ministry of Innovation Development, Uzbekistan
(No. HE-F-04).}
\thanks{\it Received  January, 1, 2015, published  March, 1,  2015.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } The inverse problem of finding a
multidimensional memory kernel of a time convolution integral
depending on a time variable $t$ and $ (n-1) $-dimensional spatial
variable $x'= \left(x_1, ..., x_ {n-1}\right)$ in the
$n$-dimensional heat equation with a time-dependent   coefficient
of thermal conductivity is studied.  The direct problem is the
Cauchy problem for this equation. The integral term has the time
convolution form of kernel and an elliptic operator of direct
problem solution. As additional information, the solution of the
direct problem on the hyperplane $x_n = 0$ is given. The problem
reduces    to an auxiliary problem which is more convenient for
further consideration.   Then the auxiliary problem is replaced by
an equivalent  system of Volterra-type integral equations with
respect to unknown functions.    Applying the method of
contraction  mappings to this system in the H\"{o}lder class of
functions,    it is  proved  the main result of the paper
representing  a local existence and uniqueness theorem.
\medskip

\noindent{\bf Keywords:} Heat equation, memory kernel, H\"{o}lder
space, convolution integral, contraction  mapping.
 \end{quote}
}


\section{Introduction. Setting up problem}

The constitutive relations for a linear non-homogeneous heat
propagation and diffusion processes in medium with memory contain
a time- and space-dependent kernel in an integral term of time
variable  convolution type  [1]-[5]. Often, in practical
applications these kernels are unknown functions and it is
required to determine them. Kernel determination inverse  problems
in parabolic integro differential equations  were the object of
studying since the end of last century. In literature are most
often found the linear inverse source and nonlinear inverse
coefficient problems with different type of over determination
conditions (see, for example [6]-[15] and references there). In
these works authors discussed the unique solvability and stability
estimates of solution as well a numerical approach for solving
such problems.

Among the works devoted to finding the kernel depending  on one
time variable (one dimensional inverse problem), we note [2], [4],
[16]. When kernel, in addition to time variable also depends on
all or part of spatial variables, i.e. multidimensional inverse
problems are few studied. In this direction we observe [4],
[17]-[19].  In the article [17] the problem of determining of the
kernel depending on time variable $t$ and $(n-1)-$dimensional
spatial variable $x'=\left(x_1,...,x_{n-1}\right)$ was
investigated. The principal part of integro--differential equation
in [17] is $n-$ dimensional heat conduction operator and integral
part has a form of time-convolution with regard to unknown
functions: the solutions of direct and inverse problems. However
in applications, the great interest is the study of kernel
determination  problems when in  convolution type integral the
kernel is multiplied by an elliptic operator of solution to the
direct problem. The present work  considers this kind of parabolic
integro--differential equations  for which will be studied inverse
problem.

Consider the problem of determining functions $u(x, t),$ $k(x',
t),$ \\  $x=\left(x', x_n\right)=\left(x_1, ..., x_{n-1},
x_n\right),$ $t>0$ from the  equations
\begin{equation}
u_t=a(t)\Delta u-\int\limits_0^tk(x',t-\tau)a(\tau)\Delta u(x,
\tau)d\tau,\quad (x,t)\in\mathbb{R}_T^n,
\end{equation}
\begin{equation}
u(x,0)=\varphi(x),\, x \in\mathbb{R}^n,
\end{equation}
\begin{equation}
u(x',0,t)=f(x',t), \, (x', t)\in\mathbb{R}_T^{n-1}, \, \ f(x',
0)=\varphi(x', 0),
\end{equation}
  where $\triangle$  is the Laplace
operator with respect to spatial
variables   $x=(x_1, ...,  x_n),$ \\
${\mathbb{R}_T^{n}}=\left\{(x, t)| x=(x', x_n)\in{\mathbb{R}^{n}},
\ 0<t<T\right\}$ is  a strip with thickness $T,$  $T>0$ is an
arbitrary fixed  number; $a(t)\in C^2[0,T],$  $0<a_0\leq a(t)\leq
a_1<\infty$, \ $a_0$ and $a_1$ are given numbers.

 Our investigations  generalize the result of  works [17], [18] to the case of the
integro-differential heat equation with a variable coefficient of
thermal conductivity and special convolution integral.


Our main result of this work is following theorem:

\begin{theorem}
Suppose that all conditions of Section 1 with respect to the given
functions $a(t),$ $\varphi(x),$ $f(x', t)$ and the matching
conditions of (3) and  (10) are satisfied. Besides $\left|f_t(x',
0)\right|>f_0=const>0,$ $f_0$ is a given number.

Then there exists sufficiently small number $T>0$ that the unique
solution to the inverse problem (1)-(3) exists  in the class of
functions $u(x, t)\in H^{l+2,
(l+2)/2}\left({\bar{\mathbb{R}}_T^{n}}\right),$ $k(x', t)\in H^{l,
l/2}\left({\bar{\mathbb{R}}_T^{n-1}}\right),\ l\in(0,1).$
\end{theorem}



In the sequel, we will use the H\"{o}lder  space $H$ with exponent
$\alpha$ for functions depending only  on spatial variables
 and for  functions depending on both spatial and time  variables -
 H\"{o}lder  space $H^{\alpha, \alpha/2}$ with exponents $\alpha$ and $\alpha/2$,
 $\alpha\in\mathbb{Z}_+:=\{0,1,...\}$. Throughout this paper  we will  assume  that
 $\varphi(x)\in H^{l+8}\left(\mathbb{R}^{n}\right),$
 $\varphi(x)\geq \varphi_0=const>0,$
 $f(x', t)\in H^{l+6, (l+6)/2}\left(\bar{\mathbb{R}}^{n-1}_T\right).$
 Spaces $H^{l}(Q),$ $H^{l, l/2}(Q_T)$ and norms in them are defined
in [20, p. 16-27]. In what follows, for norm of functions in the
space $H^{l, l/2}(Q_T)$ (in concrete cases
$Q_T={\mathbb{R}}^{n}_T$ or $Q_T={\mathbb{R}}^{n-1}_T$ ) depending
on spatial and time variables will be used notation $|\cdot|_T^{l,
l/2},$ while for functions depending only on spatial variables we
use $|\cdot|^{l}$ (in this case $Q={\mathbb{R}}^{n}$ or
$Q={\mathbb{R}}^{n-1}$).\\

The article is organized as follows. In Section 2, we reduce  the
 problem (1)-(3) into an auxiliary problem where the additional condition
contains the unknown $k$ outside integral. In Section 3, we
replace  auxiliary problem by an equivalent  system of integral
equations with respect to unknown functions. In Section 4, we
prove the main result which states the existence and uniqueness of
solution of  problem by  fixed point theorem.

\section{Preliminaries. Auxiliary problem}
We give the following assertion which will be used in obtaining
the auxiliary problem.
\begin{lemma}
 If $\{k(t),r(t)\}\in{C[0,T]}$ for a fixed $T>0$
and $k(t), r(t)$ are connected by the integral equation
\begin{equation*}
r(t)=k(t)+\int_0^tk(t-\tau)r(\tau)d\tau, \ \  t\in[0,T],
\end{equation*}
then  the solution of the integral equation
\begin{equation*}
\varphi(t)=\int_0^tk(t-\tau)\varphi(\tau)d\tau+f(t), \ \
f(t)\in{C[0,T]}
\end{equation*}
is expressed by formula
\begin{equation*}
\varphi(t)=\int_0^tr(t-\tau)f(\tau)d\tau+f(t).
\end{equation*}
\end{lemma}
\begin{proof}
The proof of Lemma follows from the general theory of
Volterra -type integral equations (see, for example [21]).

Rewriting the equation (1) in the form of Volterra integral
equation with respect to $a(t)\triangle u$
\begin{equation}
a(t)\Delta u=\int\limits_0^tk(x',t-\tau)a(\tau)\Delta u(x,
\tau)d\tau+u_t
\end{equation}
and at any fixed $x\in\mathbb{R}^n$ applying Lemma to (4), we have
\begin{equation}
u_t-a(t)\Delta u=-\int\limits_0^t r(x',t-\tau)u_\tau(x,\tau)d\tau.
\end{equation}
In (5) $r(x',t)$ is the resolvent of the kernel $k(x',t)$ and it
satisfies the integral equation
\begin{equation}
r(x',t)=k(x',t)+\int\limits_0^tk(x',t-\tau)r(x',t)d\tau, \quad
(x,t)\in\mathbb{R}_T^n.
\end{equation}
In the sequel we investigate the problem of determining the
functions $u(x, t),$ $r(x', t)$ satisfying the equations (5), (2),
(3). Then after solving this problem,  $k(x', t)$ can be found
from (6).

We introduce new function $\vartheta^{(1)}(x,t)$ by formula
$\vartheta^{(1)}(x,t)= u_{x_nx_n}(x,t)$. Then the straightaway
differentiation of equations  (5), (2) with respect to $x_n$
twice  leads us to the following relations for
$\vartheta^{(1)}(x,t)$:
\begin{equation}
\vartheta^{(1)}_t-a(t)\Delta\vartheta^{(1)}=-\int\limits_0^t
r(x',t-\tau)\vartheta^{(1)}_\tau(x,\tau)d\tau,
\end{equation}
\begin{equation}
\vartheta^{(1)}(x,0)=\varphi_{x_nx_n}(x).
\end{equation}
The over determination condition can be transformed as follows. We
allocate the term  $a(t)u_{x_nx_n}$ in the expression  $a(t)\Delta
u$ of (5) and set $x_n=0.$ Then in view of
$a(t)u_{x_nx_n}=a(t)\vartheta^{(1)}$ and using (2), we obtain
\begin{equation}
\vartheta^{(1)}(x',0,t)=\frac{1}{a(t)}f_t(x',t)-\sum\limits_{i=1}^{n-1}f_{x_ix_i}(x',t)+\frac{1}{a(t)}\int\limits_0^t
r(x',t-\tau)f_\tau(x',\tau)d\tau.
\end{equation}
Requiring the continuity of the function $\vartheta^{(1)}(x, t)$
for $x_n=t=0, \, x\in{\mathbb{R}^{n-1}}$, from (8) and (9) it
follows the matching condition
\begin{equation}
\varphi_{x_nx_n}(x',
0)=\frac{1}{a(0)}f_t(x',0)-\sum\limits_{i=1}^{n-1}f_{x_ix_i}(x',0).
\end{equation}
Hereinafter, the values of functions $a(t),$  $f(x',t)$  and their
derivatives at $t=0$ we will understand as the limit for
$t\rightarrow +0.$


 We carry out the next converting
of problem. Denoting for this purpose  the derivative of
$\vartheta^{(1)}(x,t)$ with respect to
 $t$ by $\vartheta^{(2)}(x,t), $ i.e. $\vartheta^{(2)}(x,t):=\vartheta_t^{(1)}(x,t)$ and
 $h(x',t):=r_t(x',t),$ from (7)-(9) we get
\begin{equation}
\vartheta_t^{(2)}-a(t)\Delta\vartheta^{(2)}=a'(t)\Delta
\vartheta^{(1)}-r(x',0)\vartheta^{(2)}-\int\limits_0^t
h(x',t-\tau)\vartheta^{(2)}(x,\tau)d\tau,
\end{equation}
\begin{equation}
\vartheta^{(2)}(x,0)=a(0)\Delta\varphi_{x_nx_n}(x) ,
\end{equation}
\begin{equation}
\vartheta^{(2)}(x',0,t)=-\frac{a'(t)}{a^2(t)}f_t(x',t)+\frac{1}{a(t)}f_{tt}(x',t)-\sum\limits_{i=1}^{n-1}f_{tx_ix_i}(x',t)-
\end{equation}
\begin{equation*}
-\frac{a'(t)}{a^2(t)}\int\limits_0^t
r(x',t-\tau)f_\tau(x',\tau)d\tau+\frac{1}{a(t)}\int\limits_0^t
h(x',\tau)f_\tau(x',t-\tau)d\tau+\frac{1}{a(t)}r(x', 0)f_t(x',t).
\end{equation*}
Here the initial condition (12) is derived from (7) by setting
$t=0$ and using (8). The equations  (11) and (13) also include the
unknown function $r(x', 0).$ We can determine this function by
following approach. Same as when getting the equality (10), we
require  the continuity of the function $\vartheta^{(2)}(x, t)$
for $x_n=t=0, \, x\in{\mathbb{R}^{n-1}}.$ Then,  from (12) and
(13) we have some equation the solving of which with respect to
$r(x', 0)$ yields
\begin{equation}
r(x', 0)=\frac{1}{f_t(x',
0)}\Bigg[a^2(0)\Delta\varphi_{x_nx_n}(x',
0)+\frac{a'(0)}{a(0)}f_t(x', 0) -f_{tt}(x', 0)+
\end{equation}
\begin{equation*}
+a(0)\sum\limits_{i=1}^{n-1}f_{tx_ix_i}(x',0)\Bigg]
\end{equation*}

 Further the function $r(x', 0)$ is assumed as known one.

 Introducing also function $\vartheta(x,t)$ as
$\vartheta(x,t):=\vartheta_t^{(2)}(x,t),$ in this way, we obtain
the final problem of determining $\vartheta(x,t)$ and $h(x',t)$
satisfying the equations
\begin{equation}
\vartheta_t-a(t)\Delta\vartheta=2a'(t)\Delta\vartheta^{(2)}+a''(t)\Delta
\vartheta^{(1)}-r(x',0)\vartheta-h(x',
t)a(0)\Delta\varphi_{x_nx_n}(x)-
\end{equation}
\begin{equation*}
-\int\limits_0^t h(x',\tau)\vartheta(x,t-\tau)d\tau,
\end{equation*}
\begin{equation}
\vartheta(x,0)=\Psi(x),
\end{equation}
\begin{equation}
\vartheta(x',0,t)=F(x',t)+\left(2\frac{(a'(t))^2}{a^3(t)}-\frac{a''(t)}{a^2(t)}\right)\int\limits_0^t
r(x',t-\tau)f_\tau(x',\tau)d\tau-
\end{equation}
\begin{equation*}
-2\frac{a'(t)}{a^2(t)}\int\limits_0^t
h(x',\tau)f_\tau(x',t-\tau)d\tau-\frac{1}{a(t)}\int\limits_0^t
h(x',\tau)f_{tt}(x',t-\tau)d\tau+\frac{1}{a(t)}h(x',t)f_t(x',0),
\end{equation*}
where
\begin{equation*}
\Psi(x)=a^2(0)\Delta^2\varphi_{x_nx_n}(x)+a'(0)\Delta\varphi_{x_nx_n}(x)-r(x',0)a(0)\Delta\varphi_{x_nx_n}(x),
\end{equation*}
and through $F(x',t)$ in (17)  is denoted the known function:
\begin{equation*}
F(x',t)=\left(-\frac{a''(t)}{a^2(t)}+2\frac{(a'(t))^2}{a^3(t)}\right)f_t(x',t)+\frac{1}{a(t)}f_{ttt}(x',t)-\sum\limits_{i=1}^{n-1}f_{ttx_ix_i}(x',t)-
\end{equation*}
\begin{equation*}
-2\frac{a'(t)}{a^2(t)}r(x',0)f_t(x',t)+\frac{1}{a(t)}r(x',0)f_{tt}(x',t)-2\frac{a'(t)}{a^2(t)}f_{tt}(x',t).
\end{equation*}

The equation (15) includes the expression
$2a'(t)\Delta\vartheta^{(2)}+a''(t)\Delta \vartheta^{(1)}$ on the
right side. Taking into account
$\vartheta^{(1)}_t=\vartheta^{(2)}$ and using (7) we represent it
through $\vartheta^{(2)}:$
\begin{equation}
a''(t)\Delta\vartheta^{(1)}=\frac{a''(t)}{a(t)}\vartheta^{(2)}+\frac{a''(t)}{a(t)}\int\limits_0^t
r(x',t-\tau)\vartheta^{(2)}(x,\tau)d\tau.
\end{equation}
In similar way from (7) and (11), we obtain
\begin{equation}
2a'(t)\Delta\vartheta^{(2)}=2\left(\ln
a(t)\right)'\Bigg[\vartheta-\left(\ln
a(t)\right)'\Big(\vartheta^{(2)}+\int\limits_0^t
r(x',t-\tau)\vartheta^{(2)}(x,\tau)d\tau\Big)-
\end{equation}
\begin{equation*}
-r(x', 0)\vartheta^{(2)}-\int\limits_0^t
h(x',t-\tau)\vartheta^{(2)}(x,\tau)d\tau\Bigg].
\end{equation*}

In the future, we assume that in equation (15) the expression
$2a'(t)\Delta\vartheta^{(2)}+a''(t)\Delta \vartheta^{(1)}$ is
excluded with the help of (18) and (19).

At fulfilling  the matching condition (10) and relation (14) it is
not difficult carrying out the inverse transforms to derive from
(11), (12) and (15)-(17) the equations (1)-(3) [17]. Thus the
inverse  problem (1)-(3) is equivalent to problem (11), (12) and
(15)-(17) of determining the functions $\vartheta^{(2)}(x,t),
\vartheta(x, t), h(x',t), \ r(x,'t).$
\end{proof}

\section{Reduction of the Auxiliary problem}

Here we prove the main result of Section 3, which is the following
assertion:
\begin{lemma}
 The auxiliary problems (11)-(12),
(15)-(16) and the equality  $h(x',t):=r_t(x',t),$ is equivalent to
the problem of determination functions $\vartheta^{(2)}(x,t),
\vartheta(x, t)$, $h(x',t), \ r(x,'t)$ from the following system
of integral equations:
\end{lemma}
\begin{proof}
\begin{equation}
\vartheta^{(2)}(x,t)=\int\limits_{\mathbb{R}^n}a(0)\Delta\varphi_{\xi_n\xi_n}(\xi)G\left(x-\xi,\theta(t)\right)d\xi+
\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\times
\end{equation}
\begin{equation*}
\times\int\limits_{\mathbb{R}^n}\Bigg[\left(\ln
a(\theta^{-1}(\tau))\right)'\Big(
\vartheta^{(2)}\left(\xi,\theta^{-1}(\tau)\right)+
\int\limits_0^{\theta^{-1}(\tau)}r(\xi',\theta^{-1}(\tau)-\alpha)\vartheta^{(2)}(\xi,\alpha)d\alpha\Big)-
\end{equation*}
\begin{equation*}
-r(\xi',0)\vartheta^{(2)}\left(\xi,
\theta^{-1}(\tau)\right)-\int\limits_0^{\theta^{-1}(\tau)}
h(\xi',\theta^{-1}(\tau)-\alpha)\vartheta^{(2)}(\xi,\alpha)d\alpha\Bigg]G\left(x-\xi,\theta(t)-\tau\right)
d\xi,
\end{equation*}
\begin{equation}
\vartheta(x,t)=\int\limits_{\mathbb{R}^n}\Psi(\xi)G(x-\xi,\theta(t))d\xi+
\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^n}\Bigg[\Big(\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}
-
\end{equation}
\begin{equation*}
-2((\ln
a(\theta^{-1}(\tau)))')^2\Big)\vartheta^{(2)}(\xi,\theta^{-1}(\tau))+\Big(2(\ln
a(\theta^{-1}(\tau)))'-r(\xi',0)\Big)\vartheta(\xi,\theta^{-1}(\tau))-
\end{equation*}
\begin{equation*}
-\int\limits_0^{\theta^{-1}(\tau)}
h(\xi',\alpha)\vartheta(\xi,\theta^{-1}(\tau)-\alpha)d\alpha+\Big(\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}+2(\ln
a(\theta^{-1}(\tau)))'-
\end{equation*}
\begin{equation*}
-2((\ln
a(\theta^{-1}(\tau)))')^2\Big)\int\limits_0^{\theta^{-1}(\tau)}
r(\xi',\theta^{-1}(\tau)-\alpha)\vartheta^{(2)}(\xi,\alpha)d\alpha++2(\ln
a(\theta^{-1}(\tau)))'\times
\end{equation*}
\begin{equation*}
\times\int\limits_0^{\theta^{-1}(\tau)}
h(\xi',\theta^{-1}(\tau)-\alpha)\vartheta^{(2)}(\xi,\alpha)d\alpha-h(\xi'\theta^{-1}(\tau))a(0)\Delta\varphi_{\xi_n\xi_n}(\xi)\Bigg]G(x-\xi,\theta(t)-\tau)
d\xi,
\end{equation*}
\begin{equation}
h(x',t)=\frac{a(t)}{f_t(x',0)}\Bigg[\int\limits_{\mathbb{R}^n}\Psi(\xi)G(x'-\xi',\xi_n,\theta(t))d\xi-F(x',t)\Bigg]+
\end{equation}
\begin{equation*}
+\frac{a(t)}{f_t(x',0)}\Bigg[\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^n}
\Bigg(\Big[\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}-2((\ln
a(\theta^{-1}(\tau)))')^2\Big]\vartheta^{(2)}(\xi,\theta^{-1}(\tau))+
\end{equation*}
\begin{equation*}
+\Big[2(\ln
a(\theta^{-1}(\tau)))'-r(\xi',0)\Big]\vartheta(\xi,\theta^{-1}(\tau))-\int\limits_0^{\theta^{-1}(\tau)}
h(\xi',\alpha)\vartheta(\xi,\theta^{-1}(\tau)-\alpha)d\alpha+
\end{equation*}
\begin{equation*}
+\Big[\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}+2(\ln
a(\theta^{-1}(\tau)))'-2((\ln
a(\theta^{-1}(\tau)))')^2\Big]\int\limits_0^{\theta^{-1}(\tau)}
r(\xi',\tau-\alpha)\vartheta^{(2)}(\xi,\alpha)d\alpha+
\end{equation*}
\begin{equation*}
+2(\ln a(\theta^{-1}(\tau)))'\int\limits_0^{\theta^{-1}(\tau)}
h(\xi',\theta^{-1}(\tau)-\alpha)\vartheta^{(2)}(\xi,\alpha)d\alpha-h(\xi',\theta^{-1}(\tau))a(0)\Delta\varphi_{\xi_n\xi_n}(\xi)\Bigg)\times
\end{equation*}
\begin{equation*}
\times G(x'-\xi',\xi_n,\theta(t)-\tau) d\xi
\Bigg]-f_t(x',0)(2((\ln(a(t))')^2-\frac{a''(t)}{a(t)})\int\limits_0^t
r(x',t-\tau)f_\tau(x',\tau)d\tau+
\end{equation*}
\begin{equation*}
+2f_t(x',0)(\ln(a(t))'\int\limits_0^t
h(x',\tau)f_\tau(x',t-\tau)d\tau+f_t(x',0)\int\limits_0^t
h(x',\tau)f_{tt}(x',t-\tau)d\tau,
\end{equation*}
\begin{equation}
r(x,'t)=r(x',0)+\int\limits_0^t h(x',\tau)d\tau.
\end{equation}
where $\theta(t)=\int\limits^t_0a(\tau)d\tau$ and $\theta^{-1}(t)$
is the inverse function to $\theta(t)$;
  $G(x-\xi;
\theta(t)-\tau)=\frac{1}{(2\sqrt{\pi
(\theta(t)-\tau)})^{n}}{e^{\frac{-|x-\xi|^{2}}{4(\theta(t)-\tau)}}},$
$ \ \xi=(\xi_1, ..., \xi_n), \ \xi'=(\xi_1, ..., \xi_{n-1}), \
d\xi=d\xi_1...d\xi_n, \ |x|^2=x_{1}^2+...+x_{n}^2.$


 For {\bf proof} Lemma we will use the formula
\begin{equation}
p(x,t)=\int\limits_{\mathbb{R}^{n}}\varphi(\xi)G(x-\xi;
\theta(t))d\xi+\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\times
\end{equation}
\begin{equation*}
\times\int\limits_{\mathbb{R}^{n}}F(\xi,\theta^{-1}(\tau))G(x-\xi;
\theta(t)-\tau)d\xi,
\end{equation*}
 which provides the solution of the following
Cauchy problem for the heat equation with time-variable
coefficient  of  thermal conductivity:
\begin{equation*}
p_{t}-a(t)\Delta p=F(x,t), \  x\in{\mathbb{R}^{n}}, t>0,
\end{equation*}
\begin{equation*}
p(x,0)=\varphi(x), \ x\in{\mathbb{R}}.
\end{equation*}

The equations (20) and (21) follow from the Cauchy problems (11),
(12) and (15), (16) on bases of (24), respectively. In (21) we set
$x_n=0$ and use the additional condition (17). Then we obtain the
equation (22). The equality (23) is obvious.
\end{proof}
 We join to the equations (20)-(23) the integral equation with
respect to the solution of the direct problem (1), (2), i.e. $u(x,
t).$ It can be  obtained from equalities (5) and (2). At first, we
integrate by parts in the integral on the right-hand side of (5),
then use the formula (24). As a result, one gets
\begin{equation}
u(x,t)=\int\limits_{\mathbb{R}^{n}}\varphi(\xi)G(x-\xi;
\theta(t))d\xi+\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^{n}}
\Big[r(\xi',\theta^{-1}(\tau))\varphi(\xi)-
\end{equation}
\begin{equation*}
 -r(\xi',0)u(\xi,\theta^{-1}(\tau))-\int\limits_0^{\theta^{-1}(\tau)}
h(\xi',\theta^{-1}(\tau)-\alpha)u(\xi,\alpha)d\alpha\Big]G(x-\xi;
\theta(t)-\tau)d\xi.
\end{equation*}

\section{Existence and uniqueness}

In this section we prove the existence and uniqueness theorem for
the system of the integral equations (20)-(23), (25). Then, from
this result will be followed  the existence unique solution of
inverse problem (1)-(3). Here we use  the contraction mapping
principle [22, pp. 87-97]. The idea is to write the integral
equations for unknown functions $\vartheta^{(2)}(x,t), \,
\vartheta(x,t), \, h(x',t), \, r(x',t) $ as a system with a
nonlinear operator, and prove that this operator is a contraction
mapping operator. The existence and uniqueness then follows
immediately.

We recall that some operator $F$ is  a contraction mapping
operator in a closed  set  $\Omega$ which is a subset of a Banach
space if it satisfies the following two properties:

1) if $y\in \Omega,$ then $Fx\in \Omega$ (i.e. $F$ maps $\Omega$
into itself);

2) if $y, z \in \Omega,$ then $\left\|Fy-Fz\right\|\leq
\rho\left\|y-z\right\|$ with $\rho<1$ ($\rho$ - is a constant
independent of $y$ and $z$ ).

Now, here a proof the main result:

\begin{proof}
The system of equations (20)--(23), (25) are closed system for the
unknown functions $\vartheta^{(2)}(x,t), \vartheta(x, t), h(x',t),
\ r(x,'t),$ $u(x, t)$ in the domain $\mathbb{R}^n_T.$ It can be
rewritten in a nonlinear operator equation
\begin{equation}
\psi=A\psi,
\end{equation}
where $\psi=\left(\psi_1, \ \psi_2, \
\psi_3,\psi_4\right)^{\ast}=\left(\vartheta^{(2)}(x,t),
\vartheta(x, t), h(x',t), \ r(x,'t)\right)^{\ast},$ $\ast$ is the
symbol of transposition, and according to the equations (20),
(21), (22) are operator $A\psi=\left[(A\psi)_1, \ (A\psi)_2, \
(A\psi)_3, (A\psi)_4 \right]$ has form

\begin{equation}
(A\psi)_1=\psi_{01}(x,t)+
\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^n}\Bigg[(\ln
a(\theta^{-1}(\tau)))'\Big( \psi_1(\xi,\theta^{-1}(\tau))+
\end{equation}
\begin{equation*}
+\int\limits_0^{\theta^{-1}(\tau)}
\psi_4(\xi',\theta^{-1}(\tau)-\alpha)\psi_1(\xi,\alpha)d\alpha\Big)-r(\xi',0)\psi_1(\xi,\alpha)-
\end{equation*}
\begin{equation*}
-\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\theta^{-1}(\tau)-\alpha)\psi_1(\xi,\alpha)d\alpha\Bigg]G(x-\xi,\theta(t)-\tau)
d\xi,
\end{equation*}
\begin{equation}
(A\psi)_2=\psi_{02}(x,t)+
\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^n}
\Bigg(\Big[\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}-2((\ln
a(\theta^{-1}(\tau)))')^2\Big]\times
\end{equation}
\begin{equation*}
\times\psi_1(\xi,\theta^{-1}(\tau))+\Big[2(\ln
a(\theta^{-1}(\tau)))'-r(\xi',0)\Big]\psi_2(\xi,\theta^{-1}(\tau))-\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\alpha)\times
\end{equation*}
\begin{equation*}
\times\psi_2(\xi,\theta^{-1}(\tau)-\alpha)d\alpha+\Big[\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}+2(\ln
a(\theta^{-1}(\tau)))'-2((\ln a(\theta^{-1}(\tau)))')^2\Big]\times
\end{equation*}
\begin{equation*}
\times\int\limits_0^{\theta^{-1}(\tau)}
\psi_4(\xi',\theta^{-1}(\tau)-\alpha)\psi_1(\xi,\alpha)d\alpha+2(\ln
a(\theta^{-1}(\tau)))'\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\theta^{-1}(\tau)-\alpha)\times
\end{equation*}
\begin{equation*}
\times\psi_1(\xi,\alpha)d\alpha-\psi_3(\xi',\theta^{-1}(\tau))a(0)\Delta\varphi_{\xi_n\xi_n}(\xi)\Bigg)G(x-\xi,\theta(t)-\tau)
d\xi,
\end{equation*}
\begin{equation}
(A\psi)_3=\psi_{03}(x',t)+\frac{a(t)}{f_t(x',0)}\Bigg[\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^n}
\Bigg(\Big[\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}-
\end{equation}
\begin{equation*}
-2((\ln
a(\theta^{-1}(\tau)))')^2\Big]\psi_1(\xi,\theta^{-1}(\tau))+\Big[2(\ln
a(\theta^{-1}(\tau)))'-r(\xi',0)\Big]\psi_2(\xi,\theta^{-1}(\tau))-
\end{equation*}
\begin{equation*}
-\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\alpha)\psi_2(\xi,\theta^{-1}(\tau)-\alpha)d\alpha+\Big[\frac{a''(\theta^{-1}(\tau))}{a(\theta^{-1}(\tau))}+2(\ln
a(\theta^{-1}(\tau)))'-
\end{equation*}
\begin{equation*}
-2((\ln
a(\theta^{-1}(\tau)))')^2\Big]\int\limits_0^{\theta^{-1}(\tau)}
\psi_1(\xi',\theta^{-1}(\tau)-\alpha)\psi_1(\xi,\alpha)d\alpha+2(\ln
a(\theta^{-1}(\tau)))'\times
\end{equation*}
\begin{equation*}
\times\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\theta^{-1}(\tau)-\alpha)\psi_1(\xi,\alpha)d\alpha-\psi_3(\xi',\theta^{-1}(\tau))a(0)\Delta\varphi_{\xi_n\xi_n}(\xi)\Bigg)\times
\end{equation*}
\begin{equation*}
\times G(x'-\xi',\xi_n,\theta(t)-\tau) d\xi
\Bigg]-f_t(x',0)(2((\ln(a(t))')^2-\frac{a''(t)}{a(t)})\int\limits_0^t
\psi_4(x',t-\tau)f_\tau(x',\tau)d\tau+
\end{equation*}
\begin{equation*}
+2f_t(x',0)(\ln(a(t))'\int\limits_0^t
\psi_3(x',\tau)f_\tau(x',t-\tau)d\tau+f_t(x',0)\int\limits_0^t
\psi_3(x',\tau)f_{tt}(x',t-\tau)d\tau,
\end{equation*}
\begin{equation}
(A\psi)_4=\psi_{04}(x',t)+\int\limits_0^t \psi_3(x',\tau)d\tau,
\end{equation}
\begin{equation}
(A\psi)_5=\psi_{05}(x,t)+\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^{n}}
\Big[\psi_4(\xi',\theta^{-1}(\tau))\varphi(\xi)-
\end{equation}
\begin{equation*}
-r(\xi',0)\psi_5(\xi,\theta^{-1}(\tau))-\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\theta^{-1}(\tau)-\alpha)\psi_5(\xi,\alpha)d\alpha\Big]G(x-\xi;
\theta(t)-\tau)d\xi.
\end{equation*}

In (27)--(31) we introduced notations:
\begin{equation*}
\psi_{01}(x,t)=\int\limits_{\mathbb{R}^n}a(0)\Delta\varphi_{\xi_n\xi_n}(\xi)G(x-\xi,\theta(t))d\xi,
\end{equation*}
\begin{equation*}
\psi_{02}(x,t)=\int\limits_{\mathbb{R}^n}\Psi(\xi)G(x-\xi,\theta(t))d\xi,
\end{equation*}
\begin{equation*}
\psi_{03}(x',t)=\frac{a(t)}{f_t(x',0)}\Big[\int\limits_{\mathbb{R}^n}\Psi(\xi)G(x'-\xi',\xi_n,\theta(t))d\xi-F(x',t)\Big],
\end{equation*}
\begin{equation*}
\psi_{04}(x',t)=r(x',0), \ \
\psi_{05}(x,t)=\int\limits_{\mathbb{R}^{n}}\varphi(\xi)G(x-\xi;
\theta(t))d\xi.
\end{equation*}

Denote
$\left|\psi\right|_T^{l}=\max\left(\left|\psi_1\right|_{T_0}^{l},
\ \left|\psi_2\right|_{T_0}^{l}, \ \left|\psi_3\right|_{T_0}^{l},
\left|\psi_4\right|_{T_0}^{l}, \
\left|\psi_5\right|_{T_0}^{l}\right),$ $T<T_0$ and consider in the
space $H^{l, l/2}\left(\mathbb{R}^n_T\right)$ the set $S(T)$ of
functions $\psi(x, t),$ which obey the inequality
\begin{equation}
\left|\psi-\psi_0\right|_T^{l}\leq\left|\psi_0\right|_{T_0}^{l},
\end{equation}
where $\psi_0=\left(\psi_{01}, \ \psi_{02}, \ \psi_{03}, \
\psi_{04}, \ \psi_{05}\right)$ and
\begin{equation*}
\left|\psi_0\right|_{T_0}^{l}=\max\left(\left|\psi_{01}\right|_{T_0}^{l},
\ \left|\psi_{02}\right|_{T_0}^{l}, \
\left|\psi_{03}\right|_{T_0}^{l}\
\left|\psi_{04}\right|_{T_0}^{l}, \
\left|\psi_{05}\right|_{T_0}^{l}\right).
\end{equation*}

It can be demonstrated that sufficiently small $T$ the operator
$A$ is contraction mapping operator in $S(T).$ The theorem of
existence and uniqueness then follows immediately from the
contraction mapping principle.

First it is shown that $A$ has the first property of a contraction
mapping operator. Let $\psi\in S(T), \ T<T_0.$ The from the
inequality (32), we have
\begin{equation*}
\left|\psi_i\right|_T^{l}\leq2\left|\psi_0\right|_{T_0}^{l}, \
i=1, 2, 3, 4, 5.
\end{equation*}

Let us introduce the notations:

\begin{equation*}
a_1:=\|a(t)\|_{C^2[0,T]}, \ \  a_2:=|(\ln a(t))'|_T, \ \
a_3:=|a(0)|
\end{equation*}
\begin{equation*}
r_1=|r(x',0)|^l,\ \  f_1:=|f(x',t-\tau)|^{l+6,(l+6)/2}, \ \
\varphi_1:=|\varphi(x)|^{l+6}.
\end{equation*}
It is easy to see that

\begin{equation*}
|(A\psi)_1-\psi_{01}|_T^l=
\Big|\int\limits_0^{\theta(t)}\frac{d\tau}{a(\theta^{-1}(\tau))}\int\limits_{\mathbb{R}^n}\Bigg[(\ln
a(\theta^{-1}(\tau)))'\Big( \psi_1(\xi,\theta^{-1}(\tau))+
\end{equation*}
\begin{equation*}
+\int\limits_0^{\theta^{-1}(\tau)}
\psi_4(\xi',\theta^{-1}(\tau)-\alpha)\psi_1(\xi,\alpha)d\alpha\Big)-r(\xi',0)\psi_1(\xi,\alpha)-\int\limits_0^{\theta^{-1}(\tau)}
\psi_3(\xi',\theta^{-1}(\tau)-\alpha)\times
\end{equation*}
\begin{equation*}
\times\psi_1(\xi,\alpha)d\alpha\Bigg]G(x-\xi,\theta(t)-\tau)
d\xi\Big|_T^l\leq\int\limits_0^{\theta(t)}\frac{d\tau}{|a(\theta^{-1}(\tau))|_T}\int\limits_{\mathbb{R}^n}\Bigg[|(\ln
a(\theta^{-1}(\tau)))'|_T\times
\end{equation*}
\begin{equation*}
\times\Big(
\psi_1(\xi,\theta^{-1}(\tau))|_T^l+\int\limits_0^{\theta^{-1}(\tau)}
|\psi_4(\xi',\theta^{-1}(\tau)-\alpha)|_T^l|\psi_1(\xi,\alpha)|_T^ld\alpha\Big)+|r(\xi',0)|^l|\psi_1(\xi,\alpha)|_T^l+
\end{equation*}
\begin{equation*}
+\int\limits_0^{\theta^{-1}(\tau)}
|\psi_3(\xi',\theta^{-1}(\tau)-\alpha)|_T^l|\psi_1(\xi,\alpha)|_T^ld\alpha\Bigg]G(x-\xi,\theta(t)-\tau)
d\xi\leq
\end{equation*}
\begin{equation*}
\leq|\psi_0|_{T_0}^l2T^2a_0(a_2+2Ta_2|\psi_0|_{T_0}^l+r_1+2T|\psi_0|_{T_0}^l):=|\psi_0|_{T_0}^l\beta_1,
\end{equation*}
In similar way we obtain
\begin{equation*}
|(A\psi)_2-\psi_{02}|_T^l\leq|\psi_0|_{T_0}^l[2T^2a_0(a_0a_1+2a_2^2+2a_2+r_1+2T|\psi_0|_{T_0}^l)+
\end{equation*}
\begin{equation*}
+2T|\psi_0|_{T_0}^l(a_0a_2+2a_2+2a_2^2)+4Ta_2|\psi_0|_{T_0}^l+a_3\varphi_1]:=|\psi_0|_{T_0}^l\beta_2,
\end{equation*}
\begin{equation*}
|(A\psi)_3-\psi_{03}|_T^l\leq|\psi_0|_{T_0}^l\Big(2\frac{T^2}{f_1}\Big[a_0a_1+2a_2^2+2a_2+r_1+2T|\psi_0|_{T_0}^l+2T|\psi_0|_{T_0}^l(a_0a_1+2a_2+
\end{equation*}
\begin{equation*}
+2a_2^2)+4T|\psi_0|_{T_0}^la_2+a_3\varphi_1\Big]+Tf^2_1(a_0a_1+2a_2^2+2a_2+1)\Big):=|\psi_0|_{T_0}^l\beta_3
\end{equation*}
\begin{equation*}
|(A\psi)_4-\psi_{04}|_T^l\leq2T|\psi_0|_{T_0}^l:=|\psi_0|_{T_0}^l\beta_4,
\end{equation*}
\begin{equation*}
|(A\psi)_5-\psi_{05}|_T^l\leq|\psi_0|^l_{T_0}\cdot2T^2a_0(\varphi_1+r_1+2T|\psi_0|^l_{T_0}):=|\psi_0|^l_{T_0}\beta_5,
\end{equation*}
where $\beta_i(T)\rightarrow 0$ at $T\rightarrow 0,$ $i=1,2,3, 4,
5.$ Therefore, if we choose $T$ $(T<T_0)$  so that the following
inequality  should be satisfied
$$\beta:=\max\left\{\beta_1, \ \beta_2, \ \beta_3, \ \beta_4, \beta_5\right\}<1,$$
then the operator $A$ has the first property of a contraction
mapping operator, i.e., $A\psi\in S(T).$

Consider next the second property of a contraction mapping
operator for $A.$ Let $\psi^{(1)}=\left(\psi^{(1)}_1, \
\psi^{(1)}_2, \ \psi^{(1)}_3, \ \psi^{(1)}_4, \
\psi^{(1)}_5\right)\in S(T),$ $\psi^{(2)}=\left(\psi^{(2)}_1, \
\psi^{(2)}_2, \ \psi^{(2)}_3, \ \psi^{(2)}_4, \
\psi^{(2)}_5\right)\in S(T).$ In view of inequalities

\begin{equation*}
\Bigl|\psi_{2}^{(1)}\psi_1^{(1)}-\psi_{2}^{(2)}\psi_1^{(2)}\Bigr|_T^{l}=\Bigl|\left(\psi_{2}^{(1)}-\psi_{2}^{(2)}\right)\psi_1^{(1)}
+\psi_{2}^{(2)}\left(\psi_1^{(1)}-\psi_1^{(2)}\right)\Bigr|_T^{l}\leq
\end{equation*}
\begin{equation*}
\leq2\Bigl|\psi^{(1)}-\psi^{(2)}\Bigr|_T^{l}
\max\left(\Bigl|\psi_1^{(1)}\Bigr|_T^{l}, \
\Bigl|\psi_{2}^{(2)}\Bigr|_T^{l}\right)\leq4\left|\varphi_0\right|_T^{l}\Bigl|\psi^{(1)}-\psi^{(2)}\Bigr|_T^{l},
\end{equation*}
we estimate the difference

\begin{equation*}
|((A\psi)^{(1)}-A\psi)^{(2)})_1|_T^l\leq
\end{equation*}
\begin{equation*}
|((A\psi)^{(1)}-A\psi)^{(2)})_1|_T^l\leq\int\limits_0^{\theta(t)}\frac{d\tau}{|a(\theta^{-1}(\tau))|_T}\int\limits_{\mathbb{R}^n}\Bigg[|(\ln
a(\theta^{-1}(\tau)))'|_T|
\Big((\psi_1^{(1)}(\xi,\theta^{-1}(\tau))-
\end{equation*}
\begin{equation*}
-\psi_1^{(2)}(\xi,\theta^{-1}(\tau)))|_T^l+\int\limits_0^{\theta^{-1}(\tau)}
|\Big[\psi_4^{(1)}(\xi',\theta^{-1}(\tau)-\alpha)\psi_1^{(1)}(\xi,\alpha)-
\end{equation*}
\begin{equation*}
-\psi_4^{(2)}(\xi',\theta^{-1}(\tau)-\alpha)\psi_1^{(2)}(\xi,\alpha)\Big]|_T^ld\alpha\Big)+|r(\xi',0)|^l|(\psi_1^{(1)}(\xi,\alpha)-\psi_1^{(2)}(\alpha))|_T^l+
\end{equation*}
\begin{equation*}
+\int\limits_0^{\theta^{-1}(\tau)}
|\Big[\psi_3^{(1)}(\xi',\theta^{-1}(\tau)-\alpha)\psi_1^{(1)}(\xi,\alpha)-
\psi_3^{(2)}(\xi',\theta^{-1}(\tau)-
\end{equation*}
\begin{equation*}
-\alpha)\psi_1^{(2)}(\xi,\alpha)\Big]|_T^ld\alpha\Bigg]G(x-\xi,\theta(t)-\tau)
d\xi\leq
\end{equation*}
\begin{equation*}
\leq
|\psi^{(1)}-\psi^{(2)}|_{T_0}^l2T^2a_0(a_2+4Ta_2|\psi_0|_{T_0}^l+r_1+4T|\psi_0|_{T_0}^l):=|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\mu_1.
\end{equation*}
For other differences by similar way,  we obtain
\begin{equation*}
|((A\psi)^{(1)}-A\psi)^{(2)})_2|_T^l\leq|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\Big(T^2a_0(a_0a_1+2a_2^2+2a_2+r_1+4T|\psi_0|_{T_0}^l)+
\end{equation*}

\begin{equation*}
+4T|\psi_0|_{T_0}^l(a_0a_1+2a_2+2a_2^2)+8Ta_2|\psi_0|_{T_0}^l+a_3\varphi_1\Big):=|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\mu_2
\end{equation*}
\begin{equation*}
|((A\psi)^{(1)}-A\psi)^{(2)})_3|_T^l\leq|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\Big(\frac{T^2}{f_1}\Big[a_0a_1+2a_2^2+2a_2+r_1+4T|\psi_0|_{T_0}^l+
\end{equation*}
\begin{equation*}
+4T|\psi_0|_{T_0}^l(a_0a_1+2a_2+2a_2^2)+8T|\psi_0|_{T_0}^la_2+a_3\varphi_1\Big]+
\end{equation*}
\begin{equation*}
+Tf^2_1(a_0a_1+2a_2^2+2a_2+1)\Big):=|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\mu_3
\end{equation*}
\begin{equation*}
|((A\psi)^{(1)}-A\psi)^{(2)})_4|_T^l\leq|\psi^{(1)}-\psi^{(2)}|_{T_0}^lT:=|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\mu_4
\end{equation*}
\begin{equation*}
|((A\psi)^{(1)}-A\psi)^{(2)})_5|_T^l
\leq|\psi^{(1)}-\psi^{(2)}|_{T_0}^lT^2a_0(\varphi_1+r_1+4T|\psi_0|^l_{T_0}):=|\psi^{(1)}-\psi^{(2)}|_{T_0}^l\mu_5
\end{equation*}
Hence,
$\left|\left(A\psi^{(1)}-A\psi^{(2)}\right)\right|_T^{l}<\mu\Bigl|\psi^{(1)}-\psi^{(2)}\Bigr|_T^{l},$
 if $T$ satisfies the condition
\begin{equation*}
\mu:=\max\left\{\mu_1, \ \mu_2. \ \mu_3, \ \mu_4, \mu_5\right\}<1.
\end{equation*}

It is not difficult to see if we choose $T_0$ as
$T_0=\min\left(\beta, \mu\right),$ then for $T\in(0, T_0)$ the
operator  $A$ satisfies both the properties of a contraction
mapping operator, i.e., $A$ realizes contracted mapping of the set
$S(T)$ onto itself. Hence, according to Banach theorem (see, for
instance, [22, pp. 87-97]), in the set $S(T)$ there exists only
one fixed point of transformations, i.e. there exists only one
solution of (26).

\end{proof}

\bigskip
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\end{document}
