ON THE UNIQUENESS OF SOLUTIONS OF TWO INVERSE PROBLEMS FOR THE SUBDIFFUSION
EQUATION
Ravshan Ashurov1 and Yusuf Fayziev2
1 Institute of Mathematics, Academy of Science of Uzbekistan

<EMAIL_ADDRESS>
2 National University of Uzbekistan

<EMAIL_ADDRESS>

Abstract: Let $A$ be an arbitrary positive selfadjoint operator, defined in a
separable Hilbert space $H$. The inverse problems of determining the right-
hand side of the equation and the function $\varphi$ in the non-local boundary
value problem $D_{t}^{\rho}u(t)+Au(t)=f(t)$ ($0<\rho<1$, $0<t\leq T$),
$u(\xi)=\alpha u(0)+\varphi$ ($\alpha$ is a constant and $0<\xi\leq T$), is
considered. Operator $D_{t}$ on the left-hand side of the equation expresses
the Caputo derivative. For both inverse problems $u(\xi_{1})=V$ is taken as
the over-determination condition. Existence and uniqueness theorems for
solutions of the problems under consideration are proved. The influence of the
constant $\alpha$ on the existence and uniqueness of a solution to problems is
investigated. An interesting effect was discovered: when solving the forward
problem, the uniqueness of the solution $u(t)$ was violated, while when
solving the inverse problem for the same values of $\alpha$, the solution
$u(t)$ became unique.

Keywords: Non-local problems, the Caputo derivatives, subdiffusion equation,
inverse problems.

##  1\. Introduction

Let $A:H\rightarrow H$ be an arbitrary unbounded positive selfadjoint operator
in a separable Hilbert space $H$ with the scalar product $(\cdot,\cdot)$ and
the norm $||\cdot||$. Let $A$ have a complete in $H$ system of orthonormal
eigenfunctions $\\{v_{k}\\}$ and a countable set of positive eigenvalues
$\lambda_{k}:$
$0<\lambda_{1}\leq\lambda_{2}\cdot\cdot\cdot\rightarrow+\infty$. We will also
assume that the sequence $\\{\lambda_{k}\\}$ has no finite limit points. For a
vector-valued functions (or simply functions) $h:\mathbb{R}_{+}\rightarrow H$,
we define the Caputo fractional derivative of order $0<\rho<1$ as (see, e.g.
[1])

$D_{t}^{\rho}h(t)=\frac{1}{\Gamma(1-\rho)}\int\limits_{0}^{t}\frac{h^{\prime}(\xi)}{(t-\xi)^{\rho}}d\xi,\quad
t>0,$

provided the right-hand side exists. Here $\Gamma(\sigma)$ is Euler’s gamma
function. Finally, let $C((a,b);H)$ stand for a set of continuous functions
$u(t)$ of $t\in(a,b)$ with values in $H$.

The main object studied in this work is the following non-local boundary value
problem:

(1.1) $\left\\{\begin{aligned} &D_{t}^{\rho}u(t)+Au(t)=f(t),\quad 0<t\leq
T;\\\ &u(\xi_{0})=\alpha u(0)+\varphi,\quad 0<\xi_{0}\leq
T,\end{aligned}\right.$

where $f(t)\in C((0,T];H)$, $\varphi\in H$ and $\alpha$ is a constant,
$\xi_{0}$ \- a fixed point. This problem is also called _the forward problem_.

In the case when $\xi_{0}=T$ and parameter $\alpha$ is equal to zero:
$\alpha=0$, this problem is called _the backward problem_ and it is well
studied in the works [2] \- [4] and [5]. And if $\alpha=0$ and $\rho=1$, then
we get a classical problem called the inverse heat conduction problem with
inverse time (_retrospective inverse problem_), which has been studied in
detail by various specialists (see, e.g. Chapter 8.2 of [6] and literature
therein).

It is well known that in most models described by differential (and
pseudodifferential, see e.g., [7]) equations, an initial condition is used to
select a single solution. However, there are also processes where we have to
use non-local conditions, for example, the integral over time intervals (see,
e.g. [8] for reaction diffusion equations or [9] for fractional equations), or
connection of solution values at different times, for example, at the initial
time and at the final time (see, e.g. [10] \- [11]). It should be noted that
non-local conditions model some details of natural phenomena more accurately,
since they take into account additional information in the initial conditions.

The non-local boundary value problem (1.1) for the classical diffusion
equation, namely the following problem

(1.2) $\left\\{\begin{aligned} &u^{\prime}(t)+Au(t)=f(t),\quad 0<t\leq T;\\\
&u(\xi_{0})=u(0)+\varphi,\quad 0<\xi\leq T,\end{aligned}\right.$

has been extensively studied by many researchers (see, e.g. A. O. Ashyralyev
et al. [10] \- [11]). As shown in these papers, in contrast to the
retrospective inverse problem, problem (1.2) is coersively solvable in some
spaces of differentiable functions.

Let us return to the non-local problem (1.1). The authors of this paper in
their previous work [12] studied in detail the influence of parameter
$\alpha\neq 0$ on the correctness of problem (1.1). It turned out that the
critical values of parameter $\alpha$ are in the interval $(0,1)$. In order to
formulate the main result of work [12], we recall the definition of the
Mittag-Leffler function $E_{\rho,\mu}(z)$ with two parameters (see, e.g. [13],
Chapter 1):

$E_{\rho,\mu}(z)=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\rho n+\mu)},$

where $\mu$ is an arbitrary complex number. If parameter $\mu=1$, then we have
the classical Mittag-Leffler function: $E_{\rho}(z)=E_{\rho,1}(z)$. Recall
(see, e.g. [12]), $E_{\rho}(-t)$ decreases strictly monotonically as $t>0$
and, moreover, has the following estimate

(1.3) $0<E_{\rho}(-t)<1,\,t>0.$

In work [12] it is proved that if $\alpha\in(0,1)$ and
$E_{\rho}(-\lambda_{k}t^{\rho})\neq\alpha$ for all $k$, then the solution of
problem (1.1) exists and is unique. But it may turn out that for some
eigenvalue $\lambda_{k_{0}}$ of operator $A$, with multiplicity $p_{0}$
(obviously, $p_{0}$ is a finite number), equality

(1.4) $E_{\rho}(-\lambda_{k_{0}}t^{\rho})=\alpha$

will hold. Then, as proved in [12], in order for a solution to exist, it is
necessary to require the following orthogonality conditions

(1.5) $(\varphi,v_{k})=0,\,\,(f(t),v_{k})=0,\,\,\text{for all}\,\,t>0,\,\,k\in
K_{0};\,\,K_{0}=\\{k_{0},k_{0}+1,....,k_{0}+p_{0}-1\\}.$

It should be noted that in this case there will be no uniqueness of the
solution [12].

The paper [12] also studies two inverse problems of determining the function
$\varphi$ from the non-local condition (1.1) and the source function $f$, i.e.
the right-hand side of the equation in (1.1) (in the latter case, it is
assumed that $f$ does not depend on $t$). It is proved that if
$\alpha\notin(0,1)$, then the solutions of both inverse problems exist and are
unique. The main goal of this paper is to study these inverse problems for
critical values of parameter $\alpha\in(0,1)$.

######  Problem 1.1.

Let $\alpha\in(0,1)$. Find a pair $\\{u(t),f\\}$ of function $u(t)\in
C([0,T];H)$ and $f\in H$ with the properties $D_{t}^{\rho}u(t),Au(t)\in
C((0,T];H)$ and satisfying the non-local problem (1.1) (note, $f$ does not
depend on $t$) and the over-determination condition

(1.6) $u(\xi_{1})=V,\quad 0<\xi_{1}<\xi_{0},$

where $V$ is a given element of $H$.

Note that if $\xi_{1}=\xi_{0}$, then the non-local condition in (1.1)
coincides with the Cauchy condition $u(0)=\varphi_{1}$ (note $\alpha\neq 0$).
In this case, this inverse problem was studied in [14]. If the reverse
inequality $\xi_{1}>\xi_{0}$ holds, then it will be shown that the solution
may not be unique.

######  Problem 1.2.

Let $\alpha\in(0,1)$. Find a pair $\\{u(t),\varphi\\}$ of function $u(t)\in
C([0,T];H)$ and $\varphi\in H$ with the properties $D_{t}^{\rho}u(t),Au(t)\in
C((0,T];H)$ and satisfying the non-local problem (1.1) and the over-
determination condition

(1.7) $u(\xi_{2})=W,\quad 0<\xi_{2}\leq T,\,\,\xi_{2}\neq\xi_{0},$

where $W$ is a given element of $H$.

If $\xi_{2}=\xi_{0}$, then the non-local condition $u(\xi)=\alpha
u(0)+\varphi$ coincides with the Cauchy condition $u(0)=\varphi_{1}$ (note
$\alpha\neq 0$) and we have the backward problem, considered in [2] \- [4].

Everywhere below, for the vector - function $h(t)\in H$ (which may or may not
depend on $t$) by the symbol $h_{k}(t)$ we will denote the Fourier
coefficients with respect to the system of eigenfunctions $\\{v_{k}\\}$:
$h_{k}(t)=(h(t),v_{k})$.

######  Theorem 1.3.

Let $\varphi,V\in D(A)$ and let the orthogonality conditions (1.5) be
satisfied. Then the inverse Problem 1.1 has a unique solution $\\{u(t),f\\}$
and this solution has the following form

(1.8) $f=\sum\limits_{k\notin K_{0}}\bigg{[}\frac{\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})+\xi_{1}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{1}^{\rho})[\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})]}\,V_{k}+$
$+\frac{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})+\xi_{1}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{1}^{\rho})[\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})]}\,\varphi_{k}\bigg{]}v_{k},$ (1.9)
$u(t)=\sum\limits_{k\notin
K_{0}}\left[\frac{E_{\rho}(-\lambda_{k}t^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,[{\varphi_{k}-f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}]+f_{k}t^{\rho}E_{\rho,\rho+1}(-\lambda_{k}t^{\rho})\right]v_{k}+.$
$+\sum\limits_{k\in
K_{0}}\frac{E_{\rho}(-\lambda_{k}t^{\rho})\,V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}\,v_{k}.$

Note that, due to the orthogonality condition (1.5), all Fourier coefficients
$f_{k}$ vanish for $k\in K_{0}$. Obviously, $K_{0}$ can also be an empty set;
in this case the sum $\sum_{k\notin K_{0}}$ is the same as
$\sum_{k=1}^{\infty}$.

Let $\tau$ be an arbitrary real number. In order to formulate a result on
Problem 1.2 we introduce the power of operator $A$, acting in $H$ as

$A^{\tau}h=\sum\limits_{k=1}^{\infty}\lambda_{k}^{\tau}h_{k}v_{k},$

where $h_{k}$ are the Fourier coefficients of $h\in H$. Obviously, the domain
of this operator has the form

$D(A^{\tau})=\\{h\in
H:\sum\limits_{k=1}^{\infty}\lambda_{k}^{2\tau}|h_{k}|^{2}<\infty\\}.$

For elements of $D(A^{\tau})$ we introduce the norm

$||h||^{2}_{\tau}=\sum\limits_{k=1}^{\infty}\lambda_{k}^{2\tau}|h_{k}|^{2}=||A^{\tau}h||^{2},$

and together with this norm $D(A^{\tau})$ turns into a Hilbert space.

######  Theorem 1.4.

Let $W\in D(A)$, $f\in C([0,T];D(A^{\varepsilon}))$ for some
$\varepsilon\in(0,1)$ and let the orthogonality conditions (1.5) be satisfied.
Then the inverse Problem 1.2 has a unique solution $\\{u(t),\varphi\\}$ and
this solution has the form

(1.10) $\varphi=\sum\limits_{k\notin
K_{0}}\left[\frac{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}[W_{k}-\omega_{k}(\xi_{2})]+\omega_{k}(\xi_{0})\right]v_{k},$
(1.11) $u(t)=\sum\limits_{k\notin
K_{0}}\left[\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,E_{\rho}(-\lambda_{k}t^{\rho})+\omega_{k}(t)\right]v_{k}+$
$+\sum\limits_{k\in
K_{0}}\frac{E_{\rho}(-\lambda_{k}t^{\rho})\,W_{k}}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}\,v_{k},$

where

$\omega_{k}(t)=\int\limits_{0}^{t}\eta^{\rho-1}E_{\rho,\rho}(-\lambda_{k}\eta^{\rho})f_{k}(t-\eta)d\eta.$

Note that for $k\in K_{0}$ all Fourier coefficients $\varphi_{k}$ are equal to
zero since the orthogonality condition (1.5). When $K_{0}$ is an empty set,
then the sum $\sum_{k\notin K_{0}}$ coincides with $\sum_{k=1}^{\infty}$.

######  Remark 1.5.

It should be specially noted that, as was proved in [12] and noted above, when
equality (1.4) holds, the solution to the forward problem is not unique. But
it turns out that both inverse problems have a unique solution even under
condition (1.4).

To the best of our knowledge, the inverse problem of defining the function
$\varphi$ in the non-local condition was discussed only in the paper [15]. The
authors considered this problem for the subdiffusion equation with the Caputo
fractional derivative, the elliptic part of which is a two-variable
differential expression with constant coefficients. On the other hand, it is
not difficult to simulate a real process in which we will face just such an
inverse problem. For example, in the temperature distribution process, the
initial and final temperatures are not specified, and it is not required to
find them, but information about the difference between the initial and final
temperatures is sought.

As for the inverse problems of determining the source function $f$ with final
time observation, it is well studied, both for classical partial differential
equations and for equations of fractional order. Many theoretical studies have
been published. Kabanikhin [6] and Prilepko, Orlovsky and Vasin [16] should be
mentioned as classical monographs for integer-order equations. As for
fractional differential equations, it is possible to construct theories
parallel to the works of [6], [16], and work in this direction is ongoing. In
this note, we will pay attention to only some of them, referring interested
readers to a review paper [17]. Also note the works [12], [18, 19, 20], where
there is a review of recent work in this direction.

We note right away that no one has yet proposed a method for finding the
right-hand side given in the abstract form $f(x,t)$. Known results deal with
separated source term $f(x,t)=q(t)p(x)$. The appropriate choice of the over-
determination depends on the choice whether the unknown is $q(t)$ or $p(x)$.

Quite a lot of papers are devoted to the case considered in this article,
namely $q(t)\equiv 1$ and the unknown is $p(x)$. Subdiffusion equations whose
elliptic part $A$ is an ordinary differential expression are considered, for
example, in [18, 21, 22, 23]. The authors of [19, 20], studied the inverse
problem for multi-term subdiffusion equations in which the elliptic part is
either a Laplace operator or a second order operator. Article [14] studied the
inverse problem for the subdiffusion equation (1.1) with the Cauchy condition.
Recent articles [24] \- [25] are devoted to the inverse problem for the
subdiffusion equation with Riemann-Liouville derivatives.

In [26] non-self-adjoint differential operators (with non-local boundary
conditions) were taken as A, and the solutions of the inverse problem were
found in the form of a biorthogonal series.

In their previous work [27], the authors of this article considered an inverse
problem for simultaneously determining the order of the Riemann-Liouville
fractional derivative and the source function in the subdiffusion equations.
Using the classical Fourier method, the authors proved the uniqueness and
existence of a solution to this inverse problem.

It should be noted that in all of the listed works, the Cauchy conditions in
time are considered (an exception is work [28], where the integral condition
is set with respect to the variable $t$). In the paper [12], for the best of
our knowledge, an inverse problem for subdiffusion equation with a non-local
condition in time is considered for the first time.

The most difficult case to study is the case when the function $q(t)$ is
unknown (see the survey paper [17] and [3] ). In inverse problems of this
type, the condition $u(x_{0},t)=u_{0}(t)$ is taken as an additional condition.
The authors studied mainly the uniqueness of the solution of the inverse
problem. In this regard, we note the recent papers [29], [30] where the
inverse problem for determining the right-hand side of the form $q(t)$ was
studied for the Schrodinger equation. Taking over-determination conditions of
a rather general form $Bu(\cdot,t)$, where $B:H\rightarrow R$ is a linear
bounded functional, the authors proved both the existence and uniqueness of a
solution to the inverse problem.

The papers [31] \- [32] deal with the inverse problem of determining an order
of the fractional derivative in the subdiffusion equation and in the wave
equation, respectively.

##  2\. Inverse Problem 1.1

2.1. Existence. Assume that all the conditions of Theorem 1.3 are satisfied,
i.e. $\varphi,V\in D(A)$ and let the orthogonality conditions (1.5) be
satisfied. Let us first prove the existence of a solution and that the
solution has the form (1.8) and (1.9). The fact that these series converge in
the norm $H$ and in (1.9) the summation and operators $D^{\rho}_{t}$ and $A$
can be interchanged was proved in the work of the authors [12]. Therefore, it
suffices to show that the series (1.8) and (1.9) formally satisfy the equation
and the initial condition (1.1), and the over-determination condition (1.6).
In order to do this, we rewrite the series (1.8) and (1.9) in the form $f=\sum
f_{k}v_{k}$ and $u(t)=\sum u_{k}(t)v_{k}$. Now, according to the Fourier
method, it suffices to show that the unknown coefficients $f_{k}$ and
$u_{k}(t)$ satisfy equation

(2.1) $D_{t}^{\rho}u_{k}(t)+\lambda_{k}u_{k}(t)=f_{k},$

the non-local condition

(2.2) $u_{k}(\xi_{0})=\alpha u_{k}(0)+\varphi_{k},$

and finally the over-determination condition

(2.3) $u_{k}(\xi_{1})=V_{k},$

for all $k\geq 1$.

Let us show that $u_{k}(t)$ and $f_{k}$ satisfy equation (2.1). Let $k\notin
K_{0}$. We have $u_{k}(t)=u_{k}^{1}(t)+u_{k}^{2}(t)$, where

$u_{k}^{1}(t)=\frac{E_{\rho}(-\lambda_{k}t^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,[{\varphi_{k}-f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}]$

and

$u_{k}^{2}(t)=f_{k}t^{\rho}E_{\rho,\rho+1}(-\lambda_{k}t^{\rho}).$

It is known (see, e.g. [33], p. 174) that $u_{k}^{1}(t)$ is a solution to the
homogeneous equation (2.1) with the initial condition

$u_{k}^{1}(0)=\frac{\varphi_{k}-f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}.$

It is also known (see ibid.) that the function

$\omega_{k}(t)=\int\limits_{0}^{t}\eta^{\rho-1}E_{\rho,\rho}(-\lambda_{k}\eta^{\rho})f_{k}(t-\eta)d\eta$

from Theorem 1.4 is a solution to equation (2.1) with the right-hand side
$f_{k}(t)$ and with the initial condition $\omega_{k}(0)=0$. If in this
formula $f_{k}(t)$ does not depend on $t$, then the integral can be rewritten
in the form (see e.g. [33], formula (4.4.4))

$f_{k}\int\limits_{0}^{t}\eta^{\rho-1}E_{\rho,\rho}(-\lambda_{k}\eta^{\rho})d\eta=f_{k}\,t^{\rho}E_{\rho,\rho+1}(-\lambda_{k}t^{\rho}).$

Therefore, the function $u_{k}^{2}(t)$ is a solution to the inhomogeneous
equation (2.1) with the initial condition $u_{k}^{2}(0)=0$.

Now suppose that $k\in K_{0}$. Then the function

$u_{k}(t)=\frac{E_{\rho}(-\lambda_{k}t^{\rho})\,V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}$

is a solution of homogeneous equation (2.1) with the initial data

$u_{k}(0)=\frac{V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}.$

Thus, it is proved that the functions (1.8) and (1.9) really satisfy equation
(2.1).

It remains to verify the fulfillment of the non-local condition (2.2) and the
over-determination condition (2.3).

Let $k\notin K_{0}$. Since we have calculated
$u_{k}(0)=u^{1}_{k}(0)+u^{2}_{k}(0)$, we can write

$\alpha
u_{k}(0)+\varphi_{k}=\frac{\varphi_{k}E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha
f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}.$

On the other hand, according to (1.9), $u_{k}(\xi_{0})$ has exactly the same
value:

$u_{k}(\xi_{0})=\frac{\varphi_{k}E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha
f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}.$

Let now $k\in K_{0}$. Then $\varphi_{k}=0$ (see (1.5)) and
$E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})=\alpha$. Therefore

$\alpha u_{k}(0)+\varphi_{k}=\frac{\alpha
V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})},$

and

$u_{k}(\xi_{0})=\frac{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})\,V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}=\frac{\alpha
V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}.$

Thus, the Fourier coefficients of function $u(t)$, defined by formula (1.9),
satisfy the non-local condition (2.2) for all $k\geq 1$.

Let us check the fulfillment of the over-determination condition (2.3).
Consider again the case $k\notin K_{0}$. By virtue of condition (2.3) we
obtain:

$\frac{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,[{\varphi_{k}-f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}]+f_{k}\xi_{1}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{1}^{\rho})=V_{k}.$

After simple calculations, we get

$f_{k}=\frac{\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})+\xi_{1}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{1}^{\rho})[\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})]}V_{k}+$
$+\frac{E_{\rho}(-\lambda_{k}\tau^{\rho})}{E_{\rho}(-\lambda_{k}\tau^{\rho})\xi^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi^{\rho})+\tau^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\tau^{\rho})[\alpha-
E_{\rho}(-\lambda_{k}\xi^{\rho})]}\varphi_{k},$

and this coincides with the Fourier coefficients of the function (1.8).

If $k\in K_{0}$, then

$u_{k}(\xi_{1})=\frac{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})\,V_{k}}{E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})}=V_{k}.$

This completes the proof of the existence of a solution to Problem 1.1.

2.2. Uniqueness. Let us proceed to the proof of the uniqueness of the solution
of Problem 1.1.

We proceed in the standard way: assuming the existence of two solutions, we
obtain contradictions. Let $\\{u_{1}(t),f_{1}\\}$ and $\\{u_{2}(t),f_{2}\\}$
be two solutions. It is required to prove $u(t)\equiv u_{1}(t)-u_{2}(t)\equiv
0$ and $f\equiv f_{1}-f_{2}=0$. To determine $u(t)$ and $f$ we have the
problem:

(2.4) $D_{t}^{\rho}u(t)+Au(t)=f,\quad t>0;$ (2.5) $u(\xi_{0})=\alpha
u(0),\quad 0<\xi_{0}\leq T,$ (2.6) $u(\xi_{1})=0,\quad 0<\xi_{1}<\xi_{0},$

where $\xi_{0}$ and $\xi_{1}$ are the fixed points.

Let $u(t)$ be a solution to this problem and $u_{k}(t)=(u(t),v_{k})$. Then, by
virtue of equation (2.4) and the selfadjointness of operator $A$, problem
(2.4)-(2.6) becomes the following non-local problem with respect to
$u_{k}(t)$:

(2.7) $D_{t}^{\rho}u_{k}(t)+\lambda_{k}u_{k}(t)=f_{k},\quad t>0;\quad
u_{k}(\xi_{0})=\alpha u_{k}(0),\quad u_{k}(\xi_{1})=0.$

Note that if $k\in K_{0}$ then $f_{k}=0$.

Let first $k\notin K_{0}$. Suppose that $f_{k}$ is known and use the non-local
condition to get (see, e.g. [33], p.174)

$u_{k}(t)=\frac{f_{k}\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})}{\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})}\,E_{\rho}(-\lambda_{k}t^{\rho})+f_{k}t^{\rho}E_{\rho,\rho+1}(-\lambda_{k}t^{\rho}).$

Now apply $u_{k}(\xi_{1})=0$ to have

(2.8)
$f_{k}[\xi_{0}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{0}^{\rho})E_{\rho}(-\lambda_{k}\xi_{1}^{\rho})+\xi_{1}^{\rho}E_{\rho,\rho+1}(-\lambda_{k}\xi_{1}^{\rho})(\alpha-
E_{\rho}(-\lambda_{k}\xi_{0}^{\rho}))]=0.$

Let us show that for $\xi_{1}<\xi_{0}$ the square bracket is not equal to
zero. To do this, we introduce the notations:
$a(t)=t^{\rho}E_{\rho,\rho+1}(-\lambda_{k}t^{\rho})>0$ and
$b(t)=E_{\rho}(-\lambda_{k}t^{\rho})>0$. It is known (see, e.g. [12]) that the
function $a(t)$ is increasing and the function $b(t)$ is decreasing. Now let
us rewrite the square bracket as

$c(\xi_{0},\xi_{1})=a(\xi_{0})b(\xi_{1})-a(\xi_{1})b(\xi_{0})+\alpha
b(\xi_{0}).$

Obviously, for $\xi_{1}<\xi_{0}$ this expression is strictly positive.
Therefore for all $k\notin K_{0}$ one has $f_{k}=0$ (see (2.8)).

It should be noted that if the inverse inequality $\xi_{1}>\xi_{0}$ is
satisfied, then the first term in the expression for $c(\xi_{0},\xi_{1})$
becomes less than the second one and, as a result, there is $\alpha\in(0,1)$
that turns $c(\xi_{0},\xi_{1})$ into zero. Therefore, in this case $f_{k}$ may
not vanish, i.e., the uniqueness $f_{k}$ for these $\alpha$ and $k$ is
violated.

Let us now consider the case $k\in K_{0}$. Denote $u_{k}(0)=b_{k}$. Then the
unique solution to the differential equation in (2.7) with this initial
condition has the form $u_{k}(t)=b_{k}E_{\rho}(-\lambda_{k}t^{\rho})$ (see,
e.g. [33], p.174). Since $E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})=\alpha$ in the
considering case, then the non-local condition is satisfied for an arbitrary
$b_{k}$. But the over-determination condition $u_{k}(\xi_{1})=0$ implies
$b_{k}=0$ for $k\in K_{0}$.

Therefore, from the completeness of the system of eigenfunctions
$\\{v_{k}\\}$, we finally obtain $f=0$ and $u(t)\equiv 0$, as required. The
uniqueness and hence Theorem 1.3 is completely proved.

##  3\. Inverse Problem 1.2

3.1. Existence. Suppose that $W\in D(A)$ and $f\in
C([0,T];D(A^{\varepsilon}))$ for some $\varepsilon\in(0,1)$ and let the
orthogonality conditions (1.5) be satisfied. Let us first show that series
(1.10) and (1.11) are indeed solutions to Problem 1.2. The fact that $u(t)\in
C([0,T];H)$ and $\varphi\in H$ and have properties $D_{t}^{\rho}u(t),Au(t)\in
C((0,T];H)$ was proved in our previous paper [12].Therefore, it suffices to
prove that (1.10) and (1.11) together are a formal solution to Problem 1.2. In
turn, for this it suffices to show that the Fourier coefficients $\varphi_{k}$
and $u_{k}(t)$ of functions (1.10) and (1.11) respectively, satisfy equation
(2.1), the non-local condition (2.2) and the over-determination condition

(3.1) $u_{k}(\xi_{2})=W_{k}.$

It is not hard to verify that $u_{k}(t)$ is a solution of equation (2.1).
Indeed, let first, $k\notin K_{0}$. We introduce the notation

$u_{k}^{1}(t)=\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,E_{\rho}(-\lambda_{k}t^{\rho}).$

Then $u_{k}(t)=u_{k}^{1}(t)+\omega_{k}(t)$. Here $u_{k}^{1}(t)$ is the
solution of the homogeneous equation (2.1) with the initial condition

$u_{k}^{1}(0)=\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha},$

and $\omega_{k}(t)$ is the solution of equation (2.1) with zero initial
condition (see, e.g. [33], p. 174).

If $k\in K_{0}$, then according to the orthogonality conditions $f_{k}=0$ and
the function

$u_{k}(t)=\frac{E_{\rho}(-\lambda_{k}t^{\rho})\,W_{k}}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}$

is a solution of the homogeneous equation (2.1) with the initial condition

$u_{k}(0)=\frac{W_{k}}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}.$

Thus we have shown that $u_{k}(t)$ is a solution of equation (2.1).

Let us check the non-local condition (2.2). Consider first the case $k\notin
K_{0}$. We have

$\alpha
u_{k}(0)+\varphi_{k}=\alpha\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}+\varphi_{k}.$

On the other hand,

$u_{k}(\xi_{0})=\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})+\omega_{k}(\xi_{0})=$
$=\frac{\varphi_{k}E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha\,\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}=\frac{\varphi_{k}\big{(}E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha\big{)}+\varphi_{k}\alpha-\alpha\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}=$
$=\alpha\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}+\varphi_{k}.$

Now consider the case $k\in K_{0}$. Note in this case
$E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})=\alpha$ and all Fourier coefficients
$\varphi_{k}$ are equal to zero since the orthogonality condition (1.5).
Therefore,

$\alpha
u_{k}(0)+\varphi_{k}=\alpha\frac{W_{k}}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}=E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})\frac{W_{k}}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}=u_{k}(\xi_{0}).$

Let us move on to checking the over-determination condition (3.1). Let
$k\notin K_{0}$. Then

$\frac{\varphi_{k}-\omega_{k}(\xi_{0})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})+\omega_{k}(\xi_{2})=W_{k},$

or

$\varphi_{k}=\frac{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}\,[W_{k}-\omega_{k}(\xi_{2})]+\omega_{k}(\xi_{0}),$

and this coincides with the Fourier coefficients of the function (1.10).

If $k\in K_{0}$, then

$u_{k}(\xi_{2})=\frac{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})\,W_{k}}{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}=W_{k}.$

This completes the proof of the existence of a solution to Problem 1.2.

3.2. Uniqueness. Obviously, to prove the uniqueness of the solution to Problem
1.2, it suffices to show that the solution $\\{u(t),\varphi\\}$ to the
following inverse problem:

$D_{t}^{\rho}u(t)+Au(t)=0,\quad\quad t>0;$ $u(\xi_{0})=\alpha
u(0)+\varphi,\quad 0<\xi_{0}\leq T,$ $u(\xi_{2})=0,\quad 0<\xi_{2}\leq
T,\,\,\xi_{2}\neq\xi_{0},$

is identically zero: $u(t)\equiv 0$ and $\varphi=0$.

Let $u(t)$ be a solution to this problem and let $u_{k}(t)=(u(t),v_{k})$. Then

(3.2) $D_{t}^{\rho}u_{k}(t)+\lambda_{k}u_{k}(t)=0,\quad t>0;\quad
u_{k}(\xi_{0})=\alpha u_{k}(0)+\varphi_{k},\,\,u_{k}(\xi_{2})=0.$

Let $k\notin K_{0}$. Then it is not hard to verify that the following function

$u_{k}(t)=\frac{E_{\rho}(-\lambda_{k}t^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,\,\varphi_{k}$

is the only solution to the equation and non-local condition in (3.2). The
over-determination condition in (3.2) implies

$u_{k}(\xi_{2})=\frac{E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})}{E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})-\alpha}\,\,\varphi_{k}=0.$

Since $E_{\rho}(-\lambda_{k}\xi^{\rho})\neq\alpha$ and
$E_{\rho}(-\lambda_{k}\xi_{2}^{\rho})\neq 0$, then we have $\varphi_{k}=0$ and
therefore $u_{k}(t)\equiv 0$ for all $k\notin K_{0}$.

Now consider the case $k\in K_{0}$. Denote $u_{k}(0)=b_{k}$. Then the unique
solution to the differential equation in (3.2) with this initial condition has
the form $u_{k}(t)=b_{k}E_{\rho}(-\lambda_{k}t^{\rho})$ (see, e.g. [33],
p.174). Since $E_{\rho}(-\lambda_{k}\xi_{0}^{\rho})=\alpha$ and
$\varphi_{k}=0$ in the considering case, then the non-local condition is
satisfied for an arbitrary $b_{k}$. But the over-determination condition
$u_{k}(\xi_{2})=0$ implies $b_{k}=0$ and therefore $u_{k}(t)\equiv 0$ for
$k\in K_{0}$.

Thus, from the completeness of the system of eigenfunctions $\\{v_{k}\\}$, we
finally obtain $\varphi=0$ and $u(t)\equiv 0$, as required. The uniqueness and
hence Theorem 1.4 is completely proved.

##  4\. Conclusion

In the previous paper of the authors [12] it is proved that for
$\alpha\notin(0,1)$ the solutions of the forward and two inverse problems of
determining $f$ and $\varphi$ exist and are unique. If $\alpha\in(0,1)$ and
equality (1.4) holds for some $k\in K_{0}$, then to ensure the existence of
the solution to the forward problem, it is necessary to require the
orthogonality condition (1.5). However, in this case the solution is not
unique and it is determined up to the term

$\sum\limits_{k\in K_{0}}b_{k}E_{\rho}(-\lambda_{k}t^{\rho})\,v_{k},$

where $b_{k}$ are arbitrary numbers.

In this paper, we consider the above two inverse problems for critical values
of parameter $\alpha\in(0,1)$. An interesting effect arises here: when solving
the forward problem the uniqueness of solution $u(t)$ was violated, while when
solving the inverse problem for the same values of $\alpha$, solution $u(t)$
became unique. What is the matter here? It turns out, as follows from the main
results of this paper, the over-determination condition

$u(\tau)=V$

can be rewritten in the form of two groups of conditions with respect to the
Fourier coefficients

$u_{k}(\tau)=V_{k},\,\,\,k\notin K_{0},$

and

$u_{k}(\tau)=V_{k},\,\,\,k\in K_{0}.$

With the help of the first group, the unique solutions of inverse problems are
singled out, and since the coefficients $f_{k}$ and $\varphi_{k}$ are equal to
zero for $k\in K_{0}$, the conditions from the second group are not used in
this case. And the conditions from the second group ensure the uniqueness of
the solution $u(t)$, namely, they determine uniquely the above arbitrary
coefficients $b_{k}$.

##  5\. Acknowledgement

The authors are grateful to Sh. A. Alimov for discussions of these results.

The authors acknowledge financial support from the Ministry of Innovative
Development of the Republic of Uzbekistan, Grant No F-FA-2021-424.

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