# On Stability and Convergence of a Three-layer Semi-discrete Scheme for an
Abstract Analogue of the Ball Integro-differential Equation

Jemal Rogava ID , Mikheil Tsiklauri ID and Zurab Vashakidze ID

###### Abstract

We consider the Cauchy problem for a second-order nonlinear evolution equation
in a Hilbert space. This equation represents the abstract generalization of
the Ball integro-differential equation. The general nonlinear case with
respect to terms of the equation which include a square of a norm of a
gradient is considered. A three-layer semi-discrete scheme is proposed in
order to find an approximate solution. In this scheme, the approximation of
nonlinear terms that are dependent on the gradient is carried out by using an
integral mean. We show that the solution of the nonlinear discrete problem and
its corresponding difference analogue of a first-order derivative is uniformly
bounded. For the solution of the corresponding linear discrete problem, it is
obtained high-order a priori estimates by using two-variable Chebyshev
polynomials. Based on these estimates we prove the stability of the nonlinear
discrete problem. For smooth solutions, we provide error estimates for the
approximate solution. An iteration method is applied in order to find an
approximate solution for each temporal step. The convergence of the iteration
process is proved.

Keywords and phrases: Cauchy problem, Three–layer semi–discrete scheme,
Nonlinear integro-differential equation, Stability and convergence, Abstract
analogue of beam equation, Chebyshev polynomials.
MSC 2010: 46N40, 65J08, 65M06, 65M12, 65M22, 74H15, 74K10.

## Introduction

In the present work, we consider the Cauchy problem in the Hilbert space for a
nonlinear second-order abstract differential equation. Coefficients in the
main part of the equation are self-adjoint positively defined, in general,
unbounded operators. Our goal is to find an approximate solution to this
problem. To do so, we apply a three-layer symmetrical semi-discrete scheme. In
this scheme, nonlinear terms are approximated by using integral mean.

The considered equation represents an abstract generalization of J. M. Ball
beam equation (see [6]). J. M. Ball has generalized Kirchhoff type nonlinear
equation for beam, that was obtained by S. Woinowsky-Krieger (see [37]), by
introducing damping terms, in order to account for the effect of external and
internal damping.

Investigation of the topics related to the classic Kirchhoff equation started
with Bernstein’s well-known paper (see [7]). In this paper existence and
uniqueness issues for local as well as global solutions of initial-boundary
value problem for the Kirchhoff string equation is studied. The issues of
solvability of the classical and generalized Kirchhoff equations were later
considered by many authors: A. Arosio, S. Panizzi [1], L. Berselli, R. Manfrin
[8], P. D’Ancona, S. Spagnolo [12], [13], R. Manfrin [22], L. A. Medeiros
[24], M. Matos [23], K. Nishihara [25], S. Panizzi [26]. In the works [1],
[12], [13], [22] and [25] issues of well-posedness and global solvability are
thoroughly studied for a generalized Kirchhoff equation. In [26] the existence
of a global solution with low regularity is studied for Kirchhoff-type
equations. An abstract analogue of the Kirchhoff-type beam equation is
considered in the work by L. A. Medeiros [24], where the existence and
uniqueness theorem for the regular solution of the Cauchy problem is proved.
The same abstract nonlinear equation, strengthened by the first derivative
with respect to time, is discussed in the work by P. Biler and E. H. de Brito
(see [9], [14]), where most attention is paid to study of the behaviour of
Cauchy problem. We should note that participation of the square of the main
operator in the linear part of this equation essentially helps to obtain the
necessary a priori estimates.

The following works are dedicated to approximate solutions of initial-boundary
value problems for classical and generalized Kirchhoff equations: A. I.
Christie, J. Sanz-Serna [11], T. Geveci, I. Christie [15], I.-S. Liu, M.A.
Rincon [21], J. Peradze [27], J. Rogava, M. Tsiklauri [33], [32] and in [36].
An algorithm of approximate solution for the dynamic beam equation is studied
in [15]. This algorithm represents a combination of the Galerkin method for
spatial coordinates and the finite difference method for the time coordinate.
The same combination of the methods is investigated for the classic Kirchhoff
equation in [27]. Design of algorithms for finding numerical solutions and
their investigations for initial-boundary value problems of some classes
integro-differential equations are considered in the book of T. Jangveladze,
Z. Kiguradze and B. Neta [16].

As it was mentioned before J. M. Ball - generalized the Kirchhoff beam
equation by introducing damping terms, to account for the effect of external
and internal damping. For an approximate solution of the initial-boundary
problem of this equation, S. M. Choo and S. K. Chung proposed the finite
difference method (see [10]). In this work stability and convergence of the
approximate solution is investigated.

As far as we know, issues of approximate solution of abstract analogue of
Kirchhoff-type equation for a beam are less studied. In the present paper,
investigations of stability and convergence of the designed semi-discrete
scheme for second-order (complete kind) nonlinear operator differential
equation that represents the abstract analogue of a model of J. M. Ball for
the beam is based on two facts: (a) $\left(u_{k}-u_{k-1}\right)/\tau$ and
$B^{1/2}u_{k}$ are uniformly bounded ($u_{k}$ is an approximate solution, and
$\tau$ is time step; linear operator $B$ is included in the main part of the
equation); (b) For the solution of the corresponding linear problem an a
priori estimation is obtained where on the left-hand side power $s$ and on the
right-hand side power $s-1$ of the operator $B$ is included. These facts give
the possibility to weaken the nonlinear terms in the given nonlinear equation
so much that, to make it possible to apply Grönwall’s lemma. Besides, it is
not required to impose any essential restriction for the temporal step $\tau$.

##  1 Statement of the problem and semi-discrete scheme

Let us consider the following Cauchy problem in Hilbert space $H$:

$\displaystyle\frac{d^{2}u}{dt^{2}}+a_{1}B\frac{du}{dt}+a_{2}Bu+\psi_{1}\left(\left\|A^{1/2}u\right\|^{2}\right)Au$
$\displaystyle+\frac{d}{dt}\left(\psi_{2}\left(\left\|A^{1/2}u\right\|^{2}\right)\right)Au+\psi_{3}\left(\left\|u\right\|^{2}\right)u$
$\displaystyle+Cu+N\frac{du}{dt}+M\left(u\right)=f(t),\text{\quad}t\in]0,\;\overline{t}],$
(1.1) $\displaystyle
u\left(0\right)=\varphi_{0},\text{\quad}u^{\prime}\left(t\right)|_{t=0}=\varphi_{1}\;$
(1.2)

where $A$ and $B$ are self-adjoint, positively defined (generally unbounded)
operators with the domains $D\left(A\right)$ and $D\left(B\right)$ which are
everywhere dense in $H$, besides, the following conditions are fulfilled

$\left\|Au\right\|^{2}\leq b_{0}^{2}\left(Bu,u\right),\text{\quad}\forall u\in
D(B)\subset D\left(A\right),\text{\quad}b_{0}=const>0,$ (1.3)

where by $\|\cdot\|$ and $\left(\cdot,\cdot\right)$ are defined
correspondingly the norm and scalar product in $H$; $\psi_{1}(s)$,
$\;\psi_{2}(s)$ and $\psi_{3}(s)$,$\;s\in[0,+\infty[$ are twice continuously
differentiable nonnegative functions, besides $\psi_{2}(s)$ is increasing
function; $C$ is linear operator, which satisfies the following condition

$\left\|Cu\right\|\leq a_{0}\left\|Au\right\|\;,\text{\quad}\forall u\in
D(A)\subset D(C),\text{\quad}a_{0}=const>0;\text{\quad}$ (1.4)

$N$ is linear bounded operator; nonlinear operator $M(\cdot)$ satisfies
Lipschitz condition; $a_{1}$ and $a_{2}$ are positive constants $\varphi_{0}$
and $\varphi_{1}$ are given vectors from $H$; $u\left(t\right)$ is a twice
continuously differentiable, unknown function with values in $H$ and
$f\left(t\right)$ is given continuous function with values in $H$.

As in the linear case (see S. G. Krein [19]) vector function $u\left(t\right)$
with values in $H$, defined on the interval $\left[0,\overline{t}\right]$ is
called a solution of the problem (1.1)-(1.2) if it satisfies the following
conditions: (a) $u\left(t\right)$ is twice continuously differentiable in the
interval $\left[0,\overline{t}\right]$; (b)
$u\left(t\right),u^{\prime}\left(t\right)\in D\left(B\right)$ for any $t$ from
$\left[0,\overline{t}\right]$ and $\;Bu\left(t\right)$ and
$Bu^{\prime}\left(t\right)$ are continuous functions; (c) $u\left(t\right)$
satisfies equation (1.1) on the $\left[0,\overline{t}\right]$ interval and the
initial condition (1.2). Here continuity and differentiability is meant by
metric $H$.

Equation (1.1) is an abstract analogue of the following equation

$\displaystyle\frac{\partial^{2}u}{\partial
t^{2}}+a_{1}\frac{\partial^{4}}{\partial x^{4}}\genfrac{(}{)}{}{}{\partial
u}{\partial t}+a_{2}\frac{\partial^{4}u}{\partial
x^{4}}-\left(\alpha+\beta\int\limits_{0}^{l}\left[\partial_{\xi}u\left(\xi,t\right)\right]^{2}d\xi\right)\frac{\partial^{2}u}{\partial
x^{2}}$ $\displaystyle-\gamma\frac{\partial}{\partial
t}\left(\int\limits_{0}^{l}\left[\partial_{\xi}u\left(\xi,t\right)\right]^{2}d\xi\right)\frac{\partial^{2}u}{\partial
x^{2}}+\delta\frac{\partial u}{\partial
t}=f(t),\text{\quad}(x,t)\in]0,l[\times]0,\overline{t}]\,,$ (1.5)

where $a_{1}$, $a_{2}$, $\beta$ and $\gamma$ are positive and $\alpha$ ,
$\delta$ any constants.

At the first time, equation (1) was considered by J. M. Ball in [6]. In this
paper, J. M. Ball investigated the existence, uniqueness and asymptotic
behaviour of the solution of equation (1) using the topological method.

We look for an approximate solution of the problem (1.1)-(1.2) using the
following semi-discrete scheme

$\displaystyle\frac{u_{k+1}-2u_{k}+u_{k-1}}{\tau^{2}}+a_{1}B\frac{u_{k+1}-u_{k-1}}{2\tau}+a_{2}B\frac{u_{k+1}+u_{k-1}}{2}$
$\displaystyle+a_{1,k}A\frac{u_{k+1}+u_{k-1}}{2}+d_{k}A\frac{u_{k+1}+u_{k-1}}{2}+a_{3,k}\frac{u_{k+1}+u_{k-1}}{2}$
$+Cu_{k}+N\frac{u_{k+1}-u_{k-1}}{2\tau}+M\left(u_{k}\right)=f_{k},$ (1.6)

where $f_{k}=f\left(t_{k}\right)$, $\leavevmode\nobreak\
k=1,\ldots,n-1$,$\leavevmode\nobreak\ t_{k}=k\tau$,
$\tau=\overline{t}/n\leavevmode\nobreak\ $ $\left(n>1\right)$,

$\displaystyle
a_{1,k}=\widetilde{\psi}_{1}\left(\gamma_{k-1},\gamma_{k+1}\right),\text{\quad}\gamma_{k}=\left\|A^{1/2}u_{k}\right\|^{2}\text{,}$
$\displaystyle
d_{k}=\frac{\psi_{2}(\gamma_{k+1})-\psi_{2}(\gamma_{k-1})}{2\tau}\;\text{,}\quad
a_{3,k}=\widetilde{\psi}_{3}\left(\left\|u_{k-1}\right\|^{2},\left\|u_{k+1}\right\|^{2}\right)\,,$

and where function $\widetilde{\psi}_{1}(a,b)$ (analogously of
$\widetilde{\psi}_{3}(a,b)$) is defined using the following formula

$\widetilde{\psi}_{1}(a,b)=\frac{1}{b-a}\int\limits_{a}^{b}\psi_{1}(s)ds.\text{\quad}$
(1.7)

It is clear that if interval $b-a$ is small enough, then (1.7) formula gives
good approximation of $\psi_{1}(s)$ function at $s=(a+b)/2$.

In (1.6), nonlinear terms are approximated using integral mean. This approach
first was used in [27] and [32].

As an approximate solution $u\left(t\right)$ of problem (1.1)-(1.2) at point
$t_{k}=k\tau$ we declare $u_{k}$, $u\left(t_{k}\right)\approx u_{k}\text{.}$

######  Remark 1.1.

From (1.3) condition it follows that

$\left\|Au\right\|\leq b_{0}\left\|B^{1/2}u\right\|,\text{\quad}\forall u\in
D(B)\subset D\left(A\right).$ (1.8)

It is known, that $D(B)$ is a core of $B^{1/2}$ (see [17], p. 354). It means
that, for every $u\in D(B^{1/2})$ there exists sequence $u_{n}\in D(B)$ such
that, $u_{n}\rightarrow u$ and $B^{1/2}u_{n}\rightarrow B^{1/2}u$. From here,
according to (1.8) it follows that $Au_{n}$ is Cauchy sequence and it is clear
that, since $H$ is complete, this sequence is convergent. $u\in D(A)$ and
$Au_{n}\rightarrow Au$ as $A$ is closed operator. From here and (1.8) it
follows that:

$\left\|Au\right\|\leq b_{0}\left\|B^{1/2}u\right\|,\text{\quad}\forall u\in
D(B^{1/2})\subset D\left(A\right).$ (1.9)

##  2 Uniform boundedness of solution of discrete problem and difference
analogue of the first-order derivative

The following theorem takes place (below everywhere $c$ denotes positive
constant).

######  Theorem 2.1.

For discrete problem (1.6) the vectors $(u_{k}-u_{k-1})/\tau$ and
$B^{1/2}u_{k}$ are uniformly bounded, i.e. there exist constants $c_{1}$ and
$c_{2}$ (independent of $n$) such that

$\genfrac{\|}{\|}{}{}{u_{k}-u_{k-1}}{\tau}\leq
c_{1}\;,\text{\quad}\left\|B^{1/2}u_{k}\right\|\leq
c_{2}\;,\text{\quad}k=1,\ldots,n\;\text{.}$

  * Proof.

If we multiply both sides of the equality (1.6) on vector
$u_{k+1}-u_{k-1}=\left(u_{k+1}-u_{k}\right)+\left(u_{k}-u_{k-1}\right)$, we
obtain

$\displaystyle\alpha_{k+1}-\alpha_{k}+2{a}_{1}{\tau}{\left\|{B}^{1/2}{\delta}{u}_{k}\right\|}^{2}+\frac{1}{2}{\beta}_{k+1}-\frac{1}{2}{\beta}_{k-1}+\frac{1}{2}{a}_{1,k}\left({\gamma}_{k+1}-{\gamma}_{k-1}\right)$
$\displaystyle+\frac{1}{2}{d}_{k}\left({\gamma}_{k+1}-{\gamma}_{k-1}\right)+\frac{1}{2}{a}_{3,k}\left({\vartheta}_{k+1}-{\vartheta}_{k-1}\right)=\left({g}_{k},{u}_{k+1}-{u}_{k-1}\right)\,,$
(2.1)

where

$\displaystyle\alpha_{k}=\genfrac{\|}{\|}{}{}{u_{k}-u_{k-1}}{\tau}^{2},\text{\quad}\beta_{k}=\left\|B^{1/2}u_{k}\right\|^{2}\text{,}\quad\gamma_{k}=\left\|A^{1/2}u_{k}\right\|^{2},\text{\quad}\vartheta_{k}=\left\|u_{k}\right\|^{2},$
$\displaystyle
g_{k}=f_{k}-Cu_{k}-N\delta{u}_{k}-M\left(u_{k}\right)\,,\quad\delta{u}_{k}=\frac{u_{k+1}-u_{k-1}}{2\tau}\,.$

According to (1.7) we get:

$\displaystyle a_{1,k}\left(\gamma_{k+1}-\gamma_{k-1}\right)$
$\displaystyle=\widetilde{\psi}_{1}\left(\gamma_{k-1},\gamma_{k+1}\right)\left(\gamma_{k+1}-\gamma_{k-1}\right)$
$\displaystyle=\int\limits_{0}^{\gamma_{k+1}}\psi_{1}(s)ds-\int\limits_{0}^{\gamma_{k-1}}\psi_{1}(s)ds\,,$
$a_{3,k}(\vartheta_{k+1}-\vartheta_{k-1})=\int\limits_{0}^{\vartheta_{k+1}}\psi_{3}(s)ds-\int\limits_{0}^{\vartheta_{k-1}}\psi_{3}(s)ds\,.$

Besides according to the monotonicity of function $\psi_{2}(s)$, the following
estimation is valid

$d_{k}\left(\gamma_{k+1}-\gamma_{k-1}\right)=\frac{1}{2\tau}\left(\psi_{2}(\gamma_{k+1})-\psi_{2}(\gamma_{k-1})\right)\left(\gamma_{k+1}-\gamma_{k-1}\right)\geq
0\,.$

Then from (Proof.) we get

$\displaystyle{\lambda}_{k+1}\leq{\lambda}_{k}+\left|(g_{k},u_{k+1}-u_{k-1})\right|,$
(2.2)

where
$\lambda_{k}=\alpha_{k}+\frac{1}{2}\left(\beta_{k}+\beta_{k-1}+\mu_{k}+\mu_{k-1}+\nu_{k}+\nu_{k-1}\right)$,

$\mu_{k}=\int\limits_{0}^{\gamma_{k}}\psi_{1}(s)ds,\text{\quad}\nu_{k}=\int\limits_{0}^{\vartheta_{k}}\psi_{3}(s)ds\text{.}$

If we use Schwarz inequality, condition (1.4) and Remark 1.1 we obtain

$\displaystyle\left|(g_{k},u_{k+1}-u_{k-1})\right|$
$\displaystyle\leq\left\|u_{k+1}-u_{k-1}\right\|\left(\left\|f_{k}\right\|+a_{0}b_{0}\sqrt{\beta_{k}}+\left\|M\left(u_{k}\right)\right\|+\left\|N\delta{u}_{k}\right\|\right)\text{.}$

From here follows

$\displaystyle\left|(g_{k},u_{k+1}-u_{k-1})\right|$
$\displaystyle\leq\tau\left(\sqrt{\alpha_{k+1}}+\sqrt{\alpha_{k}}\right)\left(\left\|f_{k}\right\|\,+\left\|M\left(u_{k}\right)\right\|\right.$
$\displaystyle\left.+c_{1}\sqrt{\beta_{k}}+c_{0}\left(\sqrt{\alpha_{k+1}}+\sqrt{\alpha_{k}}\right)\right),$
(2.3)

where $c_{0}=\frac{1}{2}\left\|N\right\|$, $c_{1}=a_{0}b_{0}$ .

For nonlinear operator $M\left(\cdot\right)$ due to Lipschitz condition we
have

$\displaystyle\left\|M\left(u_{k}\right)\right\|$ $\displaystyle\leq
c\left(\left\|u_{k}-u_{k-1}\right\|+\left\|u_{k-1}-u_{k-2}\right\|+\ldots+\left\|u_{1}-u_{0}\right\|\right)+\left\|M\left(u_{0}\right)\right\|$
$\displaystyle=c\tau\left(\sqrt{\alpha_{k}}+\sqrt{\alpha_{k-1}}+\ldots+\sqrt{\alpha_{1}}\right)+\left\|M\left(u_{0}\right)\right\|\,.$
(2.4)

If we insert inequality (2.4) into (2.3) we get

$\displaystyle\left|(g_{k},u_{k+1}-u_{k-1})\right|\leq\tau\left(\sqrt{\alpha_{k+1}}+\sqrt{\alpha_{k}}\right){\sigma}_{k}\,,$
(2.5)

where

$\sigma_{k}=c\tau\sum\limits_{i=1}^{k}\sqrt{\alpha_{i}}+c_{1}\sqrt{\beta_{k}}+c_{0}\left(\sqrt{\alpha_{k+1}}+\sqrt{\alpha_{k}}\right)+\left\|f_{k}\right\|\,+\left\|M\left(u_{0}\right)\right\|\,.$

From (2.2) according to (2.5) it follows

$\lambda_{k+1}\leq\lambda_{k}+\varepsilon_{k},$ (2.6)

where
$\varepsilon_{k}=\tau\left(\sqrt{\alpha_{k+1}}+\sqrt{\alpha_{k}}\right)\sigma_{k}$.

Obviously from (2.6) we obtain

$\displaystyle\lambda_{k+1}\leq\lambda_{1}+\tau\sum\limits_{i=1}^{k}\left(\sqrt{\alpha_{i}}+\sqrt{\alpha_{i+1}}\right)\sigma_{i}\text{,}$

from here we have

$\delta_{k+1}^{2}\leq\delta_{1}^{2}+\tau\sum\limits_{i=1}^{k}\left(\delta_{i}+\delta_{i+1}\right)\sigma_{i},\text{\quad\quad}\delta_{k}=\sqrt{\lambda_{k}}\,\text{.}$

from here follows the following inequality

$\delta_{k+1}\leq\delta_{1}+2\tau\sum\limits_{i=1}^{k}\sigma_{i}\leq
c\tau\sum\limits_{i=1}^{k}\delta_{i}+2c_{0}\tau\delta_{k+1}+\eta_{k},$ (2.7)

where

$\eta_{k}=\delta_{1}+c\left\|M\left(u_{0}\right)\right\|+2\tau\sum\limits_{i=1}^{k}\left\|f_{i}\right\|\text{.}$

If we assume that $1-2\tau c_{0}=1-\tau\left\|N\right\|>0$, then from (2.7) we
have

$\delta_{k+1}\leq c\tau\sum\limits_{i=1}^{k}\delta_{i}+c\eta_{k}\text{.}$

From here, according to discrete analogue of Grönwall’s lemma, we have

$\delta_{k+1}\leq c\exp(ct_{k-1})\left(\eta_{k}+\tau\delta_{1}\right)\text{.}$

From here it follows that $\alpha_{k}$ and $\beta_{k}$ are uniformly bounded.
∎

######  Remark 2.1.

From uniform boundedness of vectors $B^{1/2}u_{k}$ follows uniform boundedness
of $Au_{k}$ vectors (see inequality (1.9)). From this fact and the following
inequality

$\displaystyle\left\|A^{1/2}u\right\|=\left\|A^{-1/2}(Au)\right\|\leq\frac{1}{\sqrt{m_{A}}}\left\|Au\right\|,\text{\quad}\forall
u\in D(A)\,,$ (2.8)

where $m_{A}>0$ is lower bound of operator $A$ ($(Au,u)\geq m_{A}(u,u)$),
follows that $A^{1/2}{u}_{k}$ is uniformly bounded.

######  Remark 2.2.

According to the triangle inequality, from uniform boundedness of
$(u_{k+1}-u_{k})/\tau$ vectors follows uniform boundenss of
$(u_{k+1}-u_{k-1})/2\tau$ vectors.

######  Remark 2.3.

From uniform boundenss of $\left\|Au_{k}\right\|$and
$\left\|(u_{k+1}-u_{k})/\tau\right\|$ follows uniform boundess of

$\left|(\gamma_{k}-\gamma_{k-1})/\tau\right|\,.$

##  3 Estimations for two-variable Chebyshev polynomials

To obtain a priori estimations for the main linear part of the difference
equation (1.6) we require estimations for a specific class of polynomials that
we call two-variable Chebyshev polynomials. These polynomials are defined
using the following recurrence relation (see [31]):

$\displaystyle
U_{k+1}(x,y)=xU_{k}(x,y)-yU_{k-1}(x,y),\text{\quad}k=1,2,\ldots\text{\quad},$
(3.1) $\displaystyle U_{1}(x,y)=x,\text{\quad}U_{0}(x,y)\equiv 1.$

$U_{k}(x,y)$ we call two-variable Chebyshev polynomials, as
$U_{k}\left(2x,1\right)$ represents Chebyshev polynomials of the second kind
(see, e.g. [35]).

From recurrence relation (3.1) using induction, we get the following formula

$U_{k}(x,y)=\sqrt{y^{k}}U_{k}(\xi,1),\text{\quad}\xi=\frac{x}{\sqrt{y}},\text{\quad}y>0.$
(3.2)

The formula (3.2) is important as it relates $U_{k}(x,y)$ polynomials with
classic Chebyshev polynomials (we assume that in classic Chebyshev polynomials
$x$ variable is replaced by $x/2$). Let us introduce the following domains:

$\displaystyle\Delta=\left\\{(x,y):\;\left|y\right|<1\,\,\text{and}\,\,\left|x\right|<y+1\right\\}\text{.}$
$\displaystyle\Omega^{+}=\left\\{(x,y):\;4y-x^{2}>0\right\\},\text{\quad}\Omega^{-}=\left\\{(x,y):\;4y-x^{2}<0\right\\}\text{,}$
$\displaystyle\Delta^{+}=\left\\{(x,y)\in\Delta:\;x\geq
0\right\\},\text{\quad}\Omega_{1}=\Omega^{+}\cap\Delta^{+},\text{\quad}\Omega_{2}=\Omega^{-}\cap\Delta^{+}\text{.}$

It is well-known that roots of the classic Chebyshev polynomials are in
$\;]-1,1[$ (see, e.g., [35]). From here, according to formula (3.2), it
follows that, for any fixed positive $y$ roots of the polynomial $U_{k}(x,y)$
are inside $]-2\sqrt{y},2\sqrt{y}[$. Besides if we take into consideration,
that $U_{k}(\pm 2,1)=(-1)^{k}(k+1)$ and $\left|U_{k}(2\xi,1)\right|$ reaches
its maximum on boundary (see, e.g., [35]), then from formula (3.2) follows the
following estimation

$\left|U_{k}(x,y)\right|\leq
U_{k}(2\sqrt{y},y)=(k+1)\sqrt{y^{k}},\text{\quad}(x,y)\in\Omega^{+}.$ (3.3)

From above discussion, we conclude that for any positive $y$, $U_{k}(x,y)$ is
increasing function regarding $x$ variable, when $x\geq 2\sqrt{y}$. Besides
from recurrence relation (3.1) it follows that, for any fixed $y\leq 0$,
$U_{k}(x,y)$ is increasing function regarding to $x$ variable, when $x\geq 0$.
From here we obtain

$\left|U_{k}(x,y)\right|\leq U_{k}(1+y,y)=1+y+\ldots+y^{k},$ (3.4)

where $y\geq-1$and $\left|x\right|\leq 1+y$.

From (3.4) follows the following estimation

$\left|U_{k}(x,y)(1-y)\right|\leq 1,\text{\quad}(x,y)\in\Delta.$ (3.5)

We also need estimation for $U_{k}(x,y)-y^{m}U_{k-1}(x,y)$, $m=0,1$,
polynomials, where $(x,y)\in\Delta^{+}$.

The following inequality is simply obtained

$\left|U_{k}(x,1)-U_{k-1}(x,1)\right|\leq\frac{2}{\sqrt{2+x}},\text{\quad}x\in]-2,2].$
(3.6)

According to formula (3.2) and inequality (3.6) the following estimation is
valid

$\displaystyle\left|U_{k}(x,y)-\sqrt{y}U_{k-1}(x,y)\right|=\sqrt{y^{k}}\left|U_{k}(\xi,1)-U_{k-1}(\xi,1)\right|\leq\sqrt{2y^{k}}\;,$
(3.7)

where $\xi=x/\sqrt{y}$ , $(x,y)\in\Omega_{1}$.

Let us estimate the following difference $U_{k}(x,y)-yU_{k-1}(x,y)$, when
$(x,y)\in\Omega_{1}$. According to inequalities (3.7) and (3.3) we have

$\displaystyle\left|U_{k}(x,y)-yU_{k-1}(x,y)\right|\leq\sqrt{2},\text{\quad}(x,y)\in\Omega_{1}.$
(3.8)

Let us estimate the following difference $U_{k}(x,y)-yU_{k-1}(x,y)$, when
$(x,y)\in\Omega_{2}$ and $y>0$. For that we require the following formulas:

$U_{k}(x,y)=\sqrt{y^{k}}\sum_{i=0}^{k}C_{k+i+1}^{2i+1}(\xi-2)^{i},$ (3.9)
$U_{k}(x,y)-\sqrt{y}U_{k-1}(x,y)=\sqrt{y^{k}}\sum_{i=0}^{k}C_{k+i}^{2i}(\xi-2)^{i},$
(3.10)

where $\xi=x/\sqrt{y}$ , $\;C_{k}^{i}$ are binomial coefficients
($C_{k}^{0}=1$).

Using simple transformation from (3.9) we obtain formula (3.10). The formula
(3.9) can be obtained by using Taylor expansion of $U_{k}(\xi,1)$ at $\xi=2$.
We need to consider that $U_{k}^{(i)}(2,1)=i!C_{k+i+1}^{2i+1}$.

From the following equality

$\displaystyle
U_{k}(x,y)-yU_{k-1}(x,y)=\left(U_{k}(x,y)-\sqrt{y}U_{k-1}(x,y)\right)+\left(1-\sqrt{y}\right)\sqrt{y}U_{k-1}(x,y)\text{,}$

according to formulas (3.9) and (3.10), it follows that, for any fixed $y$
from $]0,1]$ interval $U_{k}(x,y)-yU_{k-1}(x,y)$ is increasing function
regarding $x$, when $x\geq 2\sqrt{y}$. From here follows

$\displaystyle
U_{k}\left(2\sqrt{y},y\right)-yU_{k-1}\left(2\sqrt{y},y\right)\leq
U_{k}-yU_{k-1}\leq U_{k}(1+y,y)-yU_{k-1}(1+y,y)\,,$ (3.11)

where $y>0$ and $(x,y)\in\Omega_{2}$.

If we insert $U_{k}(2\sqrt{y},y)=(k+1)\sqrt{y^{k}}$ and (3.4) in the relation
(3.11) then we obtain the following estimation

$\sqrt{y^{k}}\left((k+1)\left(1-\sqrt{y}\right)+\sqrt{y}\right)\leq
U_{k}(x,y)-yU_{k-1}(x,y)\leq 1,$ (3.12)

where $y>0$ and $(x,y)\in\Omega_{2}$.

We straightforwardly obtain the following inequality

$0\leq U_{k}(x,y)-yU_{k-1}(x,y)\leq 1,$ (3.13)

where $y\leq 0$ and $(x,y)\in\Omega_{2}$.

From estimations (3.8), (3.12) and (3.13) we have

$\left|U_{k}(x,y)-yU_{k-1}(x,y)\right|\leq\sqrt{2},\text{\quad}(x,y)\in\Delta^{+}.$
(3.14)

Analogously to (3.14) we obtain

$\left|U_{k}(x,y)-U_{k-1}(x,y)\right|\leq\sqrt{2},\text{\quad}(x,y)\in\Delta^{+}.$
(3.15)

##  4 High Order a priori estimations for three-layer semi-discrete scheme
corresponding to the second-order evolution equation

The three-layer scheme is natural for the second-order evolution equation and
the two-layer scheme is natural for the first-order evolution equation.
Investigation of a three-layer scheme is more difficult than an investigation
of a two-layer scheme. This difficulty can be somehow simplified if we reduce
the second-order evolution equation to the first-order evolution equation by
introducing additional unknowns. In this case, a self-adjoint operator is
replaced by an operator matrix that is not self-adjoint anymore. This makes it
complicated to investigate the corresponding discrete problem.

Obtaining such a priori estimations from where follows stability and
convergence of the nonlinear semi-discrete scheme (1.6), is based on high
order accuracy a priori estimations for corresponding linearized semi-discrete
scheme. In this section, we obtain a priori estimation for a three-layer semi-
discrete scheme for the second-order evolution equation. In this estimation on
the left-hand side, we have positive $s$ power for the main operator, while on
the right-hand side we have $(s-1)$. This allows us to make weaken nonlinear
part of the equation in such a way that, using the results obtained in the
previous section we are able to use Grönwall’s lemma and obtain the final
estimation. To obtain these estimations we require to construct the exact
representation of three-layer recurrence relations with operator coefficients
by using two-variable Chebyshev polynomials. These kinds of estimates were
obtained before by one of the authors of the presented papers (see [31],
[30]).

Important results for constructing and investigating approximate schemes for
the Cauchy problem for second-order evolution equations were obtained by the
following authors: G. A. Baker [2], G. A. Baker, J. H. Bramble [3], G. A.
Baker, V. A. Dougalis, S. M. Serbin [4], L. A. Bales [5], J. Kačur [18], O.
Ladyzhenskaya [20], M. Pultar [28], P. E. Sobolevskij, L. M. Chebotarova [34].

Let us consider in Hilbert space $H$ the following linear difference equation

$\frac{u_{k+1}-2u_{k}+u_{k-1}}{\tau^{2}}+a_{1}B\frac{u_{k+1}-u_{k-1}}{2\tau}+a_{2}B\frac{u_{k+1}+u_{k-1}}{2}=f_{k},$
(4.1)

where $k=1,\ldots,n-1$, $u_{0}$, $u_{1}$ and $f_{k}$ are the given vectors
from $H$.

Difference equation (4.1) represents main part of the nonlinear equation
(1.6), and obviously corresponds main part of the equation (1.1).

The following lemma takes place.

######  Lemma 4.1.

Let $B$ be self-adjoint positively defined operator and $1-\tau a>0$,
$a=a_{2}/a_{1}$. Then for scheme (4.1) the following a priori estimation is
valid:

$\displaystyle\left\|B^{s}u_{k+1}\right\|\leq\sqrt{2}\left\|B^{s}u_{0}\right\|+\frac{1}{a_{1}}\left\|B^{s-1}\frac{\Delta
u_{0}}{\tau}\right\|+\frac{\tau}{2}(1+\tau a)\left\|B^{s}\frac{\Delta
u_{0}}{\tau}\right\|$
$\displaystyle\text{\quad}+\frac{\tau}{a_{1}}\sum\limits_{i=1}^{k}\left\|B^{s-1}f_{i}\right\|,\text{\quad}u_{0},u_{1}\in
D(B^{s}),\leavevmode\nobreak\ f_{i}\in D(B^{s-1}),\leavevmode\nobreak\ $ (4.2)
$\displaystyle\genfrac{\|}{\|}{}{}{\Delta u_{k}}{\tau}\leq
a\left\|u_{0}\right\|+\sqrt{2}\genfrac{\|}{\|}{}{}{\Delta
u_{0}}{\tau}+\sqrt{2}\tau\sum\limits_{i=1}^{k}\left\|f_{i}\right\|,$ (4.3)

where $s\geq 0\leavevmode\nobreak\ $, $k=1,\ldots,n-1\text{,}$ $\Delta
u_{k}=u_{k+1}-u_{k}$ $(B^{0}=I)$.

######  Remark 4.1.

In section 2 (see Theorem 2.1 and Remark 2.1, Remark 2.2 and Remark 2.3) there
was shown that $\left\|(u_{k+1}-u_{k})/\tau\right\|$,
$\left\|A^{1/2}u_{k}\right\|$, $\left\|Au_{k}\right\|$,
$\left\|B^{1/2}u_{k}\right\|$ and
$\left|(\gamma_{k}-\gamma_{k-1})/\tau\right|$ are uniformly bounded. These
results along with (4.2) and (4.3) allows to obtain such a priori estimations
for semi-discrete scheme (1.6) from where follows stability and convergence of
the presented method.

######  Remark 4.2.

A priori estimations (4.2) and (4.3) have independent meaning, as constants on
the right-hand side are absolute constants (does not depend on interval
length). Besides these constants cannot be improved. To obtain these
estimations were possible by constructing exact representations for the
solution of difference equation (4.1).

  * Proof of Lemma 4.1.

Let us rewrite difference equation (4.1) in the following form to prove
inequality (4.2)

${B}_{0}u_{k+1}-2Iu_{k}+{B}_{1}u_{k-1}=\tau^{2}f_{k}\,,$ (4.4)

where

${B}_{0}=I+\frac{\tau}{2}a_{1}B+\frac{\tau^{2}}{2}a_{2}B\,,\quad{B}_{1}={B}_{0}-\tau{a}_{1}{B}\,.$

From (4.4) we get

$u_{k+1}=Lu_{k}-Su_{k-1}+\frac{\tau^{2}}{2}Lf_{k},$ (4.5)

where $L=2{B}_{0}^{-1}$, $S={B}_{1}{B}_{0}^{-1}$.

Let us note that $L$ and $S$ are self-adjoint, bounded linear operators in a
Hilbert space $H$. Besides it is obvious that $L$ and $S$ are commutative.

Using mathematical induction for (4.5) recurrence relation we get

$u_{k+1}=U_{k}(L,S)u_{1}-SU_{k-1}(L,S)u_{0}+\frac{\tau^{2}}{2}\sum\limits_{i=1}^{k}U_{k-i}(L,S)Lf_{i},$
(4.6)

where operator polynomial $U_{k}(L,S)$ satisfies the following recurrence
relation:

$\displaystyle
U_{k}(L,S)=LU_{k-1}(L,S)-SU_{k-2}(L,S),\text{\quad}k=1,2,\ldots\text{\quad},$
(4.7) $\displaystyle U_{0}(L,S)=I,\text{\quad}U_{-1}(L,S)=0.$

Scalar polynomials $U_{k}(x,y)$ corresponding to $U_{k}(L,S)$ satisfy (3.1)
recurrence relation.

From (4.6) using simple transformation we have

$\displaystyle u_{k+1}=\tau U_{k}(L,S)\frac{\Delta
u_{0}}{\tau}+\left(U_{k}(L,S)-SU_{k-1}(L,S)\right)u_{0}+\frac{\tau^{2}}{2}\sum\limits_{i=1}^{k}U_{k-i}(L,S)Lf_{i}.$
(4.8)

If we apply operator $B^{s}$ ($s\geq 0$) on both sides of equality (4.8) and
move on to the norm we obtain

$\displaystyle\left\|B^{s}u_{k+1}\right\|\leq\tau\left\|B^{s}U_{k}(L,S)\frac{\Delta
u_{0}}{\tau}\right\|+\left\|U_{k}(L,S)-SU_{k-1}(L,S)\right\|\left\|B^{s}u_{0}\right\|$
$\displaystyle+\frac{\tau^{2}}{2}\sum\limits_{i=1}^{k}\left\|B^{s}U_{k-i}(L,S)Lf_{i}\right\|.$
(4.9)

Using simple transformations we get:

$S=\frac{1}{1+\tau a}L-\frac{1-\tau a}{1+\tau a}I\,,$ (4.10) $I-S=\frac{\tau
a_{1}}{2}BL\,.$ (4.11)

From here we have

$\displaystyle\tau\left\|B^{s}U_{k}(L,S)Lf\right\|=\tau\left\|U_{k}(L,S)BL(B^{s-1}f)\right\|$
$\displaystyle=\frac{2}{a_{1}}\left\|U_{k}(L,S)(I-S)(B^{s-1}f)\right\|\leq\frac{2}{a_{1}}\left\|U_{k}(L,S)(I-S)\right\|\left\|B^{s-1}f\right\|,$
(4.12)

where $f\in D(B^{s-1})$.

As it is known the norm of an operator function, when the argument of the
function represents self-adjoint bounded operator, is equal to $C$-norm of the
corresponded scalar function (see, e.g., [29], Chapter VII). Using this and
representation (4.10) we have

$\displaystyle\left\|U_{k}(L,S)(I-S)\right\|\leq\max_{y\in\sigma(S)}\left|U_{k}(\eta(y),\text{\/}y)(1-y)\right|,$
(4.13)

where $\eta(y)=(1+\tau a)y+(1-\tau a)$.

Let us estimate the spectrum of operator $L$. As, according to the condition,
$B$ is a self-adjoint and positively defined operator, therefore, we have
$\sigma(L)\subset\left[0,2\right]$. According to this relation, from the
representation (4.10) we have

$\sigma(S)\subset[-1,1].$ (4.14)

If we consider relation (4.14) and estimation (3.5) we get

$\displaystyle\max_{y\in\sigma(S)}\left|U_{k}(\eta(y),\text{\/}y)(1-y)\right|\leq\max_{y\in[-1,1]}\left|U_{k}(\eta(y),\text{\/}y)(1-y)\right|$
$\displaystyle\leq\max_{(x,y)\in\Delta^{+}}\left|U_{k}(x,\text{\/}y)(1-y)\right|\leq
1.$

From here and (4.13) it follows

$\left\|U_{k}(L,S)(I-S)\right\|\leq 1.$ (4.15)

It is obvious, from (4.12) using (4.15) we have

$\tau\left\|B^{s}U_{k}(L,S)Lf\right\|\leq\frac{2}{a_{1}}\left\|B^{s-1}f\right\|,\text{\quad}f\in
D(B^{s-1}).$ (4.16)

Also from (4.16) follows the inequality

$\displaystyle\tau\left\|B^{s}U_{k}(L,S)f\right\|\leq\frac{1}{a_{1}}\left\|B^{s-1}{B}_{0}f\right\|\leq\frac{1}{a_{1}}\left\|B^{s-1}f\right\|+\frac{\tau}{2}(1+\tau
a)\left\|B^{s}f\right\|,$ (4.17)

where $f\in D(B^{s})$.

Let us estimate operator $U_{k}-SU_{k-1}$. Analogously, we have

$\left\|U_{k}(L,S)-SU_{k-1}(L,S)\right\|\leq\max_{y\in\sigma(S)}\left|U_{k}(\eta(y),y)-yU_{k-1}(\eta(y),y)\right|\text{.}$

If we consider relation (4.14) and estimation (3.14) we get

$\displaystyle\max_{y\in\sigma(S)}\left|U_{k}(\eta(y),\text{\/}y)-yU_{k-1}(\eta(y),\text{\/}y)\right|\leq\max_{(x,y)\in\Delta^{+}}\left|U_{k}(x,\text{\/}y)1-yU_{k-1}(x,\text{\/}y)\right|\leq\sqrt{2}\text{.}$

So, we have

$\left\|U_{k}(L,S)-SU_{k-1}(L,S)\right\|\leq\sqrt{2}.$ (4.18)

If we insert estimations (4.16), (4.17) and (4.18) into inequality (4.9) we
obtain estimation (4.2).

Let us prove estimation (4.3). From the formula (4.8) according to the
recurrence relation (4.7) we get

$\displaystyle\genfrac{\|}{\|}{}{}{\Delta
u_{k}}{\tau}\leq\tau^{-1}\left\|(L-S-I)U_{k-1}(L,S)\right\|\left\|u_{0}\right\|$
$\displaystyle+\left\|U_{k}-U_{k-1}\right\|\genfrac{\|}{\|}{}{}{\Delta
u_{0}}{\tau}+\frac{\tau}{2}\sum\limits_{i=1}^{k}\left\|U_{k-i}-U_{k-i-1}\right\|\left\|L\right\|\left\|f_{i}\right\|.$
(4.19)

According to equality (4.11) we have

$\displaystyle L-S-I=I-S-(2I-L)=I-S-(1+\tau a)(I-S)=-\tau a(I-S)\text{.}$

If we consider this representation and estimation (4.15), we get

$\tau^{-1}\left\|(L-S-I)U_{k-1}(L,S)\right\|=a\left\|U_{k}(L,S)(I-S)\right\|\leq
a.$ (4.20)

Analogously to (4.18), according to estimation (3.15) we obtain

$\left\|U_{k}(L,S)-U_{k-1}(L,S)\right\|\leq\sqrt{2}.$ (4.21)

From inequality (4.19) using estimations (4.20), (4.21) and
$\left\|L\right\|\leq 2$ a priori estimation (4.3) follows. ∎

######  Remark 4.3.

Analogously to (4.3) the following estimation can be obtained

$\left\|B^{s}\frac{\Delta u_{k}}{\tau}\right\|\leq
a\left\|B^{s}u_{0}\right\|+\sqrt{2}\left\|B^{s}\frac{\Delta
u_{0}}{\tau}\right\|+\sqrt{2}\tau\sum\limits_{i=1}^{k}\left\|B^{s}f_{i}\right\|,$
(4.22)

where $u_{0},\leavevmode\nobreak\ u_{1},f_{i}\in D(B^{s}),\;s\geq 0$.

##  5 The a priori estimates for perturbation of the solution of the discrete
problem

The goal of the section is to show the stability of the scheme (1.6). As
additivity does not take place for nonlinear cases, therefore it is natural
that we try to obtain the a priori estimates exactly for the solution
perturbation. From here (analogously to a linear problem) automatically
follows the stability and convergence of the nonlinear scheme.

In this section, based on the results of the previous sections, we obtain the
a priori estimates for the solution of the semi-discrete scheme (1.6) and
perturbation of the corresponding first-order difference.

The following theorem takes place (below everywhere $c$ denotes positive
constant).

######  Theorem 5.1.

Let $u_{k}$ and $\overline{u}_{k}$ be solutions of difference equation (1.6)
corresponding to initial vectors $(u_{0},\;u_{1},\;f_{k})$ and
$(\overline{u}_{0},\;\overline{u}_{1},\;\overline{f}_{k})$, components of
which are sufficiently smooth. Then for $z_{k}=u_{k}-\overline{u}_{k}$ the
following estimates are true

$\displaystyle\left\|B^{1/2}z_{k+1}\right\|+\genfrac{\|}{\|}{}{}{\Delta
z_{k}}{\tau}$ $\displaystyle\leq
c\left(\left\|B^{1/2}z_{0}\right\|+\genfrac{\|}{\|}{}{}{\Delta
z_{0}}{\tau}+\tau\left\|B^{1/2}\frac{\Delta
z_{0}}{\tau}\right\|+\tau\sum\limits_{i=1}^{k}\left\|f_{i}-\overline{f}_{i}\right\|\right)\;,$
(5.1)

where $k=1,\ldots,n-1$ , $\Delta z_{k}=z_{k+1}-z_{k}$.

In this section and the following ones the notation
$\delta{z}_{k}=\dfrac{z_{k+1}-z_{k-1}}{2\tau}$ is applied.

Let us prove the corresponding auxiliary lemma.

######  Lemma 5.1.

The following inequality is true

$\left|d_{k}-\overline{d}_{k}\right|\leq
c\left(\left\|Az_{k+1}\right\|+\left\|Az_{k-1}\right\|+\|\delta{z}_{k}\right\|),$
(5.2)

where $z_{k}=u_{k}-\overline{u}_{k}$ ,

$\displaystyle
d_{k}=\delta{\psi_{2}}\left(\gamma_{k}\right)\,,\quad\overline{d}_{k}=\delta{\psi_{2}}\left(\overline{\gamma}_{k}\right)\,,\quad\gamma_{k}=\left\|A^{1/2}u_{k}\right\|^{2}\,,\quad\overline{\gamma}_{k}=\left\|A^{1/2}\overline{u}_{k}\right\|^{2}\,.$

  * Proof.

The following representation is valid

$\displaystyle
d_{k}-\overline{d}_{k}=\left(\delta\gamma_{k}-\delta\overline{\gamma}_{k}\right)I_{1,k}+\delta\overline{\gamma}_{k}I_{2,k}\,,$
(5.3)

where

$\displaystyle
I_{1,k}=\int\limits_{0}^{1}{\psi_{2}^{\prime}\left(l_{k}\left(\xi\right)\right)}\mathrm{d}\xi\,,\quad
I_{2,k}=\int\limits_{0}^{1}{\left[\psi_{2}^{\prime}\left(l_{k}\left(\xi\right)\right)-\psi_{2}^{\prime}\left(\overline{l}_{k}\left(\xi\right)\right)\right]}\mathrm{d}\xi\,,$

and where

$l_{k}\left(\xi\right)=\gamma_{k-1}+\left(\gamma_{k+1}-\gamma_{k-1}\right)\xi\text{,}\quad\overline{l}_{k}\left(\xi\right)=\overline{\gamma}_{k-1}+\left(\overline{\gamma}_{k+1}-\overline{\gamma}_{k-1}\right)\xi\text{.}$

Using standard transformations we get

$\displaystyle\delta\gamma_{k}-\delta\overline{\gamma}_{k}=\left(Az_{k+1},\delta{u}_{k}+\delta\overline{u}_{k}\right)+\left(\delta{z}_{k},A(u_{k-1}+\overline{u}_{k-1})\right)\text{.}$

From here using the Schwarz inequality follows

$\displaystyle|\delta\gamma_{k}-\delta\overline{\gamma}_{k}|\leq\left\|Az_{k+1}\right\|\left(\|\delta{u}_{k}\|+\|\delta\overline{u}_{k}\|\right)+\|\delta{z}_{k}\|\left(\left\|Au_{k-1}\right\|+\left\|A\overline{u}_{k-1}\right\|\right).$
(5.4)

If we consider that all terms on the right-hand side of the inequality (5.4)
are bounded (see Remark 2.1 and Remark 2.2), then we obtain

$\displaystyle|\delta\gamma_{k}-\delta\overline{\gamma}_{k}|\leq
c\left(\left\|Az_{k+1}\right\|+\|\delta{z}_{k}\|\right).$ (5.5)

Let us estimate integrals $I_{1,k}$ and $I_{2,k}$. If we use the change of
variables for the integral $I_{2,k}$ we obtain

$I_{2,k}=\int\limits_{0}^{1}\int\limits_{\overline{l}_{k}(\xi)}^{l_{k}(\xi)}\psi_{2}^{\prime\prime}\left(\eta\right)d\eta
d\xi\,.$ (5.6)

From (5.6) we have

$\left|I_{2,k}\right|\leq c\int\limits_{0}^{1}\left|\chi(\xi)\right|d\xi,$
(5.7)

where

$\displaystyle\chi(\xi)=\left(\gamma_{k-1}-\overline{\gamma}_{k-1}\right)+\left(\left(\gamma_{k+1}-\overline{\gamma}_{k+1}\right)-\left(\gamma_{k-1}-\overline{\gamma}_{k-1}\right)\right)\xi\,,$
$\displaystyle
c=\max\left|\psi_{2}^{\prime\prime}\left(\eta\right)\right|<+\infty\,,\quad
0\leq\eta\leq\max_{k}(\gamma_{k},\overline{\gamma}_{k})<+\infty\,.$

Regarding inequality (5.7) it should be noted that since $\gamma_{k}$ and
$\overline{\gamma}_{k}$ are uniformly bounded (see Remark 2.1) therefore
$\max_{k}(\gamma_{k},\overline{\gamma}_{k})<+\infty$, i.e. the interval where
we have to find a maximum value of
$\left|\psi_{2}^{\prime\prime}\left(\eta\right)\right|$ is finite. From here
follows that
$\max\left|\psi_{2}^{\prime\prime}\left(\eta\right)\right|<+\infty$ (according
to the condition $\psi_{2}^{\prime\prime}\left(\eta\right)$ is continuous).
So, from (5.7) we have

$\displaystyle\left|I_{2,k}\right|\leq
c\int\limits_{0}^{1}\left|\chi(\xi)\right|d\xi\leq
c\left(\left|\gamma_{k-1}-\overline{\gamma}_{k-1}\right|+\left|\gamma_{k+1}-\overline{\gamma}_{k+1}\right|\right)\,.$
(5.8)

As vectors $A^{1/2}u_{k}$ are uniformly bounded, for the difference
$\gamma_{k}-\overline{\gamma}_{k}$ the following estimation is valid

$\displaystyle\left|\gamma_{k}-\overline{\gamma}_{k}\right|=\left(\sqrt{\gamma_{k}}+\sqrt{\overline{\gamma}_{k}}\right)\left|\sqrt{\gamma_{k}}-\sqrt{\overline{\gamma}_{k}}\right|\leq
c\left\|A^{1/2}z_{k}\right\|\leq c\left\|Az_{k}\right\|.$ (5.9)

From (5.8) according to (5.9) follows

$\left|I_{2,k}\right|\leq
c\left(\left\|Az_{k+1}\right\|+\left\|Az_{k-1}\right\|\right).$ (5.10)

For the integrals $I_{1,k}$ the following estimation is valid

$\left|I_{1,k}\right|\leq
c,\text{\quad}c=\max\left|\psi_{2}^{\prime}\left(s\right)\right|<+\infty,\text{\quad}0\leq
s\leq\max_{k}(\gamma_{k},\overline{\gamma}_{k})<+\infty\;.$ (5.11)

From (5.3) according to inequalities (5.5), (5.10), (5.11) and Remark 2.3
follows (5.2). ∎

Let us return to proof of Theorem 5.1.

  * Proof.

According to (1.6) difference $z_{k}=u_{k}-\overline{u}_{k}$ satisfies the
following equation

$\frac{z_{k+1}-2z_{k}+z_{k-1}}{\tau^{2}}+a_{1}B\frac{z_{k+1}+z_{k-1}}{2}+a_{2}B\frac{z_{k+1}+z_{k-1}}{2}=-\frac{1}{2}g_{k},\text{\quad}$
(5.12)

where $k=1,\ldots,n-1\text{,}$

$\displaystyle g_{k}=g_{1,k}+g_{2,k}+g_{3,k}+g_{4,k}+g_{5,k}\text{,}$
$\displaystyle
g_{1,k}=a_{1,k}A\left(u_{k+1}+u_{k-1}\right)-\overline{a}_{1,k}A\left(\overline{u}_{k+1}+\overline{u}_{k-1}\right)\text{,}$
$\displaystyle
g_{2,k}=d_{k}A\left(u_{k+1}+u_{k-1}\right)-\overline{d}_{k}A\left(\overline{u}_{k+1}+\overline{u}_{k-1}\right)\text{,}$
$\displaystyle
g_{3,k}=a_{3,k}A\left(u_{k+1}+u_{k-1}\right)-\overline{a}_{3,k}A\left(\overline{u}_{k+1}+\overline{u}_{k-1}\right)\text{,}$
$\displaystyle g_{4,k}=2\left(Cz_{k}+N\delta{z}_{k}\right)\,,\quad
g_{5,k}=2\left(\left(M\left(u_{k}\right)-M\left(\overline{u}_{k}\right)\right)-(f_{k}-\overline{f}_{k})\right)\text{,}$

and where
$\overline{a}_{1,k}=\widetilde{\psi}_{1}\left(\overline{\gamma}_{k+1},\overline{\gamma}_{k-1}\right)\text{,}$
$\overline{\gamma}_{k}=\left\|A^{1/2}\overline{u}_{k}\right\|^{2}$
(analogously are defined $\overline{a}_{3,k}$ and $\overline{d}_{k}$).

For scheme (5.12), according to Lemma 4.1 the following a priori estimations
are valid (see (4.2) and (4.3)):

$\displaystyle\left\|B^{1/2}z_{k+1}\right\|$ $\displaystyle\leq
c\left(\left\|B^{1/2}z_{0}\right\|+\left\|\frac{\Delta
z_{0}}{\tau}\right\|+\tau\left\|B^{1/2}\frac{\Delta
z_{0}}{\tau}\right\|+\tau\sum\limits_{i=1}^{k}\left\|g_{i}\right\|\right)\,,$
(5.13) $\displaystyle\genfrac{\|}{\|}{}{}{\Delta z_{k}}{\tau}$
$\displaystyle\leq c\left(\left\|z_{0}\right\|+\genfrac{\|}{\|}{}{}{\Delta
z_{0}}{\tau}+\tau\sum\limits_{i=1}^{k}\left\|g_{i}\right\|\right)\,.$ (5.14)

Let us estimate each $g_{.,k}$ separately. For $g_{1,k}$ we have

$g_{1,k}=\left(a_{1,k}-\overline{a}_{1,k}\right)\left(Au_{k+1}+Au_{k-1}\right)+\overline{a}_{1,k}\left(Az_{k+1}+Az_{k-1}\right).$
(5.15)

Using simple transformations for difference $a_{1,k}-\overline{a}_{1,k}$ we
get

$\displaystyle
a_{1,k}-\overline{a}_{1,k}=\int\limits_{0}^{1}\int\limits_{\overline{l}_{k}(\xi)}^{l_{k}(\xi)}\psi_{1}^{\prime}\left(\eta\right)d\eta
d\xi,$ (5.16)

From (5.16) according to Remark 2.1 we have

$\displaystyle\left|a_{1,k}-\overline{a}_{1,k}\right|\leq
c\int\limits_{0}^{1}\left|\chi(\xi)\right|d\xi\leq
c\left(\left|\gamma_{k-1}-\overline{\gamma}_{k-1}\right|+\left|\gamma_{k+1}-\overline{\gamma}_{k+1}\right|\right)\,,$
(5.17)

where
$c=\max\left|\psi_{1}^{\prime}\left(\eta\right)\right|<+\infty,\text{\quad}0\leq\eta\leq\max_{k}(\gamma_{k},\overline{\gamma}_{k})<+\infty\,.$

From (5.15) using (5.17), we get

$\displaystyle\left\|g_{1,k}\right\|$ $\displaystyle\leq
c\left(\left|\gamma_{k-1}-\overline{\gamma}_{k-1}\right|+\left|\gamma_{k+1}-\overline{\gamma}_{k+1}\right|\right)\left(\left\|Au_{k+1}\right\|+\left\|Au_{k-1}\right\|\right)$
$\displaystyle+\left|\overline{a}_{1,k}\right|\left(\left\|Az_{k+1}\right\|+\left\|Az_{k-1}\right\|\right).$
(5.18)

According to Remark 2.1, $\overline{a}_{1,k}$ is uniformly bounded. Indeed we
have

$\displaystyle\left|\overline{a}_{1,k}\right|=\left|\int\limits_{0}^{1}\psi_{1}\left(\overline{l}_{k}\left(\xi\right)\right)d\xi\right|\leq
c,$ (5.19)

where $c=\max\left|\psi_{1}\left(s\right)\right|<+\infty,\text{\quad}0\leq
s\leq\max_{k}(\gamma_{k},\overline{\gamma}_{k})\text{.}$

If we insert estimations (5.9) and (5.19) in (5.18) and take into account,
that $\left\|Au_{k}\right\|$ is uniformly bounded, we obtain

$\left\|g_{1,k}\right\|\leq
c\left(\left\|Az_{k+1}\right\|+\left\|Az_{k-1}\right\|\right).$ (5.20)

Let us estimate vector $g_{2,k}$. We have

$g_{2,k}=\left(d_{k}-\overline{d}_{k}\right)\left(Au_{k+1}+Au_{k-1}\right)+\overline{d}_{k}\left(Az_{k+1}+Az_{k-1}\right).$
(5.21)

According to Remark 2.1, for $\overline{d}_{k}$ the following estimation is
valid

$\displaystyle\left|\overline{d}_{k}\right|=\left|\delta\overline{\gamma}_{k}\right|\left|\int\limits_{0}^{1}\psi_{2}^{\prime}\left(\overline{l}_{k}\left(\xi\right)\right)d\xi\right|\leq
c\left|\delta\overline{\gamma}_{k}\right|\,,$ (5.22)

where
$c=\max\left|\psi_{2}^{\prime}\left(s\right)\right|<+\infty,\text{\quad}0\leq
s\leq\max_{k}(\gamma_{k},\overline{\gamma}_{k})\text{.}$

According to Remark 2.2, from (5.22) follows that $\overline{d}_{k}$ is
uniformly bounded. Vector $\left\|Au_{k}\right\|$ is also uniformly bounded
(see Remark 2.1). Using this and inequality (5.2) from (5.21) follows

$\left|g_{2,k}\right|\leq
c\left(\left\|Az_{k+1}\right\|+\left\|Az_{k-1}\right\|+\left\|\delta{z}_{k}\right\|\right).$
(5.23)

Let us estimate the vector $g_{3,k}$. If in representation of $g_{1,k}$
operator $A$ is replaced by identity operator, then we get $g_{3,k}$ (we
assume that $\psi_{1}$ respectivly is replaced by $\psi_{3}$). According to
this for $g_{1,k}$ analogously as for $g_{3,k}$ the following estimation is
true

$\left\|g_{3,k}\right\|\leq
c\left(\left\|z_{k+1}\right\|+\left\|z_{k-1}\right\|\right)\leq
c\left(\left\|Az_{k+1}\right\|+\left\|Az_{k-1}\right\|\right).$ (5.24)

If we consider that operator $C$ satisfies condition (1.4), and $N$ is
bounded, then for $g_{4,k}$ we get

$\left|g_{4,k}\right|\leq
c\left(\left\|Az_{k}\right\|+\left\|\delta{z}_{k}\right\|\right)\,.$ (5.25)

If we take into account that operator $M(\cdot)$ satisfies Lipschitz
condition, then for $g_{5,k}$ we obtain

$\left\|g_{5,k}\right\|\leq
c\left\|z_{k}\right\|+\left\|f_{k}-\overline{f}_{k}\right\|\leq
c\left\|Az_{k}\right\|+\left\|f_{k}-\overline{f}_{k}\right\|.$ (5.26)

Finally, according to inequalities (5.20), (5.23), (5.24), (5.25) and (5.26)
for the vector $g_{k}$ the following estimation is valid

$\displaystyle\left\|g_{k}\right\|\leq\left\|f_{k}-\overline{f}_{k}\right\|+c\left(\left\|Az_{k+1}\right\|+\left\|Az_{k}\right\|+\left\|Az_{k-1}\right\|+\genfrac{\|}{\|}{}{}{\Delta
z_{k}}{\tau}+\genfrac{\|}{\|}{}{}{\Delta z_{k-1}}{\tau}\right)\text{.}$ (5.27)

Let us introduce the following denotations

$\displaystyle\delta_{k}=\left\|B^{1/2}z_{0}\right\|+\genfrac{\|}{\|}{}{}{\Delta
z_{0}}{\tau}+\tau\left\|B^{1/2}\frac{\Delta
z_{0}}{\tau}\right\|+\tau\sum\limits_{i=1}^{k}\left\|f_{i}-\overline{f}_{i}\right\|\text{,}$
$\displaystyle\varepsilon_{k}=\left\|B^{1/2}z_{k}\right\|+\genfrac{\|}{\|}{}{}{\Delta
z_{k-1}}{\tau}\text{.}$

From (5.27) according to (1.9) we have

$\displaystyle\left\|g_{k}\right\|\leq\left\|f_{k}-\overline{f}_{k}\right\|+c\left({\varepsilon}_{k+1}+{\varepsilon}_{k}+\left\|B^{1/2}z_{k-1}\right\|\right)\;.$
(5.28)

Using this inequality, from (5.13) we obtain

$\displaystyle\left\|B^{1/2}z_{k+1}\right\|\leq
c\delta_{k}+c\tau\sum\limits_{i=1}^{k}\left(\varepsilon_{i+1}+\varepsilon_{i}+\left\|B^{1/2}z_{i-1}\right\|\right)\,.$
(5.29)

Using simple transformation from (5.29) we get

$\displaystyle\left\|B^{1/2}z_{k+1}\right\|\leq
c\delta_{k}+c\tau\sum\limits_{i=1}^{k+1}\varepsilon_{i}\,.$ (5.30)

Analogously, according to (5.28) and
$\left\|{z}_{0}\right\|\leq\left\|{B}^{1/2}{z}_{0}\right\|$, from (5.14)
follows

$\displaystyle\genfrac{\|}{\|}{}{}{\Delta z_{k}}{\tau}\leq
c{\delta}_{k}+c\tau\sum\limits_{i=1}^{k+1}\varepsilon_{i}\,.$ (5.31)

If we add inequalities (5.30) and (5.31), we get

$\displaystyle\varepsilon_{k+1}\leq
c\delta_{k}+c\tau\sum\limits_{i=1}^{k+1}\varepsilon_{i}\,.$ (5.32)

If we request that $\tau$ satisfies the condition $\tau\leq q/c$ ($0<q<1$),
then from (5.30) we get

$\displaystyle\varepsilon_{k+1}\leq
c\delta_{k}+c\tau\sum\limits_{i=1}^{k}\varepsilon_{i}\,.$

From here by the induction can be obtained (discrete analogue of Grönwall’s
lemma)

$\varepsilon_{k+1}\leq
c\left(1+c\tau\right)^{k-1}\left(\delta_{k}+\tau\varepsilon_{1}\right)\,.$
(5.33)

From (5.33), using inequality

$\left\|B^{1/2}z_{1}\right\|\leq\left\|B^{1/2}z_{0}\right\|+\tau\left\|B^{1/2}\frac{\Delta
z_{0}}{\tau}\right\|\text{,}$

follows estimation (5.1). ∎

##  6 Estimate of the error of approximate solution

In this section, using the results of the previous sections, we prove the
theorem, which considers the error estimate for the approximate solution. This
theorem represents an almost trivial result of the theorem proved in the
previous section. However, estimation of the approximation error for scheme
(1.6) because of nonlinear terms, requires additional calculations.

Before formulating the theorem regarding the convergence (error estimate of
the approximate solution) of the scheme (1.6), we would like to make a note
about the well-posedness of the problem (1.1), (1.2). We mean from the
beginning that the original continuous problem is well-posed and the solution
is sufficiently smooth. Obviously, we require the smoothness of the solution
to find a convergence order (rate). If we demand the minimal smoothness which
is necessary for the well-posedness of the problem, then the convergence is
guaranteed, but we are not able to establish an order. If we increase the
smoothness order by one unit, then the convergence rate is equal to one (in
this, as well as in the previous case, it is sufficient to take
$u_{1}=\varphi_{0}+\tau\varphi_{1}$). However, a convergence rate becomes two
if we level up smoothness by two and define starting vector $u_{1}$ using the
following formula

$u_{1}=\varphi_{0}+\tau\varphi_{1}+\frac{\tau^{2}}{2}\varphi_{2},$ (6.1)

where ${\varphi}_{2}={u}^{{\prime}{\prime}}\left(0\right)$,
${u}^{{\prime}{\prime}}\left(0\right)$ is defined from the equation (1.1) via
$\varphi_{0}$ and $\varphi_{1}$ (we assume that $\varphi_{0}\,,\varphi_{1}\in
D(B)$).

The further increase of smoothness of the solution does not make sense, as the
approximation order of the scheme (1.6) is not more than two (obviously, the
convergence order generally does not exceed the approximation order).

Let us formulate above stated as a theorem (below everywhere $c$ denotes
positive constant).

######  Theorem 6.1.

Let the problem (1.1), (1.2) be well-posed. Besides, the following conditions
are fulfilled:

  1. (a)

$\varphi_{0}\;,\varphi_{1}\;,\varphi_{2}\in D(B)$;

  2. (b)

solution $u(t)$ of problem (1.1), (1.2) is continuously differentiable to
third order including and $u^{\prime\prime\prime}\left(t\right)$ satisfies
Lipschitz condition;

  3. (c)

$u^{\prime\prime}\left(t\right)\in D(B)$ for every $t$ from
$\left[0,\overline{t}\right]$ and function $Bu^{\prime\prime}\left(t\right)$
satisfy Lipschitz condition.

Then for scheme (1.6), (6.1) the following estimates are true

$\max_{1\leq k\leq
n-1}\left(\left\|B^{1/2}\widetilde{z}_{k+1}\right\|+\genfrac{\|}{\|}{}{}{\Delta\widetilde{z}_{k}}{\tau}\right)\leq
c\tau^{2},$ (6.2)

where $\widetilde{z}_{k}=u(t_{k})-u_{k}$ is an error of approximate solution,
$\Delta\widetilde{z}_{k}=\widetilde{z}_{k+1}-\widetilde{z}_{k}$ .

  * Proof.

Let us introduce the following notations:

${\delta}{u}\left({t}_{k}\right)=\frac{u\left({t}_{k+1}\right)-u\left({t}_{k-1}\right)}{2\tau}\,,\quad\hat{u}\left({t}_{k}\right)=\frac{u\left({t}_{k+1}\right)+u\left({t}_{k-1}\right)}{2}\,.$

Let us write down the equation (1.1) at point $t=t_{k}$ in the following form

$\displaystyle\frac{\Delta^{2}u\left(t_{k-1}\right)}{\tau^{2}}+a_{1}B{\delta}{u}\left({t}_{k}\right)+a_{2}B\hat{u}\left({t}_{k}\right)+\widetilde{\psi}_{1}\left(\zeta_{k-1},\zeta_{k+1}\right)A\hat{u}\left({t}_{k}\right)$
$\displaystyle+{\delta}\psi_{2}\left(\zeta_{k}\right)A\hat{u}\left({t}_{k}\right)+\widetilde{\psi}_{3}\left(\left\|u\left(t_{k-1}\right)\right\|^{2},\left\|u\left(t_{k+1}\right)\right\|^{2}\right)\hat{u}\left({t}_{k}\right)$
$\displaystyle+Cu\left(t_{k}\right)+N{\delta}{u}\left({t}_{k}\right)+M\left(u\left(t_{k}\right)\right)=f\left(t_{k}\right)+r_{\tau}\left(t_{k}\right),$
(6.3)

where $\zeta_{k}=\left\|A^{1/2}u\left(t_{k}\right)\right\|^{2}$
($\zeta(t)=\left\|A^{1/2}u\left(t\right)\right\|^{2}$),

$r_{\tau}\left(t_{k}\right)=\sum_{j=0}^{5}r_{j,\tau}\left(t_{k}\right)\;,$
(6.4)

and where

$\displaystyle r_{0,\tau}\left(t_{k}\right)$
$\displaystyle=\frac{\Delta^{2}u\left(t_{k-1}\right)}{\tau^{2}}-u^{\prime\prime}\left(t_{k}\right)\,,\quad
r_{6,\tau}\left(t_{k}\right)=N\left({\delta}{u}\left({t}_{k}\right)-u^{\prime}\left(t_{k}\right)\right)\,,$
$\displaystyle r_{2,\tau}\left(t_{k}\right)$
$\displaystyle=\frac{1}{2}\widetilde{\psi}_{1}\left(\zeta_{k-1},\zeta_{k+1}\right)A\left(\Delta^{2}u\left(t_{k-1}\right)\right)+\left(\widetilde{\psi}_{1}\left(\zeta_{k-1},\zeta_{k+1}\right)-\psi_{1}\left(\zeta_{k}\right)\right)Au\left(t_{k}\right)\,,$
$\displaystyle r_{3,\tau}\left(t_{k}\right)$
$\displaystyle=a_{1}B\left({\delta}{u}\left({t}_{k}\right)-u^{\prime}\left(t_{k}\right)\right)\,,\quad
r_{1,\tau}\left(t_{k}\right)=\frac{1}{2}a_{2}B\left(\Delta^{2}u\left(t_{k-1}\right)\right)\,,$
$\displaystyle r_{4,\tau}\left(t_{k}\right)$
$\displaystyle={\delta}\psi_{2}\left(\zeta_{k}\right)\frac{1}{2}A\left(\Delta^{2}u\left(t_{k-1}\right)\right)+\left({\delta}\psi_{2}\left(\zeta_{k}\right)-{\left(\psi_{2}(\zeta\left(t_{k}\right)\right)}^{\prime}_{t}\right)Au\left(t_{k}\right)\,,$
$\displaystyle r_{5,\tau}\left(t_{k}\right)$
$\displaystyle=\widetilde{\psi}_{3}\left(\left\|u\left(t_{k-1}\right)\right\|^{2},\left\|u\left(t_{k+1}\right)\right\|^{2}\right)\Delta^{2}u\left(t_{k-1}\right)$
$\displaystyle+\left(\widetilde{\psi}_{3}\left(\left\|u\left(t_{k-1}\right)\right\|^{2},\left\|u\left(t_{k+1}\right)\right\|^{2}\right)\right.\left.-\psi_{3}\left(\left\|u\left(t_{k}\right)\right\|^{2}\right)\right)u\left(t_{k}\right)\,.$

From (6.3) and (1.6) according to Theorem 5.1 we obtain

$\displaystyle\left\|B^{1/2}\widetilde{z}_{k+1}\right\|+\genfrac{\|}{\|}{}{}{\Delta\widetilde{z}_{k}}{\tau}$
$\displaystyle\leq
c\left(\left\|B^{1/2}\widetilde{z}_{0}\right\|+\genfrac{\|}{\|}{}{}{\Delta\widetilde{z}_{0}}{\tau}+\tau\left\|B^{1/2}\frac{\Delta\widetilde{z}_{0}}{\tau}\right\|+\tau\sum\limits_{i=1}^{k}\left\|r_{\tau}\left(t_{k}\right)\right\|\right).$
(6.5)

If we carry out the routine calculations we obtain

$\left\|r_{\tau}\left(t_{k}\right)\right\|\leq c\tau^{2}.$ (6.6)

It is obvious that according to the conditions (a), (b) and (c) of the Theorem
6.1, and the equality (6.1) the following inequalities are true

$\displaystyle\left\|B^{1/2}\left(\Delta\widetilde{z}_{0}\right)\right\|+\genfrac{\|}{\|}{}{}{\Delta\widetilde{z}_{0}}{\tau}\leq
c\tau^{2}\,,$ (6.7)

From (6.5), taking into account (6.6) and (6.7), follows (6.2). ∎

##  7 Iterative method for discrete problem

Let us rewrite equation (1.6) in the following form

$T_{k}v_{k+1}=\frac{1}{2}\tau\psi_{2}(\gamma_{k-1})Av_{k+1}-\tau
Nv_{k+1}+\widetilde{f}_{k}\;,$ (7.1)

where $v_{k+1}=(u_{k+1}+u_{k-1})/2$,

$\displaystyle T_{k}=\left(2+\tau^{2}a_{3,k}\right)I+\tau\left(a_{1}+\tau
a_{2}\right)B+\tau{b}_{k}A\,,\quad{b}_{k}=\tau
a_{1,k}+\frac{1}{2}\psi_{2}\left(\gamma_{k+1}\right)\,,$
$\displaystyle\widetilde{f}_{k}=\tau\widetilde{g}_{k}+\tau
a_{1}Bu_{k-1}+2u_{k}\,,\quad\widetilde{g}_{k}=\tau f_{k}-\tau
M\left(u_{k}\right)+Nu_{k-1}-\tau Cu_{k}\;\text{.}$

Equation (7.1) is solved by using the following iteration

$T_{k,m}v_{k+1}^{(m+1)}=\frac{1}{2}\tau\psi_{2}(\gamma_{k-1})Av_{k+1}^{(m)}-\tau
Nv_{k+1}^{(m)}+\widetilde{f}_{k}\;,$ (7.2)

where $m=0,1,\ldots$,

$\displaystyle
T_{k,m}=\left(2+\tau^{2}a_{3,k}^{(m)}\right)I+\tau\left(a_{1}+\tau
a_{2}\right)B+\tau{b}_{k}^{\left(m\right)}A\,,\quad{b}_{k}^{\left(m\right)}=\tau
a_{1,k}^{\left(m\right)}+\frac{1}{2}\psi_{2}\left(\gamma_{k+1}^{\left(m\right)}\right)\,,$
$\displaystyle
a_{1,k}^{(m)}=\widetilde{\psi}_{1}\left(\gamma_{k-1},\gamma_{k+1}^{(m)}\right)\,,\quad\gamma_{k+1}^{(m)}=\left\|A^{1/2}u_{k+1}^{(m)}\right\|^{2},\text{\quad}u_{k+1}^{(m)}=2v_{k+1}^{(m)}-u_{k-1}\;\text{,}$
$\displaystyle
a_{3,k}^{(m)}=\widetilde{\psi}_{3}\left(\left\|u_{k-1}\right\|^{2},\left\|u_{k+1}^{(m)}\right\|^{2}\right),\text{\quad}v_{k+1}^{(0)}=(u_{k}+u_{k-1})/2\;\text{.}$

######  Remark 7.1.

We already required that $A$ and $B$ to be self-adjoint positively defined
operators. Besides condition (1.3) is valid. From here follows that
$D(A)\subset D(B^{1/2})$ (see Remark 1.1). In order to show convergence of
iterative method (7.2) we require the following condition to be fulfilled
$D(A)=D(B^{1/2})$. From here follows that operator $B^{1/2}A^{-1}$ (as it is
closed operator defined in whole Hilbert space $H$) is bounded according to
the closed graph theorem. Let us denote norm of this operator by
$c_{1}=\left\|B^{1/2}A^{-1}\right\|$.

######  Remark 7.2.

Condition $D(A)=D(B^{1/2})$ is automatically fulfilled for equation (1), if
unknown function and its derivatives satisfy homogeneous boundary conditions.
In this case $B^{1/2}=A$, where $A$ is an expansion of symmetrical operator
($-\partial_{xx}^{2}$) (with homogeneous boundary conditions) till a self-
adjoint operator.

######  Remark 7.3.

Regarding iteration (7.2) it is important that $T_{k,m}$ is self-adjoint
positively defined operator ($a_{3,k}^{(m)}$, $a_{1,k}^{(m)}$and
$\psi_{2}(\gamma_{k+1}^{(m)})$ are nonnegative, as according to the condition
$\psi_{1}(s)$, $\;\psi_{2}(s)$ and $\psi_{3}(s)$, $s\in[0,+\infty[$, are
nonnegative functions). It is well-known that from here follows that
$R(T_{k,m})=H$, i.e. equation $T_{k,m}u=g$ , for any vector $g$ from $H$ has
unique solution $u\in D(B)$ and it continuously depends on $g$.

Let us prove convergence of iteration (7.2). The prove depends on many
standard transformations. The first step is to prove uniform boundedness of
vectors ${w}_{k+1}^{\left(m\right)}={B}^{1/2}{v}_{k+1}^{\left(m\right)}$
obtained by iteration (7.2). This fact is important, as it gives a certain
opportunity that iteration (7.2) might be converged.

  1. Step 1.

Prove uniform boundedness of vector sequence ${w}_{k+1}^{(m)}$ obtained by
using iteration (7.2).

     * Proof.

Let us introduce the following notations:

$\displaystyle
S_{k,m}=\left(2+\tau^{2}a_{3,k}^{(m)}\right)I+{a}_{\tau}B\,,\quad
S_{k}=\left(2+\tau^{2}a_{3,k}\right)I+{a}_{\tau}B\,,\quad{a}_{\tau}=\tau\left(a_{1}+\tau
a_{2}\right)\,,$
$\displaystyle{P}_{k,m}={B}^{1/2}{T}^{-1}_{k,m}\,,\quad{Q}_{k,m}={B}^{1/2}{S}^{-1/2}_{k,m}\,,\quad{P}_{k}={B}^{1/2}{T}^{-1}_{k}\,,\quad{Q}_{k}={B}^{1/2}{S}^{-1/2}_{k}\,.$

If we define the vector $v^{\left(m+1\right)}_{k+1}$ from the equation (7.2),
after applying the operator $B^{1/2}$ and move on to the norm, we get

$\displaystyle\left\|{w}_{k+1}^{(m+1)}\right\|$
$\displaystyle\leq\frac{1}{2}\tau\psi_{2}(\gamma_{k-1})\left\|{P}_{k,m}\right\|\left\|AB^{-1/2}\right\|\left\|{w}_{k+1}^{(m)}\right\|$
$\displaystyle+\tau\left\|{P}_{k,m}\right\|\left\|N\right\|\left\|B^{-1/2}\right\|\left\|{w}_{k+1}^{(m)}\right\|+\left\|{P}_{k,m}\widetilde{f}_{k}\right\|.$
(7.3)

Operator $T_{k,m}$ should be written in the following form

$T_{k,m}=S_{k,m}^{1/2}G_{k,m}S_{k,m}^{1/2}\,,\quad G_{k,m}=I+\tau
b_{k}^{(m)}S_{k,m}^{-1/2}AS_{k,m}^{-1/2}\,.$ (7.4)

According to (7.4) we have

$\left\|{P}_{k,m}\right\|\leq\left\|{Q}_{k,m}\right\|\left\|G_{k,m}^{-1}\right\|\left\|S_{k,m}^{-1/2}\right\|.$
(7.5)

The following estimations can be obtained easily:

$\left\|{Q}_{k,m}\right\|\leq\frac{1}{\sqrt{\tau
a_{1}}}\,,\quad\left\|S_{k,m}^{-1/2}\right\|\leq\frac{1}{\sqrt{2}}\;,\text{\quad}\left\|G_{k,m}^{-1}\right\|\leq
1\,.$ (7.6)

If we insert inequalities (7.6) into (7.5), we get

$\left\|{P}_{k,m}\right\|=\left\|{B}^{1/2}{T}^{-1}_{k,m}\right\|\leq\frac{1}{\sqrt{2\tau
a_{1}}}\;.$ (7.7)

As $\gamma_{k}$ is uniformly bounded (see Remark 2.1), we have

$\psi_{2}(\gamma_{k})\leq M_{2},\;M_{2}=\max\psi_{2}(s)<+\infty,\;0\leq
s\leq\max_{k}\gamma_{k}\;<+\infty\,.$ (7.8)

From (7.3) according to estimations (7.7), (7.8),
$\left\|{A}{B}^{-1/2}\right\|\leq{b}_{0}$ (see (1.9)) and
$\left\|{B}^{-1/2}\right\|\leq 1/\sqrt{m_{B}}$ ($m_{B}$ is a lower bound of
the operator $B$) follows

$\left\|{w}_{k+1}^{(m+1)}\right\|\leq\sqrt{\tau}M_{3}\left\|{w}_{k+1}^{(m)}\right\|+\left\|{P}_{k,m}\widetilde{f}_{k}\right\|,$
(7.9)

where $M_{3}$ is a positive constant (independent of $m$ and $n$).

Let us estimate norm of the vector

$\displaystyle{P}_{k,m}\widetilde{f}_{k}=\tau{P}_{k,m}\widetilde{g}_{k}+\tau
a_{1}{P}_{k,m}Bu_{k-1}+2{P}_{k,m}u_{k}\;.$ (7.10)

According to Theorem 2.1, $Au_{k}$, $M\left(u_{k}\right)$ and $u_{k}$ vectors
are uniformly bounded, (uniform boundeness of vectors $M\left(u_{k}\right)$
follows from (2.4)), also according to (1.4) we have
$\left\|Cu_{k}\right\|\leq a_{0}\left\|Au_{k}\right\|$. From here follows that
vectors $\widetilde{g}_{k}$ are uniformly bounded. According to this from
(7.7) follows that there exists such constant $M_{0}$ (independent of $n$ and
$m$), that

$\sqrt{\tau}\left\|{P}_{k,m}\widetilde{g}_{k}\right\|\leq\frac{1}{\sqrt{2a_{1}}}\left\|\widetilde{g}_{k}\right\|\leq
M_{0}\;.$ (7.11)

Let us estimate second summand on the right-hand side of equality (7.10).
Using formula (7.4) and inequalities (7.6) we get the following estimation

$\displaystyle\tau\left\|{P}_{k,m}Bu_{k-1}\right\|=\tau\left\|{Q}_{k,m}G_{k,m}^{-1}{Q}_{k,m}B^{1/2}u_{k-1}\right\|\leq\frac{1}{a_{1}}\left\|B^{1/2}u_{k-1}\right\|.$
(7.12)

From here according to Theorem 2.1 follows

$\tau\left\|{P}_{k,m}Bu_{k-1}\right\|\leq M_{1}\;,$ (7.13)

where $M_{1}$ is positive constant (independent of $n$ and $m$).

Eventually, we need to estimate third summand in equality (7.10). Let us
rewrite it in the following form

${P}_{k,m}u_{k}=B^{1/2}\left(T_{k,m}^{-1}-C_{k,m}^{-1}\right)u_{k}+B^{1/2}C_{k,m}^{-1}u_{k}\;,$
(7.14)

where
$C_{k,m}=\left(2+\tau^{2}a_{3,k}^{(m)}\right)I+\tau{b}_{k}^{\left(m\right)}A\;\text{.}$

Obviously, we have

$\displaystyle
T_{k,m}^{-1}-C_{k,m}^{-1}=T_{k,m}^{-1}\left(C_{k,m}-T_{k,m}\right)C_{k,m}^{-1}=-{a}_{\tau}T_{k,m}^{-1}BC_{k,m}^{-1}\;.$
(7.15)

If we insert (7.15) into (7.14) and consider (7.4), we get

$\displaystyle{P}_{k,m}u_{k}$
$\displaystyle=\left(B^{1/2}A^{-1}\right)C_{k,m}^{-1}\left(Au_{k}\right)$
$\displaystyle-{a}_{\tau}{Q}_{k,m}G_{k,m}^{-1}{Q}_{k,m}\left(B^{1/2}A^{-1}\right)C_{k,m}^{-1}\left(Au_{k}\right).$
(7.16)

From (7.16) according to (7.6), $\left\|C_{k,m}^{-1}\right\|\leq 1/2$, Remark
7.1 and also Theorem 2.1 we have

$\left\|{P}_{k,m}u_{k}\right\|\leq M_{4}\left\|Au_{k}\right\|\leq{M}_{5}\,.$
(7.17)

where $M_{5}$ is positive constant (independent of $n$ and $m$),
$M_{4}=c_{1}\left(2+\tau a_{2}/a_{1}\right)$.

Let us insert inequalities (7.11), (7.13) and (7.17) into (7.10), we get

$\left\|{P}_{k,m}\widetilde{f}_{k}\right\|\leq M_{6}\,,\quad
M_{6}=\sqrt{\tau}M_{0}+a_{1}M_{1}+2M_{5}\,.$ (7.18)

From (7.9) according to (7.18) we have

$\left\|{w}_{k+1}^{(m+1)}\right\|\leq\sqrt{\tau}M_{3}\left\|{w}_{k+1}^{(m)}\right\|+M_{6}\;.$
(7.19)

Let $\tau$ satisfies condition $\sqrt{\tau}M_{3}\leq q<1$. Then from (7.19)
follows

$\left\|{w}_{k+1}^{(m)}\right\|\leq
q^{m}\left\|{w}_{k+1}^{(0)}\right\|+\frac{M_{6}}{1-q}\;.$ (7.20)

Inequality (7.20) shows, that vector sequence
${w}_{k+1}^{\left(m\right)}={B}^{1/2}{v}^{\left(m\right)}_{k+1}$
($v_{k+1}^{(0)}=(u_{k+1}^{(0)}+u_{k-1})/2$, $u_{k+1}^{(0)}=u_{k}$) obtained by
iterative method is uniformly bounded, i.e. there exists such constant $M_{7}$
(independent of $n$ and $m$) , that $\left\|{w}_{k+1}^{(m)}\right\|\leq M_{7}$
(here we again used Theorem 2.1 for vector ${w}_{k+1}^{(0)}$). ∎

  2. Step 2.

Let us estimate norm of error
$Z_{k+1}^{(m)}=B^{1/2}\left(v_{k+1}-v_{k+1}^{(m+1)}\right)$. Let us define
$v_{k+1}$ and $v_{k+1}^{(m+1)}$ from equalities (7.1) and (7.2), respectively.
If we apply $B^{1/2}$ to the difference of these vectors and perform
corresponding transformation, we get

$\displaystyle Z_{k+1}^{(m+1)}$
$\displaystyle=\frac{1}{2}\tau\psi_{2}(\gamma_{k-1})B^{1/2}L_{k,m}{L}{w}_{k+1}+\frac{1}{2}\tau\psi_{2}(\gamma_{k-1}){P}_{k,m}{L}Z_{k+1}^{(m)}$
$\displaystyle-\tau
B^{1/2}L_{k,m}{S}{w}_{k+1}-\tau{P}_{k,m}{S}Z_{k+1}^{(m)}+B^{1/2}L_{k,m}\widetilde{f}_{k}\;,$
(7.21)

where ${L}={A}{B}^{-1/2}$, ${S}={N}{B}^{-1/2}$ and
$L_{k,m}=T_{k}^{-1}-T_{k,m}^{-1}$.

Let us note, that in representation of (7.21) the most complicated term is
$B^{1/2}L_{k,m}\widetilde{f}_{k}$, as it does not have a small parameter
$\tau$ as a multiplier, that provides to obtain an estimation from where
follows convergence of the iteration method (7.2). For this term, using simple
transformation we get

$\displaystyle B^{1/2}L_{k,m}\widetilde{f}_{k}$
$\displaystyle=-{P}_{k}\left(T_{k}-T_{k,m}\right)T_{k,m}^{-1}\widetilde{f}_{k}=-\tau^{2}{\zeta}_{3,k}^{\left(m\right)}{P}_{k}T_{k,m}^{-1}\widetilde{f}_{k}$
$\displaystyle-\tau^{2}{\zeta}_{1,k}^{\left(m\right)}{P}_{k}AT_{k,m}^{-1}\widetilde{f}_{k}-\frac{1}{2}\tau{\zeta}_{2,k}^{\left(m\right)}{P}_{k}AT_{k,m}^{-1}\widetilde{f}_{k}\;,$
(7.22)

where
${\zeta}_{2,k}^{\left(m\right)}=\psi_{2}(\gamma_{k+1})-\psi_{2}(\gamma_{k+1}^{(m)})\,\text{,}\quad{\zeta}_{j,k}^{\left(m\right)}=a_{j,k}-a_{j,k}^{(m)}\,\text{,}\quad{j}=1,3$.

Analogously to (7.4) we can represent operator $T_{k}$ in the following form:

$T_{k}=S_{k}^{1/2}G_{k}S_{k}^{1/2}\,,\quad{G}_{k}=I+\tau
b_{k}S_{k}^{-1/2}AS_{k}^{-1/2}\,.$ (7.23)

Analogously to inequalities (7.6) we have:

$\left\|{Q}_{k}\right\|\leq\frac{1}{\sqrt{\tau
a_{1}}}\,,\quad\left\|S_{k}^{-1/2}\right\|\leq\frac{1}{\sqrt{2}}\;,\text{\quad}\left\|G_{k}^{-1}\right\|\leq
1\,.$ (7.24)

From (7.23) using inequalities (7.24) follows

$\left\|{P}_{k}\right\|=\left\|{B}^{1/2}{T}^{-1}_{k}\right\|\leq\frac{1}{\sqrt{2\tau
a_{1}}}\;.$ (7.25)

Using estimations (7.24), (7.25), (7.6), and (7.7), analogously to
inequalities (7.11), (7.12) and (7.17), we can obtain:

$\displaystyle\tau\left\|{P}_{k}AT_{k,m}^{-1}\right\|=\tau\left\|{P}_{k}{L}{P}_{k,m}\right\|\leq\frac{b_{0}}{2a_{1}}\,,$
(7.26)
$\displaystyle\tau\sqrt{\tau}\left\|{P}_{k}AT_{k,m}^{-1}Bu_{k-1}\right\|$
$\displaystyle=\tau{\nu}_{0}\left\|{L}{B}^{1/2}T_{k,m}^{-1}Bu_{k-1}\right\|\leq
b_{0}{\nu}_{0}a_{1}^{-1}\left\|B^{1/2}u_{k-1}\right\|,$ (7.27)
$\displaystyle\sqrt{\tau}\left\|{P}_{k,m}AT_{k,m}^{-1}u_{k}\right\|\leq\nu_{0}\left\|C_{k,m}^{-1}{A}{u}_{k}-{a}_{\tau}\left({L}{B}^{1/2}T_{k,m}^{-1}B{A}^{-1}C_{k,m}^{-1}\right){A}{u}_{k}\right\|$
$\displaystyle\leq\nu_{0}\left(1+(a_{1}+\tau
a_{2})b_{0}c_{1}a_{1}^{-1}\right)\left\|Au_{k}\right\|,\text{\quad}\nu_{0}=1/\sqrt{2a_{1}}.$
(7.28)

To obtain inequality (7.28) the following representations
$T_{k,m}^{-1}=C_{k,m}^{-1}-{a}_{\tau}T_{k,m}^{-1}BC_{k,m}^{-1}$ and
${B}^{1/2}{T}_{k,m}^{-1}{B}^{1/2}={Q}_{k,m}{G}_{k,m}{Q}_{k,m}$ are used.

From inequalities (7.26), (7.27) and (7.28), according to Theorem 2.1 it
follows that there exists such positive constant $M_{8}$ (independent of $n$
and $m$) that

$\sqrt{\tau}\left\|{P}_{k}AT_{k,m}^{-1}\widetilde{f}_{k}\right\|\leq M_{8}\;.$
(7.29)

According to the representation (7.4) and the estimation (7.7) we have:

$\displaystyle\sqrt{\tau}\left\|{P}_{k}T_{k,m}^{-1}\right\|\leq\frac{1}{2}\nu_{0}\,,$
(7.30)
$\displaystyle\tau\left\|{P}_{k}T_{k,m}^{-1}{B}{u}\right\|\leq\frac{1}{2a_{1}}\left\|{{B}^{1/2}}{u}\right\|\,,\quad{u}\in\operatorname{D}\left(B\right)\,.$
(7.31)

From the inequalities (7.30) and (7.31) analogously to (7.29) we have

$\sqrt{\tau}\left\|{P}_{k}T_{k,m}^{-1}\widetilde{f}_{k}\right\|\leq M_{9}\;,$
(7.32)

where $M_{9}$ is a positive constant (independent of $n$ and $m$).

If we apply norms in (7.22) and take into consideration the estimations (7.29)
and (7.32), we get

$\displaystyle\left\|B^{1/2}L_{k,m}\widetilde{f}_{k}\right\|\leq\sqrt{\tau}\left(\tau
M_{9}\left|{\zeta}_{3,k}^{\left(m\right)}\right|+\tau
M_{8}\left|{\zeta}_{1,k}^{\left(m\right)}\right|+M_{8}\left|{\zeta}_{2,k}^{\left(m\right)}\right|\right).$
(7.33)

Now, let us estimate norm of the operator $B^{1/2}L_{k,m}$. From (7.22) we
have

$\displaystyle\left\|B^{1/2}L_{k,m}\right\|\leq\tau^{2}\left|{\zeta}_{3,k}^{\left(m\right)}\right|\left\|{P}_{k}T_{k,m}^{-1}\right\|+\tau\left\|{P}_{k}AT_{k,m}^{-1}\right\|\left(\tau\left|{\zeta}_{1,k}^{\left(m\right)}\right|+\left|{\zeta}_{2,k}^{\left(m\right)}\right|\right)\text{.}$

From here according to inequalities (7.26) and (7.30) we have

$\displaystyle\left\|B^{1/2}L_{k,m}\right\|\leq\frac{1}{2}\tau^{3/2}\nu_{0}\left|{\zeta}_{3,k}^{\left(m\right)}\right|+b_{0}\nu_{0}^{2}\left(\tau\left|{\zeta}_{1,k}^{\left(m\right)}\right|+\left|{\zeta}_{2,k}^{\left(m\right)}\right|\right)\,.$
(7.34)

Let us note that, from uniform boundedness of vectors ${w}_{k+1}^{(m)}$
follows uniform boundedness of vectors $Av_{k+1}^{(m)}$ (see inequality
(1.9)). From here according to the inequality (2.8) follows uniform
boundedness of vectors $A^{1/2}v_{k+1}^{(m)}$.

If we insert the terms ${\gamma}_{k+1}^{\left(m\right)}$ and $\gamma_{k-1}$
into the inequality (5.17) instead of the following ones ${\gamma}_{k+1}$ and
$\bar{\gamma}_{k-1}$, respectively, we obtain

$\displaystyle\left|{\zeta}_{1,k}^{\left(m\right)}\right|$ $\displaystyle\leq
K_{1}\left|\gamma_{k+1}-\gamma_{k+1}^{(m)}\right|\leq
2K_{1}K_{2}\left\|A^{1/2}u_{k+1}-A^{1/2}u_{k+1}^{(m)}\right\|$
$\displaystyle=4K_{1}K_{2}\left\|A^{1/2}v_{k+1}-A^{1/2}v_{k+1}^{(m)}\right\|,$
(7.35)

where $K_{1}=\max\left|\psi_{1}^{\prime}\left(s\right)\right|<+\infty,\;0\leq
s\leq K_{2}=\max_{(k,m)}(\gamma_{k},\gamma_{k}^{(m)})<+\infty\,.$

From (2.8) using inequality (1.9) it follows that

$\left\|A^{1/2}u\right\|\leq\frac{b_{0}}{\sqrt{m_{A}}}\left\|B^{1/2}u\right\|,\text{\quad}\forall
u\in D(B^{1/2})\text{.}$

According to this inequality, from (7.35) we have

$\left|{\zeta}_{1,k}^{\left(m\right)}\right|\leq
K_{3}\left\|{w}_{k+1}-{w}_{k+1}^{(m)}\right\|.$ (7.36)

where $K_{3}$ is a positive constant (independent of $n$ and $m$).

Analogously we get

$\left|{\zeta}_{j,k}^{\left(m\right)}\right|\leq
K_{4}\left\|{w}_{k+1}-{w}_{k+1}^{(m)}\right\|,\text{\quad}j=2,3\,,\quad
K_{4}=\operatorname{const}>0.$ (7.37)

If we insert (7.36) and (7.37) inequalities into (7.33) and (7.34) we get:

$\displaystyle\left\|B^{1/2}L_{k,m}\widetilde{f}_{k}\right\|\leq\sqrt{\tau}K_{5}\left\|{w}_{k+1}-{w}_{k+1}^{(m)}\right\|,\text{\quad}K_{5}=\operatorname{const}>0,$
(7.38)

$\left\|B^{1/2}L_{k,m}\right\|\leq
K_{6}\left\|{w}_{k+1}-{w}_{k+1}^{(m)}\right\|,\text{\quad}K_{6}=\operatorname{const}>0.$
(7.39)

If we move on to norms in the equality (7.21) and consider the inequalities
(7.38), (7.39), (7.7), $\left\|{L}\right\|\leq{b}_{0}$ (see (1.9)) and Theorem
2.1 we get

$\left\|Z_{k+1}^{(m+1)}\right\|\leq\sqrt{\tau}K_{0}\left\|Z_{k+1}^{(m)}\right\|,$
(7.40)

where $K_{0}$ is a positive constant (independent of $n$ and $m$).

Let $\tau$ satisfies condition $\sqrt{\tau}K_{0}\leq q<1$. Then from (7.40)
follows that $\left\|Z_{k+1}^{(m)}\right\|\leq
q^{m}\left\|Z_{k+1}^{(0)}\right\|$, which means that (7.2) iteration is
convergent.

So, we proved the following theorem

######  Theorem 7.1.

If operators $A$ and $B$ satisfy conditions from section 1 and
$D(A)=D(B^{1/2})$, then (7.2) iteration converges with geometric progression
speed.

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## Authors’ addresses:

Jemal Rogava ID

Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State
University (TSU), Ilia Vekua Institute of Applied Mathematics (VIAM), 2
University St., Tbilisi 0186, Georgia.
E-mail<EMAIL_ADDRESS>
Mikheil Tsiklauri ID

Missouri University of Science and Technology, Electromagnetic Compatibility
Laboratory, 4000 Enterprise Drive, Rolla, MO 65409, USA.
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
Zurab Vashakidze ID

Institute of Mathematics, School of Science and Technology, The University of
Georgia (UG), 77a, M. Kostava st., Tbilisi 0171, Georgia; Ilia Vekua Institute
of Applied Mathematics (VIAM) of Ivane Javakhishvili Tbilisi State University
(TSU), 2, University St., Tbilisi 0186, Georgia.
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>