\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 33, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/33\hfil Conditional stability of a solution]
{Conditional stability of a solution of a difference scheme for an ill-posed \\
 Cauchy problem}

\author[M. A. Sultanov, M. I. Akylbaev, R. Ibragimov  \hfil EJDE-2018/33\hfilneg]
{Murat A. Sultanov, Musabek I. Akylbaev, Raskul Ibragimov}

\address{Murat A. Sultanov \newline
Department of Mathematics,
Akhmet Yasawi International Kazakh-Turkish University,
str. Sattarkhanov, 29, 161200 Turkistan,  Kazakhistan}
\email{murat.sultanov@ayu.edu.kz}

\address{Musabek I. Akylbaev \newline
Department of Computer  Science and Mathematics,
Kazakhstan Engineering and Pedagogical University
of Friendship of Peoples,
str. Tole bi  32, 160000, Shymkent,  Kazakhstan}
\email{akyl-bek@mail.ru}

\address{Raskul Ibragimov \newline
Department of Physics and Mathematics,
South Kazakhstan State Pedagogical Institute,
str. Baitursynov 13,  160000, Shymkent,  Kazakhstan}
\email{raskul1953@mail.ru}

\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted July 5, 2017. Published January 23, 2018.}
\subjclass[2010]{35R30, 65M12}
\keywords{Carleman estimates; Ill-posed Cauchy problems; finite stability;
\hfill\break\indent Difference operator; numerical solution}

\begin{abstract}
 In this article, we obtain criteria for stability of two-layer difference
 schemes for an abstract ill-posed Cauchy problem. Method of proof is
  based on obtaining a priori difference weighted Carleman type estimates.
 Stability conditions for solutions of two-layer difference schemes are
 used to prove the theorem of conditional stability of a solution of
 three-layer scheme that approximates an ill-posed Cauchy problem for an 
 integral-differential equation associated with a coefficient inverse problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article,  stability problems of difference schemes for an ill-posed
Cauchy problem and their application to investigation of coefficient
inverse problems are considered. The applied research method is based on
concept of stability of a difference scheme on functions with compact support,
 and in obtaining difference a priori weighted Carleman type estimates.
This concept was introduced and developed by Bukhgeim \cite{b7,b8}
in connection with construction of the theory of difference schemes for ill-posed
Cauchy problems,  encompassing equations with variable coefficients.

Application of a priori estimates with weight for proof of uniqueness of a
solution of the Cauchy problem originates from the work of   Carleman \cite{c1}.
Later this method was extended to a wider class of partial differential
equations by many authors \cite{h1,i1,n1}.

To inverse problems on determining coefficients of partial differential equations,
the method of weighted a priori estimates was first applied in the work of
Bukhgeim and M.V.Klibanov \cite{b9}. They proved uniqueness theorems for solutions
of multidimensional inverse problems in ``whole''.

Stability of difference schemes for an ill-posed Cauchy problem with constant
 coefficients was first investigated by  Chudov \cite{c2},  by using the Fourier
transform method. The approach based on definition of $\rho$-stability,
 introduced by Samarskii \cite{s1},  and SM (Spectral Mimetic) stability with
respect to ill-posed and inverse problems was investigated in the works of
 Vabishchevich \cite{v1,v2}. Methods of quasi-reversibility have been discussed
in \cite{b6,k3},  iterative methods have been investigated in \cite{b2,b3}.

With reduction of the method to practical numerical algorithms for solving
ill-posed inverse problems,  the Carleman estimates method was first
proposed by Klibanov and  Timonov \cite{k4}. Further development and application
of the method were considered in (see \cite{b1,b4,i2,k2} and their references).
 Current state and application of Carleman estimates in the theory of
multidimensional coefficient inverse problems is given in the review paper of
Klibanov \cite{k1}.

Conditions for conditional stability of the solution of a three-layer
difference scheme for an ill-posed Cauchy problem for an integro-differential
equation are obtained. This equation is associated with one-dimensional
coefficient inverse problem for the nonstationary Schr\"{o}dinger equation.
The stability of the difference scheme is proved by factoring the problem
into a sequence of two-layer schemes. In contrast to \cite{b8}, in which necessary
and sufficient conditions for abstract two and three-layer schemes are obtained,
we apply these results to the coefficient inverse problem and explicitly
construct the spectral decomposition of the difference version of the
operator $ - i{\partial _t},$  that occurs in the main part of the
Schr\"{o}dinger operator.

\section{Finite stability and stability of two-layer difference schemes}

Let ${\rm Z} = \{ 0,   \pm 1,   \pm 2, \dots\} $ and
$u:{\rm Z} \to H$ be a function of integer arguments
$j \in {\rm Z}$ with values in complex Hilbert space $H$,  with the norm
$\| {  u}\|$ and dot product $\langle   u, v \rangle $,
$\tau $ be an arbitrary positive number. We define the difference derivatives,
 and use the usual notation for difference schemes:
\begin{gather*}
{u_t} = ({u_{j + 1}} - {u_j})/\tau,   {u_{\bar t}}
= ({u_j} - {u_{j - 1}})/\tau,  {u_{t\bar t}}
= ({u_{j + 1}} - 2{u_j} + {u_{j - 1}})/{\tau ^2},\\
\mathop u\limits^ \wedge   = {u_{j + 1}}, \quad
 \mathop u\limits^ \vee   = {u_{j - 1}}.
\end{gather*}
Consider an abstract two-layer difference scheme with weight:
\begin{equation} \label{1}
\begin{gathered}
{( {Pu})_j} \equiv ( {{u_{j + 1}} - {u_j}} )/\tau
- A( {\sigma {u_{j + 1}} + ( {1 - \sigma } ){u_j}} ) = {f_j}, \\
{u_0} = g,\quad  j = 0, 1, \dots, N - 1.
\end{gathered}
\end{equation}

Here $A$ is a linear bounded operator,  acting in the space $H$,  and,
 possibly,  depending on $j; \sigma $ is a real parameter; $g, {f_j}$ are
given elements in the space $H$,  $\tau N = T - const$. Using the notation
introduced above,  we write the difference scheme \eqref{1} in the compact form:
\begin{equation} \label{2}
Pu \equiv {u_t} - A( {\sigma \mathop u\limits^ \wedge   + ( {1 - \sigma } )u} ) = f.
\end{equation}
Introduce the corresponding weighted norms (see \cite[p.133]{b8}).
 Let ${\rm Z}_0^N = \{ 0,  1,  \dots N\}$, $\varphi: {\rm Z}_0^N \to R$
be a real-valued monotonic decreasing weighted function,  i.e.
 $ - {\varphi _t} > 0$. Using the function $\varphi$ and  number,
we construct the function $\Psi :{\rm Z}_0^{N - 1} \to R$ so that:
$$
{\Psi _t} = s\mathop \Psi \limits^ \wedge  {\varphi _t}, \;{\Psi _0} = 1.
$$
The function $\Psi $ is a discrete analogue of the weighted function
$\exp (s\varphi (t))$. For the function $u:{\rm Z}_0^{N - 1} \to H$ we put:
\begin{equation} \label{3}
\| {  u}\|_s^2 = \tau \sum_{j = 0}^{N - 1} {\Psi _j^2(s){{\| {  {u_j}}\|}^2}}.
\end{equation}
The norm \eqref{3} is a discrete analogue:
\begin{equation} \label{4}
\int_0^T {\exp ( {2s\varphi ( t )} ){{\| {u( t )}\|}^2}dt,}
\end{equation}
moreover,  as $\tau  \to 0$ the expression \eqref{3} converges to \eqref{4}.

If we denote by ${l_2}(k, N; H)$ the Hilbert space of grid functions
$u:{\rm Z}_k^N \to H$,  ${\rm Z}_k^N = \{ k, k + 1, \dots, N\} $ with the norm
$\| {  u}\|_{{l_2}(k, N;H)}^2 = \tau {\sum_{j = k}^N {\| {  {u_j}}\|} ^2}$,
 then due to the definition of the norm
$\| {u}\|_s :  \| {u}\|_0 = \| {u}\|_{{l_2}(0, N - 1; H)} $.

We denote by ${C_0}(Z_0^N)$ the space of functions $u: Z_0^N \to H $ such that:
${u_0} = {u_N} = 0$.
The linear space ${C_0}(Z_0^N)$ is a discrete analogue of the space
${C_0}(0, T)$ of continuous functions $u(t):u(0) = u(T) = 0$ with compact
support on the interval $[ {0, T} ]$.
\smallskip

\noindent\textbf{Definition.}
Difference scheme $P$ of the form \eqref{2} is called stable on functions
with compact support,  if there exist independent from $\tau,  \| A\|$
 numbers ${s_0} > 0$, $M > 0$,  such that for all $s \ge {s_0}$,
$u \in {C_0}(Z_0^N)$ the following estimate holds:
\begin{equation} \label{5}
s \| {    u    }\|_s^2 \le M\, \| {Pu}\|_s^2.
 \end{equation}
To obtain a stability estimate on the whole grid ${\rm Z}_1^N$,
it is necessary to take into account contribution of outside integral terms
that arise when we use the formula of summation by parts,  and,  therefore,
it is necessary to work not with functions with compact support
${C_0}({\rm Z}_0^N)$,  but with arbitrary $u:{\rm Z}_0^N \to H$.
For brevity,  we introduce the  notation
$$
\| {  u}\|_{s(k, N)}^2 = \tau \sum_{j = k}^N {\Psi _j^2(s){{\| {  {u_j}}\|}^2}},
\quad k \ge 0.
$$
Consider the difference scheme
\begin{equation} \label{6}
\begin{gathered}
Pu \equiv {u_t} - ( {A + iB} )u = f, \;\;{i^2} =  - 1, \\
{u_0} = g,
\end{gathered}
\end{equation}
where $A, B$ are independent on $j$,  selfadjoint,  commuting,  positive operators,
i.e.
$$
{A^*} = A, \quad {B^*} = B, \quad [ {A, B} ] = 0, \quad A, \;B \ge 0\, .
$$
To obtain the stable estimate we will estimate $\| {Pu}\|_s^2$ below. We have
$$
\| {Pu}\|_s^2 = \tau \sum_{j = 0}^{N - 1}
{{{\| {{u_t} - ( {A + iB} )u}\|}^2}\Psi _j^2( s )}.
$$
Put $\Psi u = v$. According to the formula of difference differentiation
of product: ${u_t} = {( {{\Psi ^{ - 1}}v} )_t} = {( {{\Psi ^{ - 1}}} )_t}
\mathop v\limits^ \wedge   + {\Psi ^{ - 1}}{v_t}$.
From \eqref{3} it follows that
${( {1/\Psi } )_t} = ( { - s{\varphi _{tj}}} )/{\Psi _j}$, thus,
$$
{u_t} = ( {{v_t} - s{\varphi _t}\mathop v\limits^ \wedge  } )/\Psi.
$$
Then
\begin{equation} \label{7}
\begin{aligned}
&\| {Pu}\|_s^2 \\
&= \tau \sum_{j = 0}^{N - 1} {{{\| {{v_t} - s{\varphi _t}\mathop v\limits^ \wedge
- ( {A + iB} )v}\|}^2}}  \\
&= \tau \sum_{j = 0}^{N - 1} \Big\{ {{{\| {{v_t} - iBv}\|}^2} + }
+{{\| {Av + s{\varphi _t}\mathop v\limits^ \wedge  }\|}^2}
-  {2\operatorname{Re} \langle {v_t} - iBv,
Av + s{\varphi _t}\mathop v\limits^ \wedge  \rangle } \Big\} \\
&= \tau \sum_{j = 0}^{N - 1} \Big\{ {{\| {{v_t} - iBv}\|}^2}   
 + {\| {Av + s{\varphi _t}\mathop v\limits^ \wedge  }\|^2}
- 2\operatorname{Re} \langle {v_t}, Av\rangle
- 2\operatorname{Re} \langle {v_t}, s{\varphi _t}\mathop v\limits^ \wedge  \rangle\\
&\quad  +  {2\operatorname{Re} \langle iBv, Av\rangle
  + 2\operatorname{Re} \langle iBv, s{\varphi _t}\mathop v\limits^ \wedge
\rangle } \Big\} \equiv \sum_{k = 1}^6 {{I_k}}.
\end{aligned}
\end{equation}
Here by ${I_k}$ we denoted $\tau \sum_{j = 0}^{N - 1}$,
 corresponding to the $k$-term in curly brackets of the expression \eqref{7}.
 For numerical functions of a discrete argument,  we introduce the following
notation:
\begin{equation} \label{8}
[ {x, y} ) \equiv \tau \sum_{j = 0}^{N - 1} {{x_j}{y_j}, \quad
( {x, y} ) \equiv } \tau \sum_{j = 1}^{N - 1} {x_j}{y_j}.
\end{equation}
Using this notation,  from \eqref{7} we obtain:
\begin{gather*}
{I_1} = [ {1, {{\| {{v_t} - iBv}\|}^2}} ) \ge 0, \quad
{I_2} = [ {1,  {{\| {Av + s{\varphi _t}\mathop v\limits^ \wedge  }\|}^2}} )  \ge 0, \\
{I_3} =  - [ {1,  2\operatorname{Re} \langle {v_t}, Av\rangle } ), \quad
{I_4} =  - [ {s{\varphi _t}, 2\operatorname{Re} \langle {v_t},
 \mathop v\limits^ \wedge  \rangle } ),\\
{I_5} = [ {1, 2\operatorname{Re} \langle iBv, Av\rangle } ), \quad
{I_6} = [ {s{\varphi _t},  2\operatorname{Re} \langle iBv,
\mathop v\limits^ \wedge   \rangle } ).
\end{gather*}
For functions of discrete argument  $v, w:Z \to H$ the formula of difference
 differentiation has the form:
$$
\partial \langle v, w\rangle  = \langle {v_t}, \mathop w\limits^ \wedge  \rangle
+ \langle v, {w_t}\rangle .
$$
In particular,  when $w = v$,
$\partial {\| v\|^2} = \langle {v_t}, \mathop v\limits^ \wedge  \rangle
+ \langle v - \mathop v\limits^ \wedge, {v_t}\rangle
 + \langle \mathop v\limits^ \wedge, {v_t}\rangle
=  - \tau {\| {{v_t}}\|^2} + 2\operatorname{Re} \langle {v_t},
\mathop v\limits^ \wedge  \rangle $, i.e.
$ 2\operatorname{Re} \langle {v_t}, \mathop v\limits^ \wedge  \rangle
= \partial {\| v\|^2} + \tau {\| {{v_t}}\|^2}$.
$2\operatorname{Re} \langle {v_t}, \mathop v\limits^ \wedge  \rangle
= \partial {\| v\|^2} + \tau {\| {{v_t}}\|^2}$. Then for ${I_4}$ we obtain:
${I_4} =  - [ {s{\varphi _t}, 2\operatorname{Re} \langle {v_t},
 \mathop v\limits^ \wedge  \rangle } )
=  - [ {s{\varphi _t}, \partial {{\| v\|}^2}} )
- s\tau [ {{\varphi _t}, {{\| {{v_t}}\|}^2}} )$.
According to the formula $[ {x, \partial y} )
=  - ( {\bar \partial x, y} ) + ( {{x_{N - 1}}{y_N} - {x_0}{y_0}} )$
of summation by parts,  we obtain:
${I_4} = ( {s{\varphi _{t\bar t}}, {{\| v\|}^2}} )
- [ {s\tau {\varphi _t}, {{\| {{v_t}}\|}^2}} )
- s( {{\varphi _t}_{N - 1}{{\| {{v_N}}\|}^2}
- {\varphi _{t0}}{{\| {{v_0}}\|}^2}} )$.
Suppose that  $\tilde \mu  \ge  - {\varphi _t} \ge \mu  > 0$.
 Taking this condition into account,  we have
\begin{equation} \label{9}
{I_4} \ge ( {s{\varphi _{t\bar t}}, {{\| v\|}^2}} )
- [ {s\tau {\varphi _t}, {{\| {{v_t}}\|}^2}} )
+ s\mu {\| {{v_N}}\|^2} - s\tilde \mu {\| {{v_0}}\|^2}.
\end{equation}

We now transform the expression
$2\operatorname{Re} \langle iBv, \mathop v\limits^ \wedge  \rangle$.
$2\operatorname{Re} \langle iBv, \mathop v\limits^ \wedge  \rangle
= 2\operatorname{Re} \langle iBv, v + \tau {v_t}\rangle
= 2\operatorname{Re} \langle iBv, v\rangle  + \tau  \cdot 2\operatorname{Re}
 \langle iBv, {v_t}\rangle $. Because of self-adjointness
$B,  \operatorname{Re} \langle iBv, v\rangle  = 0$ for all $v \in H$,
 therefore, $2\operatorname{Re} \langle iBv, \mathop v\limits^ \wedge
\rangle = \tau  \cdot 2\operatorname{Re} \langle iBv, {v_t}\rangle$.
 From this equality we obtain
$$
| {2\operatorname{Re} \langle iBv, \mathop v\limits^ \wedge  \rangle } |
=  \tau | {2\operatorname{Re} \langle iBv, {v_t}\rangle } |
\le \tau \big\{ {\alpha {{\| B\|}^2}{{\| v\|}^2}
+ {\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}} \big\}.
$$
Here we used $\alpha$-inequality:
$2ab \le \alpha {a^2} + {\alpha ^{ - 1}}{b^2}, \alpha  > 0$.
Since
${I_6} = s[ {{\varphi _t}, 2\operatorname{Re}
\langle iBv, \mathop v\limits^ \wedge  \rangle } )$, we have
$| {{I_6}}| \le \tau s[ {| {{\varphi _t}} |,
\alpha {{\| B\|}^2}{{\| v\|}^2}} )
+ \tau s[ {| {{\varphi _t}} |, {\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}} )$.
Thus,  writing $| {{\varphi _t}} |$ in the form
$| {{\varphi _t}} | =  - {\varphi _t} ( { - {\varphi _t} > 0} )$,  we obtain
\begin{equation} \label{10}
\begin{aligned}
{I_6}
&\ge \tau s\big\{ {\big[ {1, {\varphi _t}\alpha {{\| B\|}^2}{{\| v\|}^2}} \big)
+ \big[ {1, {\varphi _t}{\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}} \big)} \big\}  \\
&= {\tau ^2}s{\varphi _{t0}}\alpha {\| B\|^2}{\| {{v_0}}\|^2}
 + \tau s\big( {1, {\varphi _t}\alpha {{\| B\|}^2}{{\| v\|}^2}} \big)
+ \tau s\big[ {1, {\varphi _t}{\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}}\big)  \\
&\ge - {\tau ^2}s\tilde \mu \alpha {\| B\|^2}{\| {{v_0}}\|^2}
 + \tau s\big\{ {\big( {1, {\varphi _t}\alpha {{\| B\|}^2}{{\| v\|}^2}} \big)
+ \big[ {1, {\varphi _t}{\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}} \big)} \big\}.
\end{aligned}
\end{equation}

From the	equality $2\operatorname{Re} \langle {v_t}, Av\rangle
= \partial \langle v, Av\rangle  - \tau \langle {v_t}, A{v_t}\rangle$,
and using the formula of summation by parts, we obtain
\begin{equation} \label{11}
\begin{aligned}
{I_3}& =  - [ {1, \;\partial \langle v, Av\rangle } )
 + [ {1, \;\langle {v_t}, \tau A{v_t}\rangle } )  \\
&= [ {1, \;\langle {v_t}, \;\tau A{v_t}\rangle } )
 - \langle {v_N}, A{v_N}\rangle  + \langle {v_0}, A{v_0}\rangle \\
& \ge  - \, \langle {v_N}, A{v_N}\rangle.
\end{aligned}
\end{equation}
Here we took into account the fact that ${A^*} = A$, $A \ge 0$.

Compute the term ${I_6}$. From commutativity of the operators $A$ and $B$,
we obtain
$$
2\operatorname{Re} \langle iBv, Av\rangle
 = \langle iBv, Av\rangle  + \langle Av, iBv\rangle
= \langle iABv, v\rangle  - \langle iBAv, v\rangle
= \langle i[ {A, B} ]v, v\rangle  = 0,
$$
then
$$
{I_5} = [ {1, 2\operatorname{Re} \langle iBv, Av\rangle } ) = 0.
$$
The obtained estimates yield the following result.

\begin{lemma}	\label{lemma1}
Let  $Pu \equiv u_t - ( A + iB)u, A^* = A \ge 0,  B^* = B \ge 0, [ {A, B} ] = 0$.
Then for all $u:Z_0^N \to H$:
\begin{gather*}
\| {Pu}\|_s^2 = \sum_{k = 1}^6 {I_k},  \quad
{I_1} = [ {1, {{\| {{v_t} - iBv}\|}^2}} ) \ge 0, \\
{I_2} = [ {1, {{\| {Av + s{\varphi _t}\mathop v\limits^ \wedge  }\|}^2}} ) \ge 0,\quad
{I_3}  \ge  - \langle {v_N}, A{v_N}\rangle, \; {I_5} = 0, \\
{I_4} \ge ( {s{\varphi _{t\bar t}}, {{\| v\|}^2}} )
- [ {s\tau {\varphi _t}, {{\| {{v_t}}\|}^2}} )
+ s\mu {\| {{v_N}}\|^2} - s\tilde \mu {\| {{v_0}}\|^2}, \\
{I_6} \ge \tau s\big\{ { ( {1,   {\varphi _t}\alpha {{\| B\|}^2}{{\| v\|}^2}} )
+ [ {1, {\varphi _t}{\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}} )} \big\}
- {\tau ^2}s\tilde \mu \alpha {\| B\|^2}{\| v\|^2}.
\end{gather*}
Here $v = \Psi u,  - {\varphi _t} > 0$.
\end{lemma}

This lemma implies the following theorem.

\begin{theorem}	\label{theorem1}
Let  in the conditions of  Lemma \ref{lemma1}  for all
$s \ge {s_0}$ and  some $\delta  > 0$ the following conditions hold:
\begin{gather} \label{12}
{M_1} \equiv ( {s{\varphi _{t\bar t}} + \tau s{\varphi _t}\alpha {{\| B\|}^2}} )E
\ge s\delta E, \\
 \label{13}
{M_0} \equiv  - s\tau {\varphi _t}( {1 - {\alpha ^{ - 1}}} )E \ge 0, \alpha  > 0.
\end{gather}
Then for all $u:Z_0^N \to H, s \ge {s_0}$ for the difference scheme \eqref{6}
the following stability estimate holds
\begin{equation} \label{14}
s\| u\|_{s( {1, N} )}^2 \le \mu _2^{ - 1}
\big\{ {\| {Pu}\|_s^2 + s{\mu _0}{{\| {{u_0}}\|}^2}
+ \Psi _N^2\langle {u_N}, A{u_N}\rangle } \big\}.
 \end{equation}
Here ${\mu_0}, {\mu_2}$  are certain positive constants.
\end{theorem}

\begin{proof}
Discarding the quantities ${I_1}, {I_2} \ge 0$,  and collecting separately
the terms containing  $v$ and ${v_t}$,  by Lemma \ref{lemma1} we obtain:
\begin{align*}
\| {Pu}\|_s^2
&\ge ( {1, \langle {M_1}v, v\rangle } )
 + [ {1, \langle {M_0}{v_t}, {v_t}\rangle } ) \\
&\quad - s\tilde \mu ( {1 + {\tau ^2}\alpha {{\| B\|}^2}} ){\| {{v_0}}\|^2}
 - \langle {v_N}, A{v_N}\rangle  + s\mu {\| {{v_N}}\|^2}.
\end{align*}
Thus,  from \eqref{12} and \eqref{13},  and taking into account that
$v = \Psi u$, ${\Psi _0} = 1 $, we have
\begin{equation} \label{15}
\begin{aligned}
& \delta s\| u\|_{s( {1, N - 1} )}^2 + s\mu \Psi _N^2{\| {{u_N}}\|^2}\\
& \le \| {Pu}\|_s^2 + s\tilde \mu ( {1 + {\tau ^2}
 \alpha {{\| B\|}^2}} ){\| {{u_0}}\|^2}
+ \Psi _N^2\langle {u_N}, A{u_N}\rangle.
\end{aligned}
\end{equation}
When $0 < \tau  \le {\tau _0}$ and
${\mu _2} = \min ( {\delta, \frac{\mu }{{{\tau _0}}}} )$,  we obtain
$$
s\delta \| u\|_{s( {1, N - 1} )}^2 + s\mu \Psi _N^2{\| {{u_N}}\|^2}
\ge s{\mu _2}\| u\|_{s( {1, N} )}^2.
$$
Taking this estimate into account,  and assuming
${\mu _0} = \tilde \mu ( {1 + \tau _0^2\alpha {{\| B\|}^2}} )$,
after dividing the inequality \eqref{15} by ${\mu _2}$,
we obtain \eqref{14}. The proof is complete.
\end{proof}

Furthermore,  we assume that the following conditions are satisfied:
$$
{\varphi _{t\bar t}} \ge 1, \; - {\varphi _t} \ge 1.
$$

\begin{theorem}	\label{theorem2}
Let
\begin{equation} \label{16}
\tau {\| B\|^2} \le c, \quad c > 0.
\end{equation}
Then there exists a number  ${c_1} = c( {\alpha, c} )$such that when
\begin{equation} \label{17}
{\varphi _{t\bar t}} + c{\varphi _t} \ge 1
\end{equation}
for the difference scheme \eqref{6} the estimate \eqref{14} holds.
\end{theorem}

\begin{proof}
 By choosing sufficiently large numbers ${s_0}$ and $\alpha $,  we obtain
non-negativity of the operator${M_0}$. Similarly,  due to \eqref{16} and
\eqref{17},  we have:
\begin{align*}
\langle {M_1}v, v\rangle
&= s{\varphi _{t\bar t}}{\| v\|^2} + \tau s{\varphi _t}\alpha {\| B\|^2}{\| v\|^2} \\
&\ge s{\varphi _{t\bar t}}{\| v\|^2} + s{\varphi _t}\alpha  \cdot c{\| v\|^2} \\
& = s( {{\varphi _{t\bar t}} + {c_1}{\varphi _t}} ){\| v\|^2}
\ge s \cdot \varepsilon {\| v\|^2}
\end{align*}
at a large enough number   and small $\varepsilon $. Reference to
Theorem \ref{theorem1} completes the proof of the theorem.
\end{proof}

	Consider now the difference scheme:
\begin{equation} \label{18}
\begin{gathered}
Pu \equiv {u_t} + ( {A + iB} )u = f, \;{i^2} =  - 1, \\
{u_0} = g.
\end{gathered}
\end{equation}
Here operators $A$ and $B$ satisfy the same conditions as in \eqref{6}.
Similarly to Lemma \ref{lemma1},  we establish the following result.

\begin{lemma}	\label{lemma2}
Let $ Pu \equiv {u_t} + ( {A + iB} )u$,  where ${A^*} = A \ge 0$, ${B^*} = B \ge 0$,
$ [ {A, B} ] = 0$. Then for all $u:Z_0^N \to H$ we have:
\begin{gather*}
\| {Pu}\|_s^2 = \sum_{k = 1}^6 {{{\tilde I}_k}, } \\
{\tilde I_1} = [ {1, {{\| {{v_t} + iBv}\|}^2}} ) \ge 0, \quad
 {\tilde I_2} = [ {1, {{\| {Av - s{\varphi _t}\mathop v\limits^ \wedge  }\|}^2}} )
\ge 0, \\
 {\tilde I_3} \ge  - [ {1, \langle {v_t}, \tau A{v_t}\rangle } )
- \langle {v_0}, \tau A{v_0}\rangle, \\
{\tilde I_4} \ge \big( {s{\varphi _{t\bar t}}, {{\| v\|}^2}} \big)
- \big[ {s\tau {\varphi _t}, \;{{\| {{v_t}}\|}^2}} \big)
- s\tilde \mu {\| {{v_0}}\|^2} + s\mu {\| {{v_N}}\|^2}, \quad
 {\tilde I_5} = 0, \\
{\tilde I_6} \ge \tau s\big\{ {( {1,  {\varphi _t}\alpha {{\| B\|}^2}{{\| v\|}^2}} )
+ [ {1,  {\varphi _t}{\alpha ^{ - 1}}{{\| {{v_t}}\|}^2}} )} \big\}
- {\tau ^2}s\tilde \mu \alpha {\| B\|^2}{\| v\|^2}.
\end{gather*}
\end{lemma}

This lemma yields the following two theorems.

\begin{theorem}	\label{theorem3}
Let in the conditions of Lemma \ref{lemma2} for all $s \ge {s_0}$ and for some
$\delta  > 0$ the following inequality hold:
\begin{gather*}
{\tilde M_1} \equiv \big( {s{\varphi _{t\bar t}} + \tau s{\varphi _t}\alpha
{{\| B\|}^2}} \big)E \ge s\delta E, \\
{\tilde M_0} \equiv  - s\tau {\varphi _t}( {1 - {\alpha ^{ - 1}}} )E - \tau A \ge 0.
\end{gather*}
Then for all $s \ge {s_0}, \;u:Z_0^N \to H$ to solve the difference scheme
\eqref{18} we have the stability estimate:
\begin{equation} \label{19}
s\| u\|_{s( {1, N} )}^2 \le \mu _2^{ - 1}
\big\{ {\, \| {Pu}\|_s^2 + s{\mu _0}\| {{u_0}}\|
+ \langle {u_0}, A{u_0}\rangle } \big\}.
\end{equation}
\end{theorem}

\begin{theorem}	\label{theorem4}
Let for some  $m, c > 0$ the following inequality hold:
\begin{equation} \label{20}
\tau A \le mE, \quad \tau {\| B\|^2} \le c.
\end{equation}
Then there exists a number ${c_1} = {c_1}( {\alpha, m, c} )$ such that when
$$
{\varphi _{t\bar t}} + {c_1}{\varphi _t} \ge 1
$$
to solve the difference scheme \eqref{18} we have the estimate \eqref{19}.
\end{theorem}

Theorems \ref{theorem3} and \ref{theorem4}  are proved simlarly
to Theorems \ref{theorem1} and  \ref{theorem2}.

\begin{remark} \label{remark1} \rm
To obtain Lemmas \ref{lemma1},  \ref{lemma2} we used the method of proofs
of \cite[Lemma 2.1 p. 142]{b8},  where it is used to obtain sufficient
conditions for stability with compact support of two-layer difference schemes.
\end{remark}

\begin{remark} \label{remark2} \rm
Theorem \ref{theorem2} was proved by  Bukhgeim the case of functions with
compact support without assumption of positivity of the operators \cite{b8}.
\end{remark}

\section{An ill-posed Cauchy problem for integral-differential equation}

In this section we give an ill-posed Cauchy problem related to the one-dimensional
coefficient inverse problem for the Schr\"{o}dinger equation.
Such an ill-posed Cauchy problem arises instudy of questions of uniqueness
and stability of solutions of coefficient inverse problems in non-stationary
formulation \cite{b5,b8}. Scheme of proof of these theorems consists
in reducing the inverse problem to an ill-posed Cauchy problem for
integral-differential equations,  and the subsequent application of a
priori weighted Carleman type estimates. Therefore,  justifying the
difference methods for solving these inverse problems,  it becomes
necessary to obtain stability estimates for solutions of difference
schemes that approximate an ill-posed Cauchy problem for the corresponding
integral-differential equations or inequalities.

Let $\Omega  = \{  x,  t  :   x > 0, t^2 + (x - r) < 0 \}$
(see Figure \ref{fig1}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width =0.5\textwidth]{fig1}
\end{center}
\caption{Source area.}
\label{fig1}
\end{figure}

Consider on the domain $\Omega $ the following ill-posed Cauchy problem:
\begin{gather} \label{21}
\begin{aligned}
i{v_t} + {v_{xx}}
&= {a_1}(x)v + ({b_1}\partial  + {b_0})({u_2}(x, t)f(x)/{f_2}(x)  \\
&\quad + \int_0^t {K(x,  t,  \tau )v(x, \tau )d\tau ), } \end{aligned} \\
 \label{22}
v(0, t) = {g'}(t) - {g'}_2(t)g(t)/{g_2}(t), \quad {v_x}(0, t) = 0,
\end{gather}
where $K(x, t, \tau ) = {u_2}(x, t)/{u_2}(x, \tau )$,
$\partial  = \partial /\partial x, {i^2} =  - 1$. Concerning to the coefficients
 in \eqref{21},  we assume that they are smooth enough functions of their
variables (conclusion of the integro-differential equation \eqref{21}
is given in \cite[p.40]{b8}). Writing the equation \eqref{21} in more detail,
and assuming
\begin{gather*}
\overline b (x, t) \equiv {b_0}(x, t){u_2}(x, t)/{f_2}(x)
+ {b_1}(x, t){({u_2}(x, t)/{f_2}(x))_x}, \\
 {\overline b _1}(x, t) \equiv {b_1}(x, t){u_2}(x, t)/{f_2}(x), \\
\overline K (x, t, \tau ) \equiv {b_0}(x, t)K(x, t, \tau )
 + {b_1}(x, t){K_x}(x, t, \tau ), \\
 \overline {{K_1}} (x, t, \tau ) \equiv {b_1}(x, t) K(x,     t, \tau ),
\end{gather*}
it can be rewritten in the form
\begin{equation} \label{23}
\begin{aligned}
i{v_t} + {v_{xx}}
& = {a_1}(x)v + \overline b (x, t)f(x)
+ \overline {{b_1}} (x, t){f'}(x)  \\
&\quad + \int_0^t {(\overline K (x, t, \tau )v(x, \tau )
+ \overline {{K_1}} (x, t, \tau ){v_x}(x, \tau ))d\tau.}
\end{aligned}
\end{equation}
We make the change of variables in the equation \eqref{21}.
Put $x = \xi  - {t^2},  \, t = t$. Then
$$
v(x, t) = w(\xi, t) = w(x + {t^2}, t), \quad
{v_t} = 2t \cdot {w_\xi } + {w_t}, \quad
{v_x} = {w_\xi }, {v_{xx}} = {w_{\xi \xi }}.
$$
After simple transformations and using the formula
$\int {\delta  (p(\tau ))\, u(\tau )d\tau }
= \frac{{u({\tau _0})}}{{| {{p'}({\tau _0})} |}}$,
 where ${\tau _0}$ is a unique root of the equation $p(\tau ) = 0$
(in our case  $p(\tau ) = {\tau ^2} - {t^2} + \xi  - \eta  )$,  we obtain:
\begin{equation} \label{24}
\begin{aligned}
&2it{w_\xi } + i{w_t} + {w_{\xi \xi }} \\
& = {{\tilde a}_1}(\xi, t)w(\xi, t) + \tilde b(\xi, t)\tilde f(\xi, t)
  + {{\tilde b}_1}(\xi, t){{\tilde f}_\xi }(\xi, t)  \\
&\quad \pm \int_{\xi  - {t^2}}^\xi  {\frac{{\tilde K(\eta, t,
\pm \sqrt {\eta   - \xi  + {t^2}} ) w (\eta,  \pm \sqrt {\eta  - \xi
 + {t^2}}  )}}{{2\sqrt {\eta  - \xi  + {t^2}} }}d\eta }    \\
&\quad \pm \int_{\xi  - {t^2}}^\xi  {\frac{{{{\tilde K}_1}
(\eta, t,  \pm \sqrt {\eta  - \xi  + {t^2}} ) {w_\eta }
(\eta,  \pm \sqrt {\eta \xi  + {t^2}} ) }}{{2\sqrt {\eta    - \xi  + {t^2}} }}
d\eta } .
\end{aligned}
\end{equation}

Here the sign (+) corresponds to the case $t > 0$,  and the sign (-) to the
case $t < 0$. For convenience we rename the variables
$\xi, t:\xi: = t$, $t: = x$. Then the equation \eqref{24} has the  form
\begin{gather} \label{25}
\begin{aligned}
&2ix{w_t} + i{w_x} + {w_{tt}} \\
&= {{\tilde a}_1}(t, x)w(t, x) + \tilde b(t, x)\tilde f(t, x)
 + {{\tilde b}_1}(t, x){{\tilde f}_t}(t, x) \\
&\quad \pm \int_{t - {x^2}}^t \frac{\tilde K(\eta  , x,
 \pm \sqrt {\eta  - t + {x^2}}) w(\eta  ,
 \pm \sqrt {\eta  - t + {x^2}} )} {2\sqrt {\eta  - t + {x^2}} } d\eta  \\
&\quad \pm \int_{t - {x^2}}^t \frac{{{{\tilde K}_1}(\eta, x,
 \pm \sqrt {\eta  - t + {x^2}} ){w_\eta } (\eta,
 \pm \sqrt {\eta  - t + {x^2}} )}} {{2\sqrt {\eta  - t + {x^2}} }}d\eta, 
\end{aligned} \\
\label{26}
w\big| _\gamma  = {g_x}(t, x) - {g_{2x}}(t, x)g(t, x)/
{g_2}(t, x), {w_t}\big|_\gamma
 = 0\quad,
\end{gather}
where $\gamma: t = {x^2}$. Note that the original domain $\Omega $
after the change of variables,  and renaming,  goes into the domain bounded
between the parabola $t = {x^2}$  and  line $t = r$,  which we also denote by
 $\Omega$ (see Figure \ref{fig2}).

\begin{figure}[ht]
\begin{center}
 \includegraphics[width =0.5\textwidth]{fig2} 
\end{center}
\caption{Area after the change of variables and renaming.}
\label{fig2}
\end{figure}


Assuming ${w_0} = w\big| {_\gamma } $. and introducing a new function
$\tilde w = w - {w_0}$,   the problem \eqref{25} - \eqref{26} can be
 reduced to a problem with homogeneous boundary conditions on $\gamma$:
\begin{gather} \label{27}
2ix{\tilde w_t} + i{\tilde w_x} + {\tilde w_{tt}} = F, \\
 \label{28}
\tilde w \big|_\gamma  = 0, \quad \tilde w_t  \big| _\gamma  = 0\,.
\end{gather}
where $F$ is the right side of the equation \eqref{25}.
We extend the function $\tilde w$  in \eqref{27} by continuity by zero to a
rectangular domain,  and rename $\tilde w:= w$. The condition \eqref{28},
 unbounded generality,  is replaced by $w(0, x) = g(x), {w_t}(0, x) = 0$.
Then in the domain   we obtain the following problem:
\begin{gather} \label{29}
{w_{tt}} + i{w_x} = {\tilde a_1}(t, x)w - 2ix{w_t} + Kw + f(t, x), \\
 \label{30}		
w(0, x) = g(x), {w_t}(0, x) = 0.
\end{gather}
Here
\begin{gather*}
\begin{aligned}		
Kw &= {K_0}w + {K_1}w, \; {K_0}w(t, x) \\
&= \pm \int_{t - {x^2}}^t {\frac{{\tilde K(\eta, x,
\pm \sqrt {\eta  - t + {x^2}} )\, w\, (\eta,
\pm \sqrt {\eta \; - t + {x^{}}} )}}{{2\sqrt {\eta - t + {x^2}} }}d\eta },
\end{aligned}\\
{K_1}w(t, x)
=  \pm \int_{t - {x^2}}^t \frac{{{{\tilde K}_1}(\eta, x,
\pm \sqrt {\eta  - t + {x^2}} )\, {w_\eta }(\eta,
\pm \sqrt {\eta - t + {x^2}} )}}{{2\sqrt {\eta  - t + {x^2}} }}d\eta, \\
f(t, x) = \tilde b(t, x)\tilde f(t, x) + {{\tilde b}_1}(t, x){{\tilde f}_t}(t, x).
\end{gather*}

Our goal is to obtain an estimate of conditional stability of a solution
of difference scheme that approximates the ill-posed Cauchy problem
\eqref{29}-\eqref{30}. Construction and proof of the conditional stability
of difference scheme solution for this problem are carried out in the
next section. Unconditional stability of three-layer difference scheme
for the problem \eqref{29}-\eqref{30},  depending on two parameters,
 was obtained in \cite{s2}.

\section{Stability of difference scheme for an ill-posed Cauchy problem}

We associate with the problem \eqref{29}-\eqref{30} the following three-layer
difference scheme:
\begin{gather} \label{31}
\begin{aligned}
&\frac{u_{j + 1}^k - 2u_j^k + u_{j - 1}^k}{\tau ^2} - Au_{j - 1}^k   \\
&= \tilde{a}_{1\, j - 1}^ku_{j - 1}^k - 2ikh(u_j^k - u_{j - 1}^k)/\tau
 + {K_{\tau, \, h}}u_{j - 1}^k + f_{j - 1}^k,
\end{aligned} \\
 \label{32}
\begin{gathered}
{u_0} = {g^k}, \quad {u_1} = {u_0}, \\
j = 1, 2,  \dots, N - 1,\quad \tau  N = r,\\
 k = 0, \pm 1,  \dots,   \pm ({N_1} - 1), \quad \pm h{N_1} =  \pm \sqrt r  \equiv  \pm  T.
\end{gathered}
\end{gather}
Here $Au_{j - 1}^k =  - i \frac{{u_{j - 1}^{k + 1} - u_{j - 1}^{k - 1}}}{{2h}}$,
${K_{\tau, h}}$ is an approximation of the operator $K = {K_0} + {K_1}$ such that
\begin{equation} \label{33}
\| {{K_{\tau, h}}u}\| \le c ( \| u\| + \| {{u_t}}\| ).
\end{equation}
Naturalness of this assumption follows from boundedness and Volterra property
of the integral operator ${K_0}$ by $t$. We will consider the operator
as an operator acting in the complex Hilbert space. As $H$ we take the space
of grid functions $u(x)$ defined on the grid
${\tilde \omega _h} =\{ {{x_k} = kh, \, k = 0,  \pm 1,  \dots,   \pm {N_1},
  \pm h{N_1} =  \pm  T} \} $ and vanishing for $k =  - {N_1}, \; k = {N_1}$.
The dot product and norm in space $H$are introduced in the usual form:
$$
\langle {u, v} \rangle  = h\sum_{k =  - ({N_1} - 1)}^{{N_1} - 1}
{{u^k}{{\bar v}^k}\,\quad
{{\| u\|}^2} = h\sum_{k =  - ({N_1} - 1)}^{{N_1} - 1} {{{| {{u^k}} |}^2}} }
$$
Obviously,  $A$ will be a self-adjoint operator in $H$. Furthermore,
 we shall omit the index $k$ in all the functions. We denote the right-hand
side of \eqref{31} by ${F_j}$,  and write the equation \eqref{31} with a
shift by a step to the right:
\begin{equation} \label{34}
\frac{{{u_{j + 2}} - 2{u_{j + 1}} + {u_j}}}{{{\tau ^2}}} - A{u_j} = {F_j}.
\end{equation}
Put:
\begin{gather} \label{35}
{u_{j + 1}} - R{u_j} = \tau {v_j}, \\
 \label{36}
{v_{j + 1}} - S{v_j} = \tau {F_j},
\end{gather}
and we will try to pick up the operators $R$ and $S$ so that after
exclusion $v$ the system \eqref{35} - \eqref{36} goes into
 equation \eqref{34}. Eliminating ${v_j}$ from \eqref{36} with the help
 of \eqref{35},  we obtain
$$
{u_{j + 2}} - ( {R + S} ){u_{j + 1}} + SR{u_j} = {\tau ^2}{F_{j + 1}};
$$
thus
\begin{gather} \label{37}
R + S = 2E, \\
 \label{38}
SR = E - {\tau ^2}A.
\end{gather}
We represent the operator $A$ in the form of difference of two nonnegative
commuting operators ${A_ \pm }$:
$A = {A_ + } - {A_ - }, {( {{A_ \pm }} )^*} = {A_ \pm }\,, \;{A_ + }
 = QA, \;{A_ - } = ( {Q - E} )A, $ where  $Q$
 is an orthogonal projection,  projecting the space onto the subspace
of eigenfunctions corresponding to the nonnegative part of spectrum
of the operator$A$. Since ${A_ \pm } \ge 0, $ the nonnegative operators
$\sqrt {{A_ \pm }} $ are uniquely determined. By definition,
$$
\sqrt A  = \sqrt {{A_ + }}  + i\sqrt {{A_ - }}.
$$
Since by construction ${A_ \pm }{A_ \mp } = 0$,  then
$E - {\tau ^2}A = ( {E - \tau \sqrt A } )( {E + \tau \sqrt A } )$.
Assuming now $S = ( {E - \tau \sqrt A } ), R = ( {E + \tau \sqrt A } )$,
 we find the solution of the system of equations \eqref{37}-\eqref{38}:
\begin{gather*}
{u_{j + 1}} - ( {E + \tau \sqrt A } ){u_j} = \tau {v_j}, \quad u_0 = g, \\
{v_{j + 1}} - ( {E - \tau \sqrt A } ){v_j} = \tau {F_{j + 1}}, \quad
 {v_0} =  - \sqrt A {u_0},
\end{gather*}
or
\begin{gather} \label{39}
{u_t} - \sqrt A u = v, \quad u_0 = g, \\
 \label{40}
{v_t} + \sqrt A v = \tilde F,  \quad
 {v_0} =  - \sqrt A {u_0},    ( {\tilde F = {F_{j + 1}}} ).
\end{gather}
Applying Theorem \ref{theorem2} from Section 1 to the difference
scheme \eqref{39},  and Theorem \ref{theorem4} to the scheme \eqref{40} with
$N: = N - 1$,  and taking into account that in our case
$A = \sqrt {{A_ + }} , \; B = \sqrt {{A_ - }}$,  and the conditions
\eqref{16},  \eqref{20} have the form
\begin{equation} \label{41}
\tau \sqrt {{A_ + }}  \le mE, \quad \tau {\| {\sqrt {{A_ - }} }\|^2} \le c,
\end{equation}
we obtain
$$
s\| u\|_{s( {1, N} )}^2 \le \mu _2^{ - 1}
\left\{ {{\tau _0}{{\| {{v_0}}\|}^2}
+ \| v\|_{s( {1, N - 1} )}^2 + s{\mu _0}{{\| {{u_0}}\|}^2}
+ \Psi _N^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle } \right\}
$$
(here we noted that ${\Psi _0} = 1, \;\tau  \le {\tau _0}$),
\begin{equation} \label{42}
s\| v\|_{s( {1, N - 1} )}^2
\le \mu _2^{ - 1}\big\{ {\| {\tilde F}\|_{s( {0, N - 2} )}^2
+ s{\mu _0}{{\| {{v_0}}\|}^2} + \langle {v_0},
\sqrt {{A_ + }} {v_0}\rangle } \big\}.
\end{equation}
Combining these estimates,  we have
\begin{align*}
{s^2}\| u\|_{s( {1, N} )}^2
& \le s\mu _2^{ - 1}\big\{ {s{\mu _0}{{\| {{u_0}}\|}^2}
 + \Psi _N^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle } \big\}
 + s{\tilde \mu _0}{\| {{v_0}}\|^2} \\
&\quad + \mu _2^{ - 2}\big\{ {\| {\tilde F}\|_{s( {0, N - 2} )}^2
 + \langle {v_0}, \sqrt {{A_ + }} {v_0}\rangle } \big\},
\end{align*}
where ${\tilde \mu _0} = {\tau _0}\mu _2^{ - 1} + \mu _2^{ - 2}{\mu _0}$.
Combining this inequality with \eqref{42},  and taking into account the fact
that $v = {u_t} - \sqrt A u$,  we obtain
\begin{equation} \label{43}
\begin{aligned}
&{s^2}\| u\|_{s( {1, N} )}^2 + s\| {{u_t} - \sqrt A u}\|_{s( {1, N - 1} )}^2 \\
&\le  s\mu _2^{ - 1}\big\{ {s{\mu _0}{{\| {{u_0}}\|}^2}
+ \Psi _N^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle } \big\}  \\
&\quad + s{c_1}{\| {{v_0}}\|^2} + {c_2}
 \big\{ {\| {\tilde F}\|_{s( {0, N - 2} )}^2
 + \langle {v_0}, \sqrt {{A_ + }} {v_0}\rangle } \big\}.
\end{aligned}
\end{equation}
Here  ${c_1} = {\tilde \mu _0} + \mu _2^{ - 1}{\mu _0}$,
${c_2} = \mu _2^{ - 2} + \mu _2^{ - 1}$.
Applying Theorem \ref{theorem4} to the difference problem
$$
{u_t} + \sqrt A u = v, \quad {u_0} = g, \quad
( {R = E - \tau \sqrt A } ),
$$
and Theorem \ref{theorem2} to the  difference problem
$$
{v_t} - \sqrt A v = \tilde F, \quad {v_0}
= \sqrt A {u_0}, \;\;( {S = E + \tau \sqrt A } ),
$$
we obtain
\begin{equation} \label{44}
\begin{gathered}
s\| u\|_{s( {1, N} )}^2
\le \mu _2^{ - 1}\big\{ {{\tau _0}{{\| {{v_0}}\|}^2}
+ \| v\|_{s( {1, N - 1} )}^2 + s{\mu _0}{{\| {{u_0}}\|}^2}
+ \langle {u_0}, \sqrt {{A_ + }} {u_0}\rangle } \big\},\\
\begin{aligned}
&s\| v\|_{s( {1, N - 1} )}^2 \le \\
&\le \mu _2^{ - 1}\big\{ {\| {\tilde F}\|_{s( {0, N - 2} )}^2
+ s{\mu _0}{{\| {{v_0}}\|}^2} + \Psi _{N - 1}^2\langle {v_{N - 1}},
\sqrt {{A_ + }} {v_{N - }}\rangle } \big\}.
\end{aligned}
\end{gathered}
\end{equation}
As above,  combining these estimates,  we have
\begin{align*}
{s^2}\| u\|_{s( {1, N} )}^2 
&\le s\mu _2^{ - 1}
\big\{ {s{\mu _0}{{\| {{u_0}}\|}^2} + \langle {u_0}, \sqrt {{A_ + }} 
{u_0}\rangle } \big\} + s{\tilde \mu _0}{\| {{v_0}}\|^2} \\
&\quad + \mu _2^{ - 2}\big\{ {\| {\tilde F}\|_{s( {0, N - 2} )}^2 
+ \Psi _{N - 1}^2\langle {v_{N - 1}}, \sqrt {{A_ + }} {v_{N - 1}}\rangle }
 \big\}.
\end{align*}
Combining this inequality with \eqref{44} and taking into account that 
$v = {u_t} + \sqrt A u$,  we have
\begin{equation} \label{45}
\begin{aligned}
&{s^2}\| u\|_{s( {1, N} )}^2 + s\| {{u_t} + \sqrt A u}\|_{s( {1, N - 1} )}^2\\
& \le s\mu _2^{ - 1}\big\{ {s{\mu _0}{{\| {{u_0}}\|}^2}
 + \langle {u_0}, \sqrt {{A_ + }} {u_0}\rangle } \big\} 
 + s{c_1}{\| {{v_0}}\|^2} \\
&\quad +  {c_2} \big\{ {\| {\tilde F}\|_{s( {0, N - 2} )}^2
+ \Psi _{N - 1}^2\langle {v_{N - 1}}, \sqrt {{A_ + }} {v_{N - 1}}\rangle } \big\}.
\end{aligned}
\end{equation}
Adding now the estimate \eqref{45} with the estimate \eqref{43},  and taking
into account the identity:
$$
{\| {{u_t} - \sqrt A u}\|^2} + {\| {{u_t} + \sqrt A u}\|^2}
= 2{\| {{u_t}}\|^2} + 2{\| {\sqrt A u}\|^2},
$$
we obtain
\begin{equation} \label{46}
\begin{aligned}
&2{s^2}\| u\|_{s( {1, N} )}^2 + 2s( {\| {{u_t}}\|_{s({1, N - 1} )}^2
 + \| {\sqrt A u}\|_{s( {1, N - 1} )}^2} )  \\
&\le s\mu _2^{ - 1}\big\{ {2s{\mu _0}{{\| {{u_0}}\|}^2}
 + \langle {u_0}, \sqrt {{A_ + }} {u_0}\rangle
 + \Psi _N^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle } \big\}
 + 2s{c_1}{\| {{v_0}}\|^2}   \\
&\quad + {c_2}\big\{ {2\| {\tilde F}\|_{s( {0, N - 2} )}^2
 + \langle {v_0}, \sqrt {{A_ + }} {v_0}\rangle
 + \Psi _{N - 1}^2\langle {v_{N - 1}}, \sqrt {{A_ + }} {v_{N - 1}}\rangle } \big\}.
\end{aligned}
\end{equation}
Estimate the term $\| \tilde F\|_{s( {0, N - 2} )}^2$ in the right-hand side
of \eqref{46}. Noting that $\tilde F= \, =F_{j + 1}$ in the right-hand side
 of \eqref{31},  due to the obvious inequality
${\| u\|_{s( {0, N - 2} )}}  \le  {\| u\|_s} \le {\| u\|_{s( {0, N} )}}$
and the condition \eqref{33},  we have
\begin{equation} \label{47}
\| {{F_{j + 1}}}\|_{s( {0, N - 2} )}^2
\le c( {\| u\|_{s( {1, N} )}^2 + \| {{u_t}}\|_{s( {1, N - 1} )}^2
+ \| f\|_s^2 + {{\| {{u_0}}\|}^2}} )
\end{equation}
with some constant $c$. Here we used the condition ${u_1} = {u_0}$ and
${\Psi _0} = 1, \;\tau  \le {\tau _0}$. From \eqref{47} and \eqref{46} we obtain
\begin{align*}
&2{s^2}( {1 - \frac{{{c_2}{c_0}}}{{{s^2}}}} )\| u\|_{s( {1, N} )}^2
 + 2s( {1 - \frac{{{c_2}{c_0}}}{s}} )\| {{u_t}}\|_{s( {1, N - 1} )}^2
 + 2s\| {\sqrt A u}\|_{s( {1, N - 1} )}^2 \\
&\le s\mu _2^{ - 1}\big\{ {2s{\mu _0}{{\| {{u_0}}\|}^2}
 + \langle {u_0}, \sqrt {{A_ + }} {u_0}\rangle
 + \Psi _N^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle } \big\}
 + 2{c_2}{c_0}{\| {{u_0}}\|^2} \\
&\quad  + 2s{c_1}{\| {{v_0}}\|^2} + {c_2}
 \big\{ {\langle {v_0}, \sqrt {{A_ + }} {v_0}\rangle
 + \Psi _{N - 1}^2\langle {v_{N - 1}}, \sqrt {{A_ + }} {v_{N - 1}}\rangle } \big\}
 + 2{c_2}{c_0}\| f\|_s^2.
\end{align*}
Choosing ${s_0}$ large enough,  it is possible to achieve the condition
$1 - \frac{{{c_2}{c_0}}}{s} \ge \frac{1}{2}$ when $s \ge {s_0}$
(especially $1 - \frac{{{c_2}{c_0}}}{{{s^2}}} \ge \frac{1}{2}$). Therefore,
\begin{equation} \label{48}
\begin{aligned}
{s^2}\| u\|_{s( {1, N} )}^2
&\le s\mu _2^{ - 1}\big\{ 2s{\mu _0}{{\| {{u_0}}\|}^2}
 + \langle {u_0}, \sqrt {{A_ + }} {u_0}\rangle 
 + \Psi _N^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle  \big\} \\ 
&\quad + 2{c_2}{c_0}{\| {{u_0}}\|^2}
 + {c_2}\big\{ \langle {v_0}, \sqrt {{A_ + }} {v_0}\rangle \\
&\quad + \Psi _{N - 1}^2\langle {v_{N - 1}}, \sqrt {A_ + } {v_{N - 1}}\rangle a 
 \big\}  + 2{c_2}{c_0}\| f\|_s^2.
\end{aligned}
\end{equation}

Note that ${v_{N - 1}} = {u_{tN - 1}} - \sqrt A {u_{N - 1}}, {v_0} 
=  - \sqrt A {u_0}$, by \eqref{39} and \eqref{40}.
 Taking into account that $\sqrt A  = \sqrt {{A_ + }}  + i\sqrt {{A_ - }}$, 
${( {\sqrt {{A_ \pm }} } )^*} = \sqrt {{A_ \pm }}  \ge 0$, 
$ \sqrt {{A_ \pm }} \sqrt {{A_ \mp }}  = 0$ it is easy to establish the 
following equalities:
\begin{gather*}
{\| {{v_0}}\|^2} = \langle {A_ + }{u_0}, {u_0}\rangle  
+ \langle {A_ - }{u_0}, {u_0}\rangle,  \langle {v_0}, \sqrt {{A_ + }} {v_0}\rangle  
= \langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_0}, \sqrt {{A_ + }} {u_0}\rangle,  \\
\begin{aligned}
\langle {v_{N - 1}}, \sqrt {{A_ + }} {v_{N - 1}}\rangle  
&= \langle \sqrt {{A_ + }} {u_{tN - 1}}, {u_{tN - 1}}\rangle  
- 2\operatorname{Re} \langle {A_ + }{u_{tN - 1}}, {u_{N - 1}}\rangle  \\
&\quad + \langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_{N - 1}}, 
\sqrt {{A_ + }} {u_{N - 1}}\rangle 
\end{aligned}
\end{gather*}
According to these equalities,  and taking into account the obvious 
inequalities 
\begin{gather*}
{\Psi _{j + 1}} < {\Psi _j}, \| u\|_{s( {1, N} )}^2 
\ge  \| u\|_{s( {1, N - 1} )}^2 
\ge \Psi _{N - 1}^2\| u\|_{{l_2}( {1, N - 1; H} )}^2, \\
\| f\|_s^2 \le \| f\|_{{l_2}( {1, N - 1;H} )}^2,
\end{gather*}  
from \eqref{48} we obtain the estimate
\begin{align*}
&{s^2}\Psi _{N - 1}^2\| u\|_{{l_2}( {1, N - 1;H} )}^2 \\
&\le s\mu _2^{ - 1}\left\{ {2s{\mu _0}{{\| {{u_0}}\|}^2} 
 + \langle {u_0}, \sqrt {{A_ + }} {u_0}\rangle  
 + \Psi _{N - 1}^2\langle {u_N}, \sqrt {{A_ + }} {u_N}\rangle } \right\} 
 + 2{c_2}{c_0}{\| {{u_0}}\|^2} \\
&\quad +2s{c_1}( {\langle {A_ + }{u_0}, {u_0}\rangle  
 + \langle {A_ - }{u_0}, {u_0}\rangle } ) 
 + {c_2}\Big\{ {\langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_0}, 
 \sqrt {{A_ + }} {u_0}\rangle  } \\
&\quad + \Psi _{N - 1}^2\Big( \langle \sqrt {{A_ + }} {u_{tN - 1}}, 
 {u_{tN - 1}}\rangle  - 2\operatorname{Re} \langle {A_ + }{u_{tN - 1}}, 
 {u_{N - 1}}\rangle \\ 
&\quad  +   \langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_{N - 1}}, 
\sqrt {{A_ + }} {u_{N - 1}}\rangle  \Big) \Big\}+ 2{c_2}{c_0}
\| f\|_{{l_2}( {0, N - 1;H} )}^2.
\end{align*}
Assuming $s \ge 1$ and
$$
{\tilde c_1} = \max \big( {\frac{1}{{{\mu _2}}}, {c_2}} \big), \quad
{\tilde c_2} = \frac{{2{\mu _0}}}{{{\mu _2}}} + 2{c_2}{c_0}, \quad
{\tilde c_3} = \max \big( {\frac{1}{{{\mu _2}}}, \;2{c_1}, \;{c_2}, \;2{c_2}{c_0}} 
\big)
$$
from the above estimate we have
\begin{align*}
&{s^2}\Psi _{N - 1}^2\| u\|_{{l_2}( {1, N - 1;H} )}^2 \\
&\le s{\tilde c_1}\Psi _{N - 1}^2\Big\{ {\langle \sqrt {{A_ + }} {u_N}, {u_N}\rangle 
 + } \langle \sqrt {{A_ + }} {u_{tN - 1}}, {u_{tN - 1}}\rangle  \\
&\quad - 2\operatorname{Re} \langle {A_ + }{u_{tN - 1}}, {u_{N - 1}}\rangle
  +  {\langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_{N - 1}}, \sqrt {{A_ + }} 
{u_{N - 1}}\rangle } \Big\} \\
&\quad + {s^2}{\tilde c_2}{\| {{u_0}}\|^2}  
  + s{\tilde c_3}\Big\{ {\langle \sqrt {{A_ + }} {u_0}, {u_0}\rangle  + } 
 \langle {A_ + }{u_0}, {u_0}\rangle  + \langle {A_ - }{u_0}, {u_0}\rangle  \\
&\quad + \langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_0}, 
 \sqrt {{A_ + }} {u_0}\rangle  + {\| f\|_{{l_2}( {0, N - 1;H} )}^2} \Big\}.
\end{align*}
Dividing both sides of this inequality by ${s^2}\Psi _{N - 1}^2$,  and supposing
$$
{\varepsilon ^2} = {\tilde c_1}/s, \quad 
{c^2}( \varepsilon  ) = \max \big\{ {{{\tilde c}_2}  
\exp ( {2s\, \tilde m\, \tau } ), \;{{\tilde c}_3}{s^{ - 1}}
 \exp ( {2s\, \tilde m\, \tau } )} \big\},
$$
from inequality $1/{\Psi _{N - 1}} < 1/{\Psi _N} \le \exp ( {s\tilde m\tau } )$ 
(see  \cite[Lemma 1.1,  p.132]{b8}) we obtain
\begin{align*}
&\| u\|_{{l_2}( {1, N - 1;H} )}^2 \\
&\le {\varepsilon ^2}\Big\{ {\langle \sqrt {{A_ + }} {u_N}, {u_N}\rangle  
+ } \langle \sqrt {{A_ + }} {u_{tN - 1}}, {u_{tN - 1}}\rangle  \\
&\quad - 2\operatorname{Re} \langle {A_ + }{u_{tN - 1}}, {u_{N - 1}}\rangle  
 +  {\langle \sqrt {{A_ + }} \sqrt {{A_ + }} {u_{N - 1}}, 
 \sqrt {{A_ + }} {u_{N - 1}}\rangle } \Big\} \\
&\quad + {c^2}( \varepsilon  ) \Big\{ {{{\| {{u_0}}\|}^2} 
+ \langle \sqrt {{A_ + }} {u_0}, {u_0}\rangle  + } 
  \langle {A_ + }{u_0}, {u_0}\rangle  + \langle {A_ - }{u_0}, {u_0}\rangle  \\
&\quad + \langle \sqrt {{A_ + }}\sqrt {{A_ + }} {u_0}, \sqrt {{A_ + }} {u_0}\rangle 
 +  {\| f\|_{{l_2}( {0, N - 1;H} )}^2} \Big\}.
\end{align*}
Denoting $\| u\|_D^2 = \langle Du, u\rangle $, where the operator $D \ge 0$,
 we rewrite the last inequality in the  form
\begin{equation} \label{49}
\begin{aligned}
\| u\|_{{l_2}( {1, N - 1;H} )}^2 
&\le {\varepsilon ^2}\Big\{ {\| {{u_N}}\|_{\sqrt {{A_ + }} }^2
 + \| {{u_{tN - 1}}}\|_{\sqrt {{A_ + }} }^2}   \\
&\quad - 2\operatorname{Re} \langle {A_ + }{u_{tN - 1}}, {u_{N - 1}}\rangle 
 +  {\| {\sqrt {{A_ + }} {u_{N - 1}}}\|_{\sqrt {{A_ + }} }^2} \Big\}  \\
&\quad + {c^2}( \varepsilon  )\Big\{ {{{\| {{u_0}}\|}^2} 
 + \| {{u_0}}\|_{\sqrt {{A_ + }} }^2 +\| {A_ + ^{1/2}{u_0}}\| +}  \\
&\quad +  { \| {A_ - ^{1/2}{u_0}}\| 
 + \| {\sqrt {{A_ + }} {u_0}}\|_{\sqrt {{A_ + }} }^2 
 + \| f\|_{{l_2}( {0, N - 1;H} )}^2} \Big\}.
\end{aligned}
\end{equation}
We now turn to the construction of the operators 
$A_+$,  $A_-$, $\sqrt {A_+}$,  $\sqrt {A_-} $. Consider for the operator
$$
{( {Au} )^k} =  - i\frac{{{u^{k + 1}} - {u^{k - 1}}}}{{2h}}, \quad {i^2} =  - 1, 
$$
and the  eigenvalue problem
\begin{equation} \label{50}
\begin{gathered}
A{u^k} = \lambda {u^k}, \quad k = 0,  \pm 1, \dots,  \pm ( {{N_1} - 1} ), \quad
 \pm h{N_1} =  \pm T, \\
{u^{{N_1}}} = {u^{ - {N_1}}} = 0.
\end{gathered}
\end{equation}
By direct calculations it is not difficult to show that eigenvalues of 
the operator $A$ and the corresponding eigenfunctions are determined 
by the  formulas
$$
{\lambda _m} = \frac{1}{h}\sin \frac{{\pi   m}}{{2{N_1}}}, \quad
 u_m^k = {e^{ik\frac{{\pi   m}}{{2{N_1}}}}} 
 - {( - 1)^{k - {N_1}}}{e^{i(2{N_1} - k)\frac{{\pi   m}}{{2{N_1}}}}},
$$
for $k,  m = 0,  \pm 1, \dots,  \pm ({N_1} - 1)$,
and norm of the eigenfunctions $ u_m^k$ in the sense of the above dot 
product is $ \| u_m^k\|^2 = 4T$.
	
Since the eigenfunctions $u_m^k$ are orthogonal,  and consequently,  
linearly independent,  then the functions $\mu _m^k = \frac{1}{{2\sqrt T }}u_m^k$ 
for orthonormal basis in the space $H$,  consisting  of  eigenfunctions of 
 the operator $A$,  corresponding to ${\rm{\{ }}{\lambda _{\rm{m}}}{\rm{\} }}$.

Since ${A^*} = A$,   we have spectral decomposition of the operator $A$:
$$
A = \sum_{m =  - ({N_1} - 1)}^{{N_1} - 1} {{\lambda _m}{P_m}}  
= \sum_{m = 0}^{{N_1} - 1} {{\lambda _m}{P_m}}  
 - \sum_{m =  - ({N_1} - 1)}^{ - 1} {( - {\lambda _m}){P_m}},
$$
where ${P_m}$ projector,  defined by the relation 
${P_m}u = \langle     u $, $\mu _m^k \rangle  \mu _m^k$, $u \in H$. 
Hence we see that the operators ${A_ \pm }$ have the form:
$$
{A_ + } = \sum_{m = 0}^{{N_1} - 1} {{\lambda _m}{P_m}} \,, \quad 
{A_ - } = \sum_{m = 1 - {N_1}}^{ - 1} {( - {\lambda _m}){P_m}}.
$$
We define the operators  $\sqrt {{A_ \pm }} $ by 
$$
\sqrt {{A_ + }}  = \sum_{m = 0}^{{N_1} - 1} {\lambda _m^{1/2}} {P_m}\,, \quad 
\sqrt {{A_ - }}  = \sum_{m = 1 - {N_1}}^{ - 1} {{{( { - {\lambda _m}} )}^{1/2}}} 
{P_m}.
$$
It is obvious that
\begin{equation} \label{51}
\begin{gathered}
\| A\| = \| {{A_ + }}\| = \| {{A_ - }}\| 
= \frac{1}{h}\sin \frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}} \le \frac{1}{h}, \\
\| {\sqrt {{A_ + }} }\| = \| {\sqrt {{A_ - }} }\| 
= \sqrt {\frac{1}{h}\sin \frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}}} 
 \le \frac{1}{{\sqrt h }}\, .
\end{gathered}
\end{equation}

We transform the conditions
\begin{equation} \label{52}
\tau {\| {\sqrt {{A_ + }} }\|^2} \le c,  \tau \sqrt {{A_ + }}  \le mE,  c, m > 0,
\end{equation}
under which we obtained stability of the two-layer difference schemes 
\eqref{39},  \eqref{40}. Note that the condition 
$\tau \sqrt {{A_ + }}  \le mE$ is satisfied if  
$\tau \| {\sqrt {{A_ + }} }\| \le m$. Indeed,
\begin{align*}
\langle ( {mE - \tau \sqrt {{A_ + }} } )u, u\rangle  
&= m{\| u\|^2} - \tau \langle \sqrt {{A_ + }} u, u\rangle  \\
&\ge m{\| u\|^2} - \tau \| {\sqrt {{A_ + }} }\|{\| u\|^2} \\
&= ( {m - \tau \| {\sqrt {{A_ + }} }\|} ){\| u\|^2} \ge 0,
\end{align*}
if  $\tau \| {\sqrt {{A_ + }} }\| \le m$. By \eqref{51} and this remark,  
instead of the condition \eqref{52} we obtain the  conditions
\begin{equation} \label{53}
\begin{gathered}
\frac{\tau }{h}\sin \frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}} \le c, \quad 
\tau  \le \frac{{ch}}{{sin\frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}}}}, \\
\frac{{{\tau ^2}}}{h}\sin \frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}} \le {m^2}, \quad 
{\tau ^2} \le \frac{{{m^2}h}}{{sin\frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}}}}.
\end{gathered}
\end{equation}
Since $\sin \frac{{\pi ( {{N_1} - 1} )}}{{2{N_1}}} 
= \sin \frac{\pi }{2}( {\frac{{T - h}}{T}} ) = O( 1 )$ at small $h$,
 from conditions \eqref{53} we obtain the condition:
$$
{\tau ^2} \le c\,  \cdot h, 
$$
where $c = \min ( {c,;m} )$, $c$ and $m$  are constants from \eqref{52}.
Therefore,  the following theorem of conditional stability of solution 
of the difference scheme \eqref{31}.

\begin{theorem}
Let ${\tau ^2} \le c h $ and $c > 0$.
Then for all $\tau  \in ( 0,\tau _0 ]$, $(\tau _0 = \sqrt {ch} )$, 
$\varepsilon  > 0$, $u:Z_0^N \to H$,  to solve the difference scheme 
\eqref{31} we have the stability estimate \eqref{49}.
\end{theorem}

\subsection*{Conclusions}
On the basis of notion of stability of a difference scheme on functions with 
compact support,  stability criteria for two-layer difference schemes,  
that approximate an ill-posed abstract Cauchy problem,  are obtained. 
Stability of difference schemes is based on obtaining a priori difference
 weighted Carleman type estimates. Obtained stability criteria are used to 
prove conditional stability of the solution of a three-layer difference 
scheme for an ill-posed Cauchy problem. In connection with cumbersomeness 
and technical complexity of obtaining a difference analogue of weighted 
stability estimates for three-layer schemes,  preliminary factorization 
of the problem into a sequence of two-layer schemes was carried out.

\subsection{Acknowledgments} 
 Murat Sultanov was supported by the Ministry of Education and Science of the 
Republic of Kazakhstan (project 3630/GF4).
The authors would like to thank the anonymous referees for the careful 
reading of this paper and for the valuable suggestions to improve the paper.

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\end{document}