A novel approach to solving a multipoint boundary value problem for an
integro-differential equation

Anar T. Assanova111Corresponding author E-mail addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>(Anar T. Assanova<EMAIL_ADDRESS>(Elmira A.
Bakirova<EMAIL_ADDRESS>(Roza E. Uteshova)
The authors have received funding from the European Unions Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 873071. , Elmira A. Bakirova, Roza E. Uteshova

Institute of Mathematics and Mathematical Modeling,
125, Pushkin Str., 050010, Almaty, Kazakhstan
International Information Technology University, Almaty, Kazakhstan

Abstract

In the present paper, we study a multipoint boundary value problem for a
system of Fredholm integro-differenial equations by the method of
parameterization.The case of a degenerate kernel is studied separately, for
which we obtain well-posedness conditions and propose some algorithms to find
approximate and numerical solutions of the problem. We then establish
necessary and sufficient conditions for the well-posedness of the multipoint
problem for a system of Fredholm integro-differential equations and develop
some algorithms for finding its approximate solutions. These algorithms are
based on the solutions of an approximating problem for the system of integro-
differential equations with degenerate kernel.

MSC: 45J05, 45L05; 47G20; 65Q99.

Keywords: Fredholm integro-differential equation, multipoint problem,
parameterization method, algorithm, solvability criteria.

1\. Introduction

Various types of multipoint problems for differential and integro-differential
equations have been studied by many researchers, see [1-4, 7-11, 20-25]. A
number of methods have been applied to solve these problems, e.g., methods of
qualitative theory of differential equations, the method of Green’s functions,
the method of upper and lower solutions, numerical-analytical methods.
However, the problem of establishing effective criteria for the unique
solvability of multipoint problems for integro-differential equations, as well
as developing algorithms for finding their approximate and numerical
solutions, still remains open.

One of constructive methods of investigation and solving boundary value
problems for ordinary differential equations and integro-differential
equations is the method of parameterization proposed by Dzhumabaev [12]. This
method was originally developed for studying and solving boundary value
problems for systems of ordinary differential equations. In [12], coefficient
criteria were established for the unique solvability of linear boundary value
problems. An algorithm for finding their approximate solutions was developed.
The method of parameterization was later extended to linear multipoint
boundary value problems [20-21], for which necessary and sufficient conditions
were obtained for the unique solvability in terms of initial data and an
algorithm for finding their approximate solutions was proposed. In [13-15,
19], the method of parameterization was applied to two-point boundary value
problems for Fredholm integro-differential equations to establish criteria for
their solvability and unique solvability. For these problems, based on the
method of parameterization and a new concept of general solution, novel
algorithms for approximate and numerical solutions were developed, see
[16-18]. The results obtained in above-mentioned papers were used to
investigate a multipoint boundary value problem for loaded differential
equations [4] and a boundary value problem with a parameter for Fredholm
integro-differential equations [3].

Consider the multipoint boundary value problem for the system of integro-
differential equations

$\frac{dx}{dt}=A(t)x+\int^{T}_{0}K(t,\tau)x(\tau)d\tau+f(t),\qquad x\in
R^{n},\quad t\in(0,T),$ $None$ $\sum\limits^{m}_{i=0}B_{i}x(t_{i})=d,\qquad
d\in R^{n}.$ $None$

Here $x(t)=col(x_{1}(t),x_{2}(t),...,x_{n}(t))$ is an unknown function,
$(n\times n)$ matrix $A(t)$ and $n$-vector $f(t)$ are continuous on $[0,T]$,
$(n\times n)$ matrix $K(t,\tau)$ is continuous on $[0,T]\times[0,T]$, $B_{i}$
are constant $(n\times n)$ matrices,
$0=t_{0}<t_{1}<t_{2}<...<t_{m-1}<t_{m}=T$,
$\|x\|=\max\limits_{i=\overline{1,n}}|x_{i}|$.

A solution to multipoint problem (1.1), (1.2) is a function
$x^{\ast}(t):[0,T]\to\mathbb{R}^{n}$ that is continuous on $[0,T]$,
continuously differentiable on $(0,T)$ and satisfies integro-differential
equations (1.1) and multipoint condition (1.2).

The aim of the present paper is to obtain criteria for the unique solvability
of problem (1.1), (1.2) and develop algorithms for finding its approximate
solutions. To this end, the parameterization method is used. The interval
$[0,T]$ is partitioned and additional parameters are introduced as the values
of the desired solution at the left endpoints of the partition subintervals.
When applying the method of parameterization to problem (1.1), (1.2), some
intermediate problems occur, so called special Cauchy problems for integro-
differential equations with parameters. The questions of solvability and
unique solvability of such problems were thoroughly investigated in [13-19].

Section 2 is devoted to the study of Fredholm integro-differential equations
with degenerate kernel. We divide $[0,T]$ into $m$ parts and introduce
additional parameters as the values of the desired solution at the left
endpoints $t=t_{i}$, $i=\overline{0,m-1},$ of the subintervals. The unique
solvability of a special Cauchy problem for the $\Delta_{m}$ partition is
equivalent to the invertibility of a matrix $I-G(\Delta_{m})$ constructed
through a fundamental matrix of the differential part and the matrices of the
integral kernel. The $\Delta_{m}$ partition is called regular if the matrix
$I-G(\Delta_{m})$ is invertible (see [16]).

For a regular $\Delta_{m}$ partition, a system of linear algebraic equations
in the parameters introduced is constructed using $[I-G(\Delta_{m})]^{-1},$
the multipoint condition (1.2), and the continuity conditions at the interior
partition points $t=t_{i}$, $i=\overline{1,m-1}.$ It is shown that the
invertibility of the matrix of the system constructed is equivalent to the
unique solvability of the multipoint boundary value problem under
consideration.

In Section 3, we develop the algorithms for finding a solution to a multipoint
boundary value problem for the integro-differential equation with degenerate
kernel. For a chosen $\Delta_{m}$ partition, the matrix $G(\Delta_{m})$ is
calculated. If there is an inverse of $I-G(\Delta_{m})$, then we construct a
system of linear algebraic equations. The elements of $G(\Delta_{m}),$ the
coefficients and right-hand side of the system are determined by the solutions
of the Cauchy problems for ordinary differential equations and the values of
the definite integrals of some functions over the partition subintervals. By
solving the system of algebraic equations, we determine the values of the
solution at the left endpoints of the subintervals. Next, using the values
obtained and the data of the integro-differential equation we compose a
function $\mathcal{F}^{\ast}(t)$ that is continuous on $[0,T]$. Solving the
Cauchy problems for ordinary differential equations with the right-hand side
$\mathcal{F}^{\ast}(t)$, we get the values of the desired solution at the
remaining points of the interval $[0,T]$.

If a fundamental matrix of the differential part is found explicitly and the
integrals are evaluated exactly, then the algorithm allows us to find a
closed-form solution as well. As is known, it is usually impossible to
explicitly find a fundamental matrix for a system of ordinary differential
equations with variable coefficients, and, in general, only approximate values
of definite integrals can be obtained. For this reason, in this section we
propose a numerical implementation of the algorithm. The Cauchy problems for
ordinary differential equations on the subintervals are solved by the fourth-
order Runge-Kutta method; the integrals are calculated by the Simpson formula.
It should be noted that the elements of the matrix $G(\Delta_{m}),$ the
coefficients and the right-hand side of the system of algebraic equations in
parameters can be evaluated by parallel computing on the partition
subintervals.

In Section 4, a multipoint boundary value problem is considered for a Fredholm
integro-differential equation when the integral kernel is not degenerate. We
approximate the kernel by the degenerate one and then use the results obtained
in Section 2. At each step of the process, a multipoint boundary value problem
for the integro-differential equation with degenerate kernel is solved. We
establish sufficient conditions for the convergence of the iterative process
to a solution of the multipoint boundary value problem for the Fredholm
integro-differential equation with non-degenerate kernel. The accuracy of the
approximate solution depends on the choice of the approximating kernel and the
number of iteration steps. The necessary and sufficient conditions for the
well-posedness of the multipoint problem (1.1), (1.2) are obtained in terms of
properties of solutions to approximating problems for integro-differential
equations with degenerate kernels.

2\. The well-posedness of multipoint problems for Fredholm integro-
differential equations with degenerate kernel

Consider the integro-differential equation

$\frac{dx}{dt}=A(t)x+\sum\limits_{j=1}^{k}\int^{T}_{0}\varphi_{j}(t)\psi_{j}(\tau)x(\tau)d\tau+f(t),\quad
t\in(0,T),\qquad x\in R^{n},$ $None$

subject to the multipoint condition

$\sum\limits^{m}_{i=0}B_{i}x(t_{i})=d,\qquad d\in R^{n},$ $None$

where the matrices $A(t),$ $\varphi_{j}(t),$ $\psi_{j}(\tau),$
$j=\overline{1,k},$ and the vector $f(t)$ are continuous on $[0,T],$
$\|x\|=\max\limits_{i=\overline{1,n}}|x_{i}|.$

The interval $[0,T)$ is divided into $m$ parts by the points
$t_{0}=0<t_{1}<\ldots<t_{m}=T$, and the partition
$\displaystyle{[0,T)=\bigcup\limits_{r=1}^{m}[t_{r-1},t_{r})}$ is denoted by
$\Delta_{m}$. The case of no partitioning the interval $[0,T]$ is denoted by
$\Delta_{1}$.

We introduce the following spaces: $C([0,T],R^{n})$ is the space of continuous
functions $x:[0,T]\rightarrow R^{n}$ with the norm
$\|x\|_{1}=\max\limits_{t\in[0,T]}\|x(t)\|;$

$C([0,T],\Delta_{N},R^{nm})$ is the space of function systems
$x[t]=(x_{1}(t),x_{2}(t),\ldots,x_{m}(t)),$ where functions
$x_{r}:[t_{r-1},t_{r})\rightarrow R^{n}$ are continuous and have finite left-
handed limits $\lim\limits_{t\rightarrow t_{r}-0}x_{r}(t)$ for all
$r=\overline{1,m},$ with the norm
$\|x[\cdot]\|_{2}=\max\limits_{r=\overline{1,m}}\sup\limits_{t\in[t_{r-1},t_{r})}\|x_{r}(t)\|.$

Let $x_{r}(t)$ be the restriction of the function $x(t)$ to the $r$th
subinterval $[t_{r-1},t_{r}),$ i.e. $x_{r}(t)=x(t),$ $t\in[t_{r-1},t_{r}),$
$r=\overline{1,m}.$

We introduce additional parameters $\lambda_{r}=x_{r}(t_{r-1})$ and make the
substitution $x_{r}(t)=u_{r}(t)+\lambda_{r}$ on each $r$th subinterval. The
multipoint problem (2.1), (2.2) is then reduced to the following problem with
parameters:

$\frac{du_{r}}{dt}=A(t)(u_{r}+\lambda_{r})+\sum\limits_{s=1}^{m}\sum\limits_{j=1}^{k}\int^{t_{s}}_{t_{s-1}}\varphi_{j}(t)\psi_{j}(\tau)[u_{r}(\tau)+\lambda_{r}]d\tau+f(t),\,\,t\in(t_{r-1},t_{r}),\,\,r=\overline{1,m},$
$None$ $u_{r}(t_{r-1})=0,\qquad r=\overline{1,m},$ $None$
$\sum\limits^{m-1}_{i=0}B_{i}\lambda_{i+1}+B_{m}\lambda_{m}+B_{m}\lim\limits_{t\rightarrow
T-0}u_{m}(t)=d,$ $None$ $\lambda_{p}+\lim\limits_{t\to
t_{p}-0}u_{p}(t)-\lambda_{p+1}=0,\qquad p=\overline{1,m-1},$ $None$

where (2.6) are the continuity conditions for the solution at the interior
points of the partition $\Delta_{m}.$ Note that conditions (2.6) and integro-
differential equations (2.3) ensure the continuity of the derivative of the
solution at those points.

If $x^{\ast}(t)$ is a solution to multipoint problem (2.1),(2.2), then the
pair $(\lambda^{\ast},u^{\ast}[t])$ with elements
$\lambda^{\ast}=(\lambda_{1}^{\ast},\lambda_{2}^{\ast},\ldots,\lambda_{m}^{\ast})\in
R^{nm},$
$u^{\ast}[t]=\bigl{(}u_{1}^{\ast}(t),u_{2}^{\ast}(t),\ldots,u_{m}^{\ast}(t)\bigr{)}\in
C([0,T],\Delta_{m},R^{nm}),$ where $\lambda_{r}^{\ast}=x^{\ast}(t_{r-1}),$
$u_{r}^{\ast}(t)=x^{\ast}(t)-x^{\ast}(t_{r-1}),$ $[t_{r-1},t_{r}),$
$r=\overline{1,m}$, is a solution to the problem with parameters (2.3)-(2.6).
Vice versa, if a pair $(\widetilde{\lambda},\widetilde{u}[t])$ with elements
$\widetilde{\lambda}=(\widetilde{\lambda}_{1},\widetilde{\lambda}_{2},\ldots,\widetilde{\lambda}_{m})\in
R^{nm},$
$\widetilde{u}[t]=\bigl{(}\widetilde{u}_{1}(t),\widetilde{u}_{2}(t),\ldots,\widetilde{u}_{m}(t)\bigr{)}\in
C([0,T],\Delta_{m},R^{nm}),$ is a solution to problem with parameters
(2.3)-(2.6), then the function $\widetilde{x}(t)$ defined as
$\widetilde{x}(t)=\widetilde{\lambda}_{r}+\widetilde{u}_{r}(t),$
$t\in[t_{r-1},t_{r}),$ $r=\overline{1,m},$
$\widetilde{x}(T)=\widetilde{\lambda}_{m}+\lim\limits_{t\rightarrow
T-0}\widetilde{u}_{m}(t),$ is a solution to the original problem (2.1),(2.2).

If $X_{r}(t)$ is a fundamental matrix of the differential equation
$\displaystyle{\frac{dx_{r}}{dt}=A(t)x_{r}}$ on $[t_{r-1},t_{r}]$, then the
special Cauchy problem for the system of integro-differential equations with
parameters (2.3), (2.4) is reduced to the equivalent system of integral
equations

$u_{r}(t)=X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)A(\tau)d\tau\lambda_{r}+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\sum\limits_{s=1}^{m}\sum\limits_{j=1}^{k}\int_{t_{s-1}}^{t_{s}}\varphi_{j}(\tau)\psi_{j}(\tau_{1})u_{s}(\tau_{1})d\tau_{1}d\tau+$
$+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\sum\limits_{s=1}^{m}\sum\limits_{j=1}^{k}\int_{t_{s-1}}^{t_{s}}\varphi_{j}(\tau)\psi_{j}(\tau_{1})d\tau_{1}d\tau\lambda_{s}+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)f(\tau)d\tau,$
$None$

$\qquad\qquad t\in[t_{r-1},t_{r}),\qquad r=\overline{1,m}.$

Setting
$\displaystyle\mu_{j}=\sum\limits_{s=1}^{m}\int_{t_{s-1}}^{t_{s}}\psi_{j}(\tau)u_{s}(\tau)d\tau$,
rewrite (2.7) in the following way:

$u_{r}(t)=\sum\limits_{j=1}^{k}X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\varphi_{j}(\tau)d\tau\mu_{j}+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\biggl{[}A(\tau)\lambda_{r}+$
$+\sum\limits_{j=1}^{k}\varphi_{j}(\tau)\sum\limits_{s=1}^{m}\int_{t_{s-1}}^{t_{s}}\psi_{j}(\tau_{1})d\tau_{1}\lambda_{s}+f(\tau)\biggr{]}d\tau,\qquad
t\in[t_{r-1},t_{r}),\qquad r=\overline{1,m}.$ $None$

Multiplying both sides of (2.8) by $\psi_{p}(t)$, integrating them over
$[t_{r-1},t_{r}]$, and summing up with respect to $r$, we get the following
system of linear algebraic equations in $\mu=(\mu_{1},\ldots,\mu_{k})\in
R^{nk}$:

$\mu_{p}=\sum\limits_{l=1}^{k}G_{p,l}(\Delta_{m})\mu_{l}+\sum\limits^{m}_{r=1}V_{p,r}(\Delta_{m})\lambda_{r}+g_{p}(f,\Delta_{m}),\quad
p=\overline{1,k},$ $None$

with $(n\times n)$ matrices

$G_{p,l}(\Delta_{m})=\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)X_{r}(\tau)\int_{t_{r-1}}^{\tau}X_{r}^{-1}(\tau_{1})\varphi_{l}(\tau_{1})d\tau_{1}d\tau,$
$None$
$V_{p,r}(\Delta_{m})=\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)X_{r}(\tau)\int_{t_{r-1}}^{\tau}X^{-1}_{r}(\tau_{1})A(\tau_{1})d\tau_{1}d\tau+$
$+\sum\limits_{s=1}^{m}\sum\limits_{j=1}^{k}\int_{t_{s-1}}^{t_{s}}\psi_{p}(\tau)X_{s}(\tau)\int_{t_{s-1}}^{\tau}X_{s}^{-1}(\tau_{1})\varphi_{j}(\tau_{1})d\tau_{1}d\tau\int_{t_{r-1}}^{t_{r}}\psi_{j}(\tau_{2})d\tau_{2},$
$None$

and vectors of dimension $n$

$g_{p}(f,\Delta_{m})=\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)X_{r}(\tau)\int_{t_{r-1}}^{\tau}X_{r}^{-1}(\tau_{1})f(\tau_{1})d\tau_{1}d\tau,\quad
p=\overline{1,k},\quad j=\overline{1,k}.$ $None$

Using the matrices $G_{p,l}(\Delta_{m})$ and $V_{p,r}(\Delta_{m})$, we
construct the matrices $G(\Delta_{m})=(G_{p,l}(\Delta_{m})),$
$p,l=\overline{1,k},$ and $V(\Delta_{m})=(V_{p,r}(\Delta_{m})),$
$p=\overline{1,k},$ $r=\overline{1,m}$. Then, system (2.9) can be rewritten in
the form

$[I-G(\Delta_{m})]\mu=V(\Delta_{m})\lambda+g(f,\Delta_{m}),$ $None$

where $I$ is the identity matrix of dimension $nk,$
$g(f,\Delta_{m})=(g_{1}(f,\Delta_{m}),\ldots,g_{k}(f,\Delta_{m}))\in R^{nk}.$

Definition 2.1 The partition $\Delta_{m}$ is said to be regular if the matrix
$I-G(\Delta_{m})$ is invertible.

Any fundamental matrix of the differential equation
$\displaystyle{\frac{dx_{r}}{dt}=A(t)x_{r}}$ on $[t_{r-1},t_{r}]$ can be
represented as $X_{r}(t)=X_{r}^{0}(t)\cdot C_{r},$ where $X_{r}^{0}(t)$ is the
normalized fundamental matrix $(X_{r}^{0}(t_{r-1})=I)$ and $C_{r}$ is an
arbitrary invertible matrix. Thus, the following equalities hold true

$G_{p,l}(\Delta_{m})=\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)X_{r}^{0}(\tau)C_{r}\int_{t_{r-1}}^{\tau}[X_{r}^{0}(\tau_{1})C_{r}]^{-1}\varphi_{l}(\tau_{1})d\tau_{1}d\tau=$
$=\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)X_{r}^{0}(\tau)\int_{t_{r-1}}^{\tau}[X_{r}^{0}(\tau_{1})]^{-1}\varphi_{l}(\tau_{1})d\tau_{1}d\tau,$

and the regularity of the $\Delta_{m}$ partition does not depend on the choice
of a fundamental matrix for the differential part of the equation.

Let us denote by $\sigma(k,[0,T])$ the set of regular partitions $\Delta_{m}$
of the interval $[0,T]$ for Eq. (2.1).

Definition 2.2 The special Cauchy problem (2.3),(2.4) is called uniquely
solvable if it has a unique solution for any $\lambda\in R^{nm}$ and $f(t)\in
C([0,T],R^{n})$ .

The special Cauchy problem (2.3), (2.4) is equivalent to the system of
integral equations (2.7). Since the kernel of (2.7) is degenerate, this
system, in turn, is equivalent to the system of algebraic equations (2.9) in
$\mu=(\mu_{1},\ldots,\mu_{k})\in R^{nk}.$ Therefore, the special Cauchy
problem is uniquely solvable if and only if the $\Delta_{m}$ partition,
generating this problem, is regular.

Since the special Cauchy problem is uniquely solvable for a partition with a
sufficiently small step size $h>0$ (see [13, p.1152]), the set
$\sigma(k,[0,T])$ is not empty.

Let us take a partition $\Delta_{m}\in\sigma(k,[0,T])$ and represent the
matrix $[I-G(\Delta_{m})]^{-1}$ in the form
$[I-G(\Delta_{m})]^{-1}=\Big{(}M_{j,p}(\Delta_{m})\Big{)},$
$j,p=\overline{1,k},$ where $M_{j,p}(\Delta_{m})$ are square matrices of
dimension $n.$

Then, taking into account (2.13), we can determine the elements of the vector
$\mu\in R^{nk}$ from the equalities

$\mu_{j}=\sum\limits_{i=1}^{m}\Big{(}\sum\limits_{p=1}^{k}M_{j,p}(\Delta_{m})V_{p,i}(\Delta_{m})\Big{)}\lambda_{i}+\sum\limits^{k}_{p=1}M_{j,p}(\Delta_{m})g_{p}(f,\Delta_{m}),\quad
j=\overline{1,k}.$ $None$

In (2.8), by replacing $\mu_{j}$ with the right-hand side of (2.14) we get the
representation of the functions $u_{r}(t)$ through $\lambda_{i},$
$i=\overline{1,m}:$

$u_{r}(t)=\sum\limits_{i=1}^{m}\biggl{\\{}\sum\limits_{j=1}^{k}X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\varphi_{j}(\tau)d\tau\biggl{[}\sum\limits_{p=1}^{k}M_{j,p}(\Delta_{m})V_{p,i}(\Delta_{m})+\int_{t_{i-1}}^{t_{i}}\psi_{j}(\tau_{1})d\tau_{1}\biggr{]}\biggr{\\}}\lambda_{i}+$
$+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)A(\tau)d\tau\lambda_{r}+$
$+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\biggl{[}\sum\limits_{j=1}^{k}\varphi_{j}(\tau)\sum\limits^{k}_{p=1}M_{j,p}(\Delta_{m})g_{p}(f,\Delta_{m})+f(\tau)\biggr{]}d\tau,\quad
t\in[t_{r-1},t_{r}),\ r=\overline{1,m}.$ $None$

We will use the following notation:

$D_{r,i}(\Delta_{m})=\sum\limits_{j=1}^{k}X_{r}(t_{r})\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\varphi_{j}(\tau)d\tau\biggl{[}\sum\limits_{p=1}^{k}M_{j,p}(\Delta_{m})V_{p,i}(\Delta_{m})+$
$+\int_{t_{i-1}}^{t_{i}}\psi_{j}(\tau_{1})d\tau_{1}\biggr{]},\qquad i\neq
r,\qquad r,j=\overline{1,m},$ $None$
$D_{r,r}(\Delta_{m})=\sum\limits_{j=1}^{k}X_{r}(t_{r})\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\varphi_{j}(\tau)d\tau\biggl{[}\sum\limits_{p=1}^{k}M_{j,p}(\Delta_{m})V_{p,r}(\Delta_{m})+\int_{t_{r-1}}^{t_{r}}\psi_{j}(\tau_{1})d\tau_{1}\biggr{]}+$
$+X_{r}(t_{r})\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)A(\tau)d\tau,$ $None$
$F_{r}(\Delta_{m})=X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\biggl{[}\sum\limits_{j=1}^{k}\varphi_{j}(\tau)\sum\limits^{k}_{p=1}M_{j,p}(\Delta_{m})g_{p}(f,\Delta_{m})+f(\tau)\biggr{]}d\tau,\quad
r=\overline{1,m}.$ $None$

From (2.15), we find the limits

$\lim\limits_{t\rightarrow
t_{r}-0}u_{r}(t)=\sum\limits_{i=1}^{m}D_{r,i}(\Delta_{m})\lambda_{i}+F_{r}(\Delta_{m}).$
$None$

Substituting the right-hand side of (2.19) into condition (2.5) and continuity
conditions (2.6), we get the following system of linear algebraic equations in
parameters $\lambda_{r},$ $r=\overline{1,m}$:

$\sum\limits^{m-2}_{i=0}[B_{i}+B_{m}D_{r,i+1}(\Delta_{m})]\lambda_{i+1}+[B_{m-1}+B_{m}+B_{m}D_{r,m}(\Delta_{m})]\lambda_{m}=d-B_{m}F_{m}(\Delta_{m}),$
$None$
$[I+D_{p,p}(\Delta_{m})]\lambda_{p}-[I-D_{p,p+1}(\Delta_{m})]\lambda_{p+1}+\sum\limits_{{}^{\,\,\,\,\,\,\,\,\,\,i=1}_{i\neq
p,\,i\neq p+1}}^{m}D_{p,i}(\Delta_{m})\lambda_{i}=-F_{p}(\Delta_{m}),\quad
p=\overline{1,m-1}.$ $None$

Let $Q_{\ast}(\Delta_{m})$ be the matrix corresponding to the left-hand side
of system (2.20), (2.21). Then the system (2.20), (2.21) can be written in the
form

$Q_{\ast}(\Delta_{m})\lambda=-F_{\ast}(\Delta_{m}),\quad\lambda\in R^{nm},$
$None$

where
$F_{\ast}(\Delta_{m})=\Big{(}-d+B_{m}F_{m}(\Delta_{m}),F_{1}(\Delta_{m}),\ldots,F_{m-1}(\Delta_{m})\Big{)}\in
R^{nm}.$

Lemma 2.1 The following statements hold true for
$\Delta_{m}\in\sigma(k,[0,T])$:
(a) the vector
$\lambda^{\ast}=(\lambda_{1}^{\ast},\lambda_{2}^{\ast},\ldots,\lambda_{m}^{\ast})\in
R^{nm},$ composed of the values of a solution $x^{\ast}(t)$ to problem
(2.1),(2.2) at the partition points, $\lambda_{r}^{\ast}=x^{\ast}(t_{r-1}),$
$r=\overline{1,m}$, satisfies system (2.22);
(b) if
$\widetilde{\lambda}=(\widetilde{\lambda}_{1},\widetilde{\lambda}_{2},\ldots,\widetilde{\lambda}_{m})\in
R^{nm}$ is a solution to system (2.22) and a function system
$\widetilde{u}[t]=(\widetilde{u}_{1}(t),\widetilde{u}_{2}(t),...,\widetilde{u}_{m}(t))$
is a solution to the special Cauchy problem (2.3), (2.4) with
$\lambda_{r}=\widetilde{\lambda}_{r},$ $r=\overline{1,m},$ then the function
$\widetilde{x}(t),$ defined as
$\widetilde{x}(t)=\widetilde{\lambda}_{r}+\widetilde{u}_{r}(t),$
$t\in[t_{r-1},t_{r}),$ $r=\overline{1,m},$
$\widetilde{x}(T)=\widetilde{\lambda}_{m}+\lim\limits_{t\rightarrow
T-0}\widetilde{u}_{m}(t),$ is a solution to problem (2.1),(2.2).

The proof of Lemma 2.1 repeats the proof of Lemma 1 in [6, p. 1155] with minor
changes.

We will use the following notation:

$\alpha=\max\limits_{t\in[0,T]}\|A(t)\|,\quad\overline{\omega}=\max\limits_{r=\overline{1,m}}(t_{r}-t_{r-1}),$
$\overline{\varphi}(k)=\max\limits_{r=\overline{1,m}}\int\limits_{t_{r-1}}^{t_{r}}\sum\limits_{j=1}^{k}\|\varphi_{j}(t)\|dt,\quad\overline{\psi}(T)=\max\limits_{p=\overline{1,k}}\int\limits_{0}^{T}\|\psi_{p}(t)\|dt.$

Theorem 2.1 Let $\Delta_{m}\in\sigma(k,[0,T])$ and the matrix
${Q}_{\ast}(\Delta_{m}):R^{nm}\to R^{nm}$ be invertible. Then problem
(2.1),(2.2) has a unique solution $x^{\ast}(t)$ for any $f(t)\in
C([0,T],R^{n}),$ $d\in R^{n},$ and the following estimate holds:

$\|x^{\ast}\|_{1}\leq\mathcal{N}(k,\Delta_{m})\max(\|d\|,\|f\|_{1}),$ $None$

where

$\mathcal{N}(k,\Delta_{m})=e^{\alpha\overline{\omega}}\Big{\\{}\overline{\varphi}(k)\Big{[}\|[I-G(\Delta_{m})]^{-1}\|\cdot\overline{\psi}(T)\Big{(}e^{\alpha\overline{\omega}}-1+e^{\alpha\overline{\omega}}\cdot\overline{\varphi}(k)\cdot\overline{\psi}(T)\Big{)}+\overline{\psi}(T)\Big{]}+1\Big{\\}}\times$
$\times\gamma_{\ast}(\Delta_{m})(1+\|C\|)\max\Big{\\{}1,\overline{\omega}e^{\alpha\overline{\omega}}\Big{[}1+e^{\alpha\overline{\omega}}\cdot\overline{\varphi}(k)\cdot\|[I-G(\Delta_{m})]^{-1}\|\cdot\overline{\psi}(T)\Big{\\}}+$
$+e^{\alpha\overline{\omega}}\overline{\omega}\Big{[}\overline{\varphi}(k)\cdot\|[I-G(\Delta_{m})]^{-1}\|\cdot\overline{\psi}(T)\cdot
e^{\alpha\overline{\omega}}+1\Big{]}.$ $None$

Proof. Take a partition $\Delta_{m}\in\sigma(k,[0,T])$. Let $f(t)\in
C([0,T],R^{n})$ and $d\in R^{n}$. Since the matrix ${Q}_{\ast}(\Delta_{m})$ is
invertible, we can find the unique solution to the system of linear algebraic
equations (2.22):

$\lambda^{\ast}=-[Q_{\ast}(\Delta_{m})]^{-1}F_{\ast}(\Delta_{m}).$

By solving the special Cauchy problem (2.3), (2.4) with
$\lambda=\lambda^{\ast}$, we get the function system
$u^{\ast}[t]=(u^{\ast}_{1}(t),u^{\ast}_{2}(t),\ldots,u^{\ast}_{m}(t)).$ It
follows from the regularity of the $\Delta_{m}$ partition that there exists a
unique function system $u^{\ast}[t]$ with the elements $u^{\ast}_{r}(t)$ that
are determined from the right-hand side of representation (2.15) with
$\lambda=\lambda^{\ast}=(\lambda^{\ast}_{1},\lambda^{\ast}_{2},\ldots,\lambda^{\ast}_{m})\in
R^{nm}.$ Then, by Lemma 2.1, the function $x^{\ast}(t)$ defined as
$x^{\ast}(t)=\lambda_{r}^{\ast}+u_{r}^{\ast}(t),$ $\ t\in[t_{r-1},t_{r}),$
$r=\overline{1,m},$ $x^{\ast}(T)=\lambda_{m}^{\ast}+\lim\limits_{t\rightarrow
T-0}u_{m}^{\ast}(t),$ is a solution to problem (2.1), (2.2). The uniqueness of
a solution can be proved by contradiction.

Let us verify the validity of the estimate (2.23).

Using the equalities

$X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)P(\tau)d\tau=\int_{t_{r-1}}^{t}P(\tau_{1})d\tau_{1}+\int_{t_{r-1}}^{t}A(\tau_{1})\int_{t_{r-1}}^{\tau_{1}}P(\tau_{2})d\tau_{2}d\tau_{1}+$
$+\int_{t_{r-1}}^{t}A(\tau_{1})\int_{t_{r-1}}^{\tau_{1}}A(\tau_{2})\int_{t_{r-1}}^{\tau_{2}}P(\tau_{3})d\tau_{3}d\tau_{2}d\tau_{1}+\ldots,\quad
t\in[t_{r-1},t_{r}],$

we get the estimates

$\|X_{r}(t_{r})\int_{t_{r-1}}^{t_{r}}X_{r}^{-1}(\tau)\varphi_{j}(\tau)d\tau\|\leq
e^{\alpha(t_{r}-t_{r-1})}\int_{t_{r-1}}^{t_{r}}\|\varphi_{j}(t)\|dt,\quad
r=\overline{1,m}.$ $None$

It follows from (2.12), (2.18), (2.25) that

$\|g_{p}(f,\Delta_{m})\|\leq\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\|\psi_{p}(\tau)\|\cdot\|X_{r}(\tau)\int_{t_{r-1}}^{\tau}X_{r}^{-1}(\tau_{1})f(\tau_{1})d\tau_{1}\|d\tau\leq$
$\leq\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\|\psi_{p}(\tau)\|d\tau\cdot
e^{\alpha\overline{\omega}}\cdot\overline{\omega}\cdot\|f\|_{1}=\int_{0}^{T}\|\psi_{p}(t)\|dt\cdot
e^{\alpha\overline{\omega}}\cdot\overline{\omega}\cdot\|f\|_{1},\quad
p=\overline{1,k},$ $None$ $\|F_{r}(\Delta_{m})\|\leq
e^{\alpha(t_{r}-t_{r-1})}\sum\limits_{j=1}^{k}\int_{t_{r-1}}^{t_{r}}\|\psi_{j}(t)\|dt\|[I-G(\Delta_{m})]^{-1}\|\max\limits_{p=\overline{1,k}}\|g_{p}(f,\Delta_{m})\|+e^{\alpha(t_{r}-t_{r-1})}\omega_{r}\|f\|_{1}.$

By using

$\|F_{\ast}(\Delta_{m})\|\leq(1+\|B_{m}\|)\max\Bigl{(}\|d\|,\max\limits_{r=\overline{1,m}}\|F_{r}(\Delta_{m})\|\Bigr{)},$

and taking into account (2.16),(2.25), and (2.26), we get

$\|F_{\ast}(\Delta_{m})\|\leq(1+\|B_{m}\|)\max\Big{\\{}1,\overline{\omega}e^{\alpha\overline{\omega}}\Big{[}1+$
$+e^{\alpha\overline{\omega}}\max\limits_{r=\overline{1,m}}\int_{t_{r-1}}^{t_{r}}\sum\limits_{j=1}^{k}\|\varphi_{j}(t)\|dt\|[I-G(\Delta_{m})]^{-1}\|\max\limits_{p=\overline{1,k}}\int_{0}^{T}\|\psi_{p}(t)\|dt\Big{]}\Big{\\}}\max(\|d\|,\|f\|_{1}).$
$None$

The inequalities (2.22), (2.27) and the invertibility of
$Q_{\ast}(\Delta_{m})$ yield the following estimate:

$\|\lambda^{\ast}\|\leq\|[Q_{\ast}(\Delta_{m})]^{-1}\|\|F_{\ast}(\Delta_{m})\|\leq\gamma_{\ast}(\Delta_{m})(1+\|B_{m}\|)\max\Big{\\{}1,\overline{\omega}e^{\alpha\overline{\omega}}\Big{[}1+$
$+e^{\alpha\overline{\omega}}\max\limits_{r=\overline{1,m}}\int_{t_{r-1}}^{t_{r}}\sum\limits_{j=1}^{k}\|\varphi_{j}(t)\|dt\|[I-G(\Delta_{m})]^{-1}\|\max\limits_{p=\overline{1,k}}\int_{0}^{T}\|\psi_{p}(t)\|dt\Big{\\}}\max(\|d\|,\|f\|_{1}).$
$None$

By (2.15) and (2.11), we get

$\|u^{\ast}[\cdot]\|_{2}\leq\biggl{\\{}e^{\alpha\overline{\omega}}\max\limits_{r=\overline{1,m}}\int_{t_{r-1}}^{t_{r}}\sum\limits_{j=1}^{k}\|\varphi_{j}(t)\|dt\|[I-G(\Delta_{m})]^{-1}\|\max\limits_{p=\overline{1,k}}\int_{0}^{T}\|\psi_{p}(t)\|dt\Big{(}e^{\alpha\overline{\omega}}-1+$
$+e^{\alpha\overline{\omega}}\max\limits_{r=\overline{1,m}}\int_{t_{r-1}}^{t_{r}}\max\limits_{j=\overline{1,k}}\|\varphi_{j}(t)\|dt\max\limits_{p=\overline{1,k}}\int_{0}^{T}\|\psi_{p}(t)\|dt\Big{)}+\max\limits_{p=\overline{1,k}}\int_{0}^{T}\|\psi_{p}(t)\|dt\Big{]}+(e^{\alpha\overline{\omega}}-1)\biggr{\\}}\|\lambda\|+$
$+e^{\alpha\overline{\omega}}\overline{\omega}\Big{[}\max\limits_{r=\overline{1,m}}\int_{t_{r-1}}^{t_{r}}\sum\limits_{j=1}^{k}\|\varphi_{j}(t)\|dt\|[I-G(\Delta_{m})]^{-1}\|\max\limits_{p=\overline{1,k}}\int_{0}^{T}\|\psi_{p}(t)\|dt\cdot
e^{\alpha\overline{\omega}}+1\Big{]}\cdot\|f\|_{1}.$ $None$

Finally, by using (2.28), (2.29) and
$\|x^{\ast}\|_{1}\leq\|\lambda^{\ast}\|+\|u^{\ast}[\cdot]\|_{2}$, we arrive at
the estimate (2.23).

Theorem 2.1 is proved.

Definition 2.3 Problem (2.1), (2.2) is said to be well-posed if, for any pair
$(f(t),d),$ with $f(t)\in C([0,T],R^{n})$ and $d\in R^{n}$, it has a unique
solution $x(t)$, and the estimate

$\|x\|_{1}\leq K\max(\|f\|_{1},\|d\|)$

holds, where $K$ is a constant independent of $f(t)$ of $d.$

Theorem 2.2 Problem (2.1),(2.2) is well-posed if and only if the matrix
$Q_{\ast}(\Delta_{m}):$ $R^{nm}\rightarrow R^{nm}$ is invertible for any
partition $\Delta_{m}\in\sigma(k,[0,T])$.

Proof. For a fixed $k$ and $\Delta_{m}\in\sigma(k,[0,T])$ the number
$\mathcal{N}(k,\Delta_{m})$, defined by (2.24), does not depend on $f(t)$ and
$d.$ Thus the sufficiency of the conditions of Theorem 2.2 for the well-
posedness of problem (2.1), (2.2) follows from Theorem 2.1.

Necessity. Let problem (2.1), (2.2) be well-posed and
$\Delta_{m}\in\sigma(k,[0,T]).$ Suppose to the contrary that the matrix
$Q_{\ast}(\Delta_{m}):$ $R^{nm}\rightarrow R^{nm}$ is not invertible. This is
possible only if the homogeneous system of equtions

$Q_{\ast}(\Delta_{N})\lambda=0,\quad\lambda\in R^{nN},$ $None$

has a nonzero solution. Assuming that
$\widetilde{\lambda}=(\widetilde{\lambda}_{1},\widetilde{\lambda}_{2},\ldots,\widetilde{\lambda}_{m})$
is a nonzero solution (i.e. $\|\widetilde{\lambda}\|\neq 0$) to system (2.30),
consider the homogeneous problem (2.1), (2.2) with $f(t)=0$ and $d=0.$ For
this problem, system (2.22) coincides with (2.30). Then, by Lemma 2.1, the
function $\widetilde{x}(t)$ defined as
$\widetilde{x}(t)=\widetilde{\lambda}_{r}+\widetilde{u}_{r}(t),$ $\
t\in[t_{r-1},t_{r}),$ $r=\overline{1,m},$
$\widetilde{x}(T)=\widetilde{\lambda}_{m}+\lim\limits_{t\rightarrow
T-0}\widetilde{u}_{m}(t),$ is a nonzero solution to the homogeneous problem.
Here the function system
$\widetilde{u}[t]=(\widetilde{u}(t),\widetilde{u}_{2}(t),\ldots,\widetilde{u}_{m}(t))$
is a solution to the special Cauchy problem (2.3), (2.4) with
$\lambda=\widetilde{\lambda}$ and $f(t)=0.$ This contradicts the well-
posedness of problem (2.1), (2.2).

Theorem 2.2 is proved.

3. An algorithm for solving multipoint problems for Fredholm integro-differential equations with degenerate kernel and its numerical implementation

The Cauchy problems for ordinary differential equations on the subintervals

$\frac{dx}{dt}=A(t)x+P(t),\qquad x(t_{r-1})=0,\quad t\in[t_{r-1},t_{r}],\quad
r=\overline{1,m},$ $None$

are an essential part of the algorithm proposed. Here $P(t)$ is an $(n\times
n)$ matrix or $n$ vector that is continuous on $[t_{r-1},t_{r}],$
$r=\overline{1,m}$. Hence, a solution to problem (3.1) is a matrix or a vector
of dimension $n$.

Let $E_{\ast,r}(A(\cdot),P(\cdot),t)$ denote a solution to the Cauchy problem
(3.1). Clearly,

$E_{\ast,r}(A(\cdot),P(\cdot),t)=X_{r}(t)\int_{t_{r-1}}^{t}X^{-1}(\tau)P(\tau)d\tau,\quad
t\in[t_{r-1},t_{r}],$ $None$

where $X_{r}(t)$ is a fundamental matrix of differential equation (3.1) on the
$r$th subinterval.

The choice of a regular partition is another important part of the algorithm.
We can start with $\Delta_{1},$ when the interval $[0,T]$ is not partitioned.

I. We divide $[0,T]$ into $m$ parts by the points
$t_{0}=0<t_{1}<\ldots<t_{m-1}<t_{m}=T,$ involved in the multipoint condition.
The resulting partition we denote by $\Delta_{m},m=1,2,\ldots$.

II. By solving $mk$ Cauchy problems for ordinary differential matrix equations

$\frac{dx}{dt}=A(t)x+\varphi_{k}(t),\qquad x(t_{r-1})=0,\quad
t\in[t_{r-1},t_{r}],$ $None$

we obtain the matrix functions

$E_{\ast,r}(A(\cdot),\varphi_{j}(\cdot),t),\quad t\in[t_{r-1},t_{r}],\quad
r=\overline{1,m},\quad j=\overline{1,k}.$ $None$

III. We multiply each $(n\times n)$ matrix (3.4) by $(n\times n)$ matrix
$\psi_{p}(t),$ $p=\overline{1,k},$ and integrate the products over
$[t_{r-1},t_{r}]:$

$\widehat{\psi}_{p,r}(\varphi_{j})=\int_{t_{r-1}}^{t_{r}}\psi_{p}(t)E_{\ast,r}(A(\cdot),\varphi_{j}(\cdot),t)dt.$
$None$

Summing up (3.5) with respect to $r$ and taking into account (2.10), (3.2), we
get the $(n\times n)$ matrices

$G_{p,j}(\Delta_{m})=\sum\limits_{r=1}^{m}\widehat{\psi}_{p,r}(\varphi_{j}),\quad
p,j=\overline{1,k}.$

We then construct the $(nk\times nk)$ matrix
$G(\Delta_{m})=(G_{p,j}(\Delta_{m})),$ $p,j=\overline{1,k},$ and determine
whether the matrix $[I-G(\Delta_{m})]:R^{nk}\rightarrow R^{nk}$ is invertible.
If so, we find its inverse and represent it as
$[I-G(\Delta_{m})]^{-1}=(M_{p,j}(\Delta_{m})),$ where $M_{p,j}(\Delta_{m}))$
are $(n\times n)$ matrices, $p,j=\overline{1,k}$. We then move on to the next
step of the algorithm.

If $[I-G(\Delta_{m})]$ is not invertible, i.e. the $\Delta_{m}$ partition is
not regular, then we take a new partition of $[0,T]$ and start the algorithm
again. A simple way to choose a new partition is to take $\Delta_{2m},$
dividing each subinterval $\Delta_{m}$ in half. We add to the points $t=t_{i}$
of the multipoint condition the points $(t_{i}-t_{i-1})/2$,
$i=\overline{1,m}$. Then, redesignating all points as $theta_{0}=t_{0}=0,$
$\theta_{1}=(t_{1}-t_{0})/2$, $\theta_{2}=t_{1}$,
$\theta_{3}=(t_{2}-t_{1})/2$, $\theta_{4}=t_{2}$, …,
$\theta_{2m-1}=(t_{m}-t_{m-1})/2$, $\theta_{2m}=t_{m}=T$, we again get problem
(2.1), (2.2) with multipoint conditions at the points $t=\theta_{i}$,
$i=\overline{0,2m}$.

IV. Solving again the Cauchy problems for ordinary differential equations

$\frac{dx}{dt}=A(t)x+A(t),\quad x(t_{r-1})=0,\quad t\in[t_{r-1},t_{r}],$
$\frac{dx}{dt}=A(t)x+f(t),\quad x(t_{r-1})=0,\quad t\in[t_{r-1},t_{r}],\quad
r=\overline{1,m},$

we obtain $E_{\ast,r}(A(\cdot),A(\cdot),t)$ and
$E_{\ast,r}(A(\cdot),f(\cdot),t),$ $r=\overline{1,m}.$

V. We evaluate the integrals

$\widehat{\psi}_{p,r}=\int_{t_{r-1}}^{t_{r}}\psi_{p}(t)dt,\quad\widehat{\psi}_{p,r}(A)=\int_{t_{r-1}}^{t_{r}}\psi_{p}(t)E_{\ast,r}(A(\cdot),A(\cdot),t)dt,$
$\widehat{\psi}_{p,r}(f)=\int_{t_{r-1}}^{t_{r}}\psi_{p}(t)E_{\ast,r}(A(\cdot),f(\cdot),t)dt.$

From (2.11), (2.12), and (3.2) we determine the $(n\times n)$ matrices

$V_{p,r}(\Delta_{m})=\widehat{\psi}_{p,r}(A)+\sum\limits_{i=1}^{m}\sum\limits_{k=1}^{m}\widehat{\psi}_{p,i}(\varphi_{j})\cdot\widehat{\psi}_{j,r}$

and the $n$ vectors

$g_{p}(f,\Delta_{m})=\sum\limits_{r=1}^{m}\widehat{\psi}_{p,r}(\Delta_{m}),\quad
p=\overline{1,k},\quad r=\overline{1,m}.$

VI. We construct the system of linear algebraic equations in parameters

$Q_{\ast}(\Delta_{m})\lambda=-F_{\ast}(\Delta_{m}),\qquad\lambda\in R^{nm}.$
$None$

The elements of the matrix $Q_{\ast}(\Delta_{m})$ and the vector
$F_{\ast}(\Delta_{m})=(-d+B_{m}F_{m}(\Delta_{m}),F_{1}(\Delta_{m}),\ldots,$
$F_{m-1}(\Delta_{m}))\in R^{nm}$ are determined by the equalities (2.16),
(2.17), (2.18), where, by (3.2), we replace
$X_{r}(t_{r})\int_{t_{r-1}}^{t_{r}}X_{r}^{-1}(\tau)\varphi_{j}(\tau)d\tau$ and
$X_{r}(t_{r})\int_{t_{r-1}}^{t_{r}}X_{r}^{-1}(\tau)f(\tau)d\tau$ with
$E_{\ast,r}(A(\cdot),\varphi_{j}(\cdot),t_{r})\ $ and $\
E_{\ast,r}(A(\cdot),f(\cdot),t_{r}),$ respectively.

It follows from Theorem 2.2 that the invertibility of the matrix
$Q_{\ast}(\Delta_{m})$ is equivalent to the well-posedness of problem
(2.1),(2.2).

By solving system (3.6), we get
$\lambda^{\ast}=(\lambda_{1}^{\ast},\lambda_{2}^{\ast},\ldots,\lambda_{m}^{\ast})\in
R^{nm}.$

VII. From the equalities

$\mu^{\ast}_{s}=\sum\limits_{j=1}^{m}\Big{(}\sum\limits_{p=1}^{k}M_{s,p}(\Delta_{m})V_{p,j}(\Delta_{m})\Big{)}\lambda^{\ast}_{j}+\sum\limits_{p=1}^{k}M_{s,p}(\Delta_{m})g_{p}(f,\Delta_{m})$
$None$

we determine the components
$\mu^{\ast}=(\mu^{\ast}_{1},\mu^{\ast}_{2},\ldots,\mu^{\ast}_{k})\in R^{nk}$
and construct the function

$\mathcal{F}^{\ast}(t)=\sum\limits_{s=1}^{k}\varphi_{s}(t)\Big{[}\mu^{\ast}_{s}+\sum\limits_{r=1}^{m}\int_{t_{r-1}}^{t_{r}}\psi_{s}(\tau)d\tau\lambda^{\ast}_{r}\Big{]}+f(t).$
$None$

Recall that $\lambda_{r}^{\ast}=x^{\ast}(t_{r-1}),$ where $x^{\ast}(t)$ is a
solution to problem (2.1), ( 2.2). Hence, by solving system (3.6), we get the
values of the desired solution at the left endpoints of the partition
subintervals.

In order to determine the values of the function $x^{\ast}(t)$ at the
remaining points of the subintervals $[t_{r-1},t_{r})$, we solve the following
Cauchy problems for the ordinary differential equation:

$\frac{dx}{dt}=A(t)x+\mathcal{F}^{\ast}(t),\quad
x(t_{r-1})=\lambda_{r}^{\ast},\quad t\in[t_{r-1},t_{r}),\quad
r=\overline{1,m}.$

Thus, the proposed algorithm contains seven interrelated parts.

If the fundamental matrices $X_{r}(t),$ $r=\overline{1,m},$ are known, then
the equalities (2.16),(2.17), and (2.18) enable us to construct the system
(3.6). Let
$\lambda^{\ast}=(\lambda_{1}^{\ast},\lambda_{2}^{\ast},\ldots,\lambda_{m}^{\ast})\in
R^{nm}$ be a solution to (3.6). Then, using (3.7) and (3.8), we construct the
function $\mathcal{F}^{\ast}(t)$ and determine a solution to problem (2.1),
(2.2) by the equalities

$x^{\ast}(t)=X_{r}(t)X_{r}^{-1}(t_{r-1})\lambda_{r}^{\ast}+X_{r}(t)\int_{t_{r-1}}^{t}X_{r}^{-1}(\tau)\mathcal{F}^{\ast}(\tau)d\tau,\quad
t\in[t_{r-1},t_{r}),\quad r=\overline{1,m},$ $None$
$x^{\ast}(T)=X_{m}(T)X_{m}^{-1}(t_{m-1})\lambda_{m}^{\ast}+X_{m}(T)\int_{t_{m-1}}^{T}X_{m}^{-1}(\tau)\mathcal{F}^{\ast}(\tau)d\tau.$
$None$

So, in this case the proposed algorithm provides the solution to the linear
multipoint boundary value problem for integro-differential equations (2.1),
(2.2) in the form (3.9), (3.10).

As is known, it is not always possible to construct a fundamental matrix for a
system of ordinary differential equations with variable coefficients. For this
reason, we propose the following numerical implementation of the algorithm
that is based on the fourth-order Runge-Kutta method and Simpson’s rule.

I. Let us take a partition $\Delta_{m}:$
$t_{0}=0<t_{1}<\ldots<t_{m-1}<t_{m}=T$. We divide each $r$th subinterval
$[t_{r-1},t_{r}],$ $r=\overline{1,m},$ into $m_{r}$ parts with the step size
$h_{r}=(t_{r}-t_{r-1})/m_{r}.$

Suppose that on each subinterval $[t_{r-1},t_{r}]$ a variable $\widehat{t}$
takes on discrete values: $\widehat{t}=t_{r-1},$ $\widehat{t}=t_{r-1}+h_{r},$
$\ldots,$ $\widehat{t}=t_{r-1}+(m_{r}-1)h_{r},$ $\widehat{t}=t_{r}$. Let
$\\{t_{r-1},t_{r}\\}$ denote the set of such points.

II. Using the fourth-order Runge-Kutta method, we obtain numerical solutions
to the Cauchy problems (3.1) and determine the values of the $(n\times n)$
matrix $E_{\ast,r}^{h_{r}}(A(\cdot),\varphi_{j}(\cdot),\widehat{t})$ on the
set $\\{t_{r-1},t_{r}\\}$, $r=\overline{1,m}$, $j=\overline{1,k}.$

III. Using the values of the $(n\times n)$ matrices $\psi_{j}(s)$ and
$E_{\ast,r}^{h_{r}}\Big{(}A(\cdot),\varphi_{j}(\cdot),\widehat{t}\Big{)}$ on
$\\{t_{r-1},t_{r}\\}$ and applying Simpson’s rule, we determine the $(n\times
n)$ matrices

$\widehat{\psi}_{p,r}^{h_{r}}(\varphi_{j})=\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)E_{\ast,r}^{h_{r}}(A(\cdot),\varphi_{j}(\cdot),\tau)d\tau,\quad
p,j=\overline{1,k},\quad r=\overline{1,m}.$

Summing up the matrices $\widehat{\varphi}_{p,r}^{h_{r}}(\psi_{j})$ with
respect to $r$, we get the $(n\times n)$ matrices
$G_{p,j}^{\widetilde{h}}(\Delta_{m})=\sum\limits_{r=1}^{m}\widehat{\varphi}_{p,r}^{h_{r}}(\psi_{j})$,
where $\widetilde{h}=(h_{1},h_{2},\ldots,h_{m})\in R^{n}$. We then construct
the $(nk\times nk)$ matrix
$G^{\widetilde{h}}(\Delta_{m})=(G_{p,j}^{\widetilde{h}}(\Delta_{m})),$
$p,j=\overline{1,k}$.

Determine whether the matrix
$[I-G^{\widetilde{h}}(\Delta_{m})]:R^{nk}\rightarrow R^{nk}$ is invertible. If
so, we find
$[I-G^{\widetilde{h}}(\Delta_{m})]^{-1}=(M_{p,j}^{\widetilde{h}}(\Delta_{m})),$
$p,j=\overline{1,k}.$

In the case $[I-G^{\widetilde{h}}(\Delta_{m})]$ is not invertible, we choose a
new partition. In particular, as shown above, each subinterval can be divided
in half.

IV. Solving the Cauchy problem (3.5), (3.6) by the fourth-order Runge-Kutta
method, we get the values of the $(n\times n)$ matrix
$E_{\ast,r}(A(\cdot),A(\cdot),\widehat{t})$ and $n$ vector
$E_{\ast,r}(A(\cdot),f(\cdot),\widehat{t})$ on $\\{t_{r-1},t_{r}\\},$
$r=\overline{1,m}.$

V. By using Simpson’s rule on the grid $\\{t_{r-1},t_{r}\\}$, we evaluate the
definite integrals

$\widehat{\psi}_{p,r}^{h_{r}}=\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)d\tau,\quad\widehat{\psi}_{p,r}^{h_{r}}(A)=\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)E_{\ast,r}^{h_{r}}(A(\cdot),A(\cdot),\tau)d\tau,$
$\widehat{\psi}_{p,r}^{h_{r}}(f)=\int_{t_{r-1}}^{t_{r}}\psi_{p}(\tau)E_{\ast,r}^{h_{r}}(A(\cdot),f(\cdot),\tau)d\tau,\quad
r=\overline{1,N},\quad p=\overline{1,m}.$

We the determine the $(n\times n)$ matrices
$V_{p,r}^{\widetilde{h}}(\Delta_{m})$ and $n$ vectors
$g_{p}^{\widetilde{h}}(f,\Delta_{m})$, $r=\overline{1,m}$, $p=\overline{1,k}$,
by the equalities

$V_{p,r}^{\widetilde{h}}(\Delta_{m})=\widehat{\psi}_{p,r}^{h_{r}}(A)+\sum\limits_{j=1}^{m}\sum\limits_{i=1}^{k}\widehat{\psi}_{p,j}^{h_{j}}(\varphi_{i})\cdot\widehat{\psi}_{i,r}^{h_{r}},\quad
g_{p}^{\widetilde{h}}(f,\Delta_{m})=\sum\limits_{r=1}^{m}\widehat{\psi}_{p,r}^{h_{r}}(f).$

VI. We construct the system of linear algebraic equations in parameters

$Q_{\ast}^{\widetilde{h}}(\Delta_{m})\lambda=-F_{\ast}^{\widetilde{h}}(\Delta_{m}),\qquad\lambda\in
R^{nm}.$ $None$

The elements of the matrix $Q_{\ast}^{\widetilde{h}}(\Delta_{m})$ and the
vector
$F_{\ast}^{\widetilde{h}}(\Delta_{m})=(-d+B_{m}F_{m}^{\widetilde{h}}(\Delta_{m}),F_{1}^{\widetilde{h}}(\Delta_{m}),\ldots,F_{m-1}^{\widetilde{h}}(\Delta_{m}))$
are determined by the equalities

$D_{r,i}^{\widetilde{h}}(\Delta_{m})=\sum\limits_{j=1}^{k}E_{\ast,r}^{h_{r}}(A(\cdot),\varphi_{j}(\cdot),t_{r})\Big{[}\sum\limits_{p=1}^{k}M_{j,p}^{\widetilde{h}}(\Delta_{m})V_{p,i}^{\widetilde{h}}(\Delta_{m})+\widehat{\psi}_{j,i}^{h_{i}}\Big{]},\quad
i\neq r,\quad r,i=\overline{1,m},$
$D_{r,r}^{\widetilde{h}}(\Delta_{m})=\sum\limits_{j=1}^{k}E_{\ast,r}^{h_{r}}(A(\cdot),\varphi_{j}(\cdot),t_{r})\Big{[}\sum\limits_{p=1}^{k}M_{j,p}^{\widetilde{h}}(\Delta_{m})V_{p,r}^{\widetilde{h}}(\Delta_{m})+\widehat{\psi}_{j,r}^{h_{r}}\Big{]}+E_{\ast,r}^{h_{r}}(A(\cdot),A(\cdot),t_{r}),\quad
r=\overline{1,m},$
$F_{r}^{\widetilde{h}}(\Delta_{m})=\sum\limits_{j=1}^{k}E_{\ast,r}^{h_{r}}(A(\cdot),\varphi_{j}(\cdot),t_{r})\sum\limits_{p=1}^{k}M_{j,p}^{\widetilde{h}}(\Delta_{m})g_{p}^{\widetilde{h}}(\Delta_{m})+E_{\ast,r}^{h_{r}}(A(\cdot),f(\cdot),t_{r}),\quad
r=\overline{1,m}.$

Using the constructed matrix $(Q_{\ast}^{\widetilde{h}}(\Delta_{m}),$ we can
establish the well-posedness of problem (2.1), (2.2). Suppose the matrix
$Q_{\ast}^{\widetilde{h}}(\Delta_{m})$ is invertible and the estimate
$||Q_{\ast}(\Delta_{m})-Q_{\ast}^{\widetilde{h}}(\Delta_{m})||\leq\varepsilon(\widetilde{h})$
holds. If the inequality
$||[Q_{\ast}^{\widetilde{h}}(\Delta_{m})]^{-1}||\cdot\varepsilon(\widetilde{h})<1$
is true, then, by Theorem 4 [11, p.212], the matrix $Q_{\ast}(\Delta_{m})$ is
invertible. It follows then from Theorem 2.2 that problem (2.1),(2.2) is well-
posed.

By solving (3.11) we determine $\lambda^{\widetilde{h}}\in R^{nm}.$ As noted
above, the elements
$\lambda^{\widetilde{h}}=(\lambda^{\widetilde{h}}_{1},\lambda^{\widetilde{h}}_{2},\ldots,\lambda^{\widetilde{h}}_{m})$
are the values of an approximate solution to problem (2.1), (2.2) at the left
endpoints of the subintervals:
$x^{\widetilde{h}_{r}}(t_{r-1})=\lambda^{\widetilde{h}}_{r},$
$r=\overline{1,m}.$

VII. In order to calculate the values of the approximate solution at the
remaining points of the set $\\{t_{r-1},t_{r}\\}$, we first find

$\mu_{i}^{\widetilde{h}}=\sum\limits_{j=1}^{m}\Big{(}\sum\limits_{p=1}^{k}M_{i,p}^{\widetilde{h}}(\Delta_{m})V_{p,j}^{\widetilde{h}}(\Delta_{m})\Big{)}\lambda_{i}^{h}+\sum\limits_{p=1}^{k}M_{i,p}^{\widetilde{h}}(\Delta_{m})g_{p}^{\widetilde{h}}(f,\Delta_{m}),\quad
i=\overline{1,k},$

and then, using the fourth-order Runge-Kutta method, solve the Cauchy problems

$\frac{dx}{dt}=A(t)x+\mathcal{F}^{\widetilde{h}}(t),\quad
x(t_{r-1})=\lambda_{r}^{\widetilde{h}},\quad t\in[t_{r-1},t_{r}],\quad
r=\overline{1,m},$

where

$\mathcal{F}^{\widetilde{h}}(t)=\sum\limits_{i=1}^{k}\varphi_{i}(t)\Big{(}\mu_{i}^{\widetilde{h}}+\sum\limits_{j=1}^{m}\widehat{\psi}_{i,j}^{{h}_{j}}\lambda_{j}^{h}\Big{)}+f(t).$

Thus the algorithm allows us to find a numerical solution to problem (2.1),
(2.2).

4. A multipoint problem for an integro-differential equation with degenerate kernel

Let us now turn to the original multipoint problem (1.1), (1.2). To solve the
problem, we will approximate the kernel of the integral summand by a
degenerate kernel [5, 6, 8, 10].

By the Weierstrass polynomial approximation theorem, for any $\varepsilon>0$
there exist a number $k=k(\varepsilon)$ and continuous on $[0,T]$ matrices
$\varphi_{j}(t),$ $\psi_{j}(\tau),$ $j=\overline{1,k}$, such that the
following inequality holds

$\max\limits_{t\in[0,T]}\int\limits_{0}^{T}\|K(t,\tau)-\sum\limits_{j=1}^{k}\varphi_{j}(t)\psi_{j}(\tau)\|d\tau<\varepsilon.$
$None$

The set of matrices $\\{\varphi_{j}(t),\psi_{j}(\tau),j=\overline{1,m}\\},$
satisfying (4.1), we will call the $\varepsilon$-approximating set for
$K(t,\tau)$. The multipoint problem with degenerate kernel (2.1), (2.2),
corresponding to (1.1),(1.2), we will call the $\varepsilon$-approximating
problem for problem (1.1), (1.2).

Assuming the $\varepsilon$-approximating multipoint problem (2.1), (2.2) to be
well-posed with constant $C_{k}$, we find the solution to problem (1.1), (1.2)
according to the following algorithm.

Step 0. By solving problem (2.1), (2.2) we get a function $x^{(0)}(t)$, which
we take as an initial approximation to the solution to problem (1.1), (1.2).

Step 1. Using $x^{(0)}(t)$ and solving the $\varepsilon$-approximating problem

$\frac{dx}{dt}=A(t)x+\sum\limits_{j=1}^{k}\varphi_{j}(t)\int^{T}_{0}\psi_{j}(\tau)x(\tau)d\tau+f(t)+$
$+\int^{T}_{0}\Bigl{[}K(t,\tau)-\sum\limits_{j=1}^{k}\varphi_{j}(t)\psi_{j}(\tau)\Bigr{]}x^{(0)}(\tau)d\tau,\qquad
t\in(0,T),$ $None$ $\sum\limits^{m}_{i=0}B_{i}x(t_{i})=d,\qquad d\in R^{n},$
$None$

we get the function $x^{(1)}(t)$.

Continue the algorithm, in the $i$th step $(i=1,\ldots)$ we solve the problem

$\frac{dx}{dt}=A(t)x+\sum\limits_{j=1}^{k}\varphi_{j}(t)\int^{T}_{0}\psi_{j}(\tau)x(\tau)d\tau+f(t)+$
$+\int^{T}_{0}\Bigl{[}K(t,\tau)-\sum\limits_{j=1}^{k}\varphi_{j}(t)\psi_{j}(\tau)\Bigr{]}x^{(i-1)}(\tau)d\tau,\qquad
t\in(0,T),$ $None$ $\sum\limits^{m}_{i=0}B_{i}x(t_{i})=d,$ $None$

and get the function $x^{(i)}(t)$.

The well-posedness of the approximating problem ensures the feasibility of the
algorithm and allows us to construct the sequence $(x^{(i)}(t)),$
$i=0,1,\ldots.$

The following assertion provides conditions for the convergence of the
algorithm to the unique solution of multipoint problem (1.1), (1.2) and the
estimates for the difference between the exact and approximate solutions to
the problem.

Theorem 4.1 Let the $\varepsilon$-approximating problem (2.1), (2.2) be well-
posed with constant $C_{k}$. Suppose that the following inequality holds:

$q_{m}^{\varepsilon}=K_{m}\cdot\varepsilon<1.$ $None$

Then the algorithm converges to $x^{\ast}(t)$ and the estimate

$\|x^{\ast}-x^{(i)}\|_{1}\leq\frac{1}{1-q_{k}^{\varepsilon}}(q_{k}^{\varepsilon})^{i}\cdot
C_{k}\max(\|f\|_{1},\|d\|)$ $None$

is valid, where $x^{\ast}(t)$ and $x^{(i)}(t)$ are the unique solutions to
problems (1.1), (1.2) and (4.4), (4.5), respectively.

Proof. By assumption, there exists a unique solution to problem (2.1),(2.2)
and it satisfies the inequality

$\|x^{(0)}\|_{1}\leq C_{k}\max(\|f\|_{1},\|d\|).$

By solving problem (4.2), (4.3), we get $x^{(1)}(t).$ The difference $\Delta
x^{(1)}(t)=x^{(1)}(t)-x^{(0)}(t)$ satisfies the inequality

$\|\Delta x^{(1)}\|_{1}\leq
C_{k}\cdot\max\limits_{t\in[0,T]}\int_{0}^{T}\|K(t,s)-\sum\limits_{j=1}^{k}\varphi_{j}(t)\psi_{j}(\tau)\|d\tau\|x^{(0)}\|_{1}\leq
C_{k}\cdot\varepsilon\cdot C_{k}\cdot\max(\|f\|_{1},\|d\|).$ $None$

In the same way, by solving problem (4.4), (4.5), we get $x^{(i)}(t)$, and for
the difference $\Delta x^{(i)}(t)=x^{(i)}(t)-x^{(i-1)}(t)$ we have

$\|\Delta x^{(i)}\|_{1}\leq C_{k}\cdot\varepsilon\cdot\|\Delta
x^{(i-1)}\|_{1}=q_{k}^{\varepsilon}\|\Delta x^{(i-1)}\|_{1},\quad
i=2,3,\ldots.$ $None$

The convergence of the sequence $(x^{(i)}(t)),$ $i=0,1,\ldots$ to the solution
$x^{\ast}(t)$ of problem (1.1), (1.2), as well as the uniqueness of this
solution, follow from the inequalities (4.6) and (4.9). The estimate is
derived from (4.8) and (4.9).

Theorem 4.1 is proved.

The conditions of Theorem 2.1 ensure the existence of a unique solution to
problem (2.1), (2.2.) and the validity of the estimate (2.23). The number
$\mathcal{N}(k,\Delta_{m})$ in (2.23), as mentioned above, does not depend on
$f(t)$ and $d$. We therefore can treat this number as a constant of well-
posedness of problem (2.1), (2.2). Hence, by Theorems 2.1 and 4.1, the
following statement holds true.

Theorem 4.2 Suppose that

(a) the set of the matrices
$\\{\varphi_{j}(t),\psi_{j}(\tau),j=\overline{1,k}\\}$ is an
$\varepsilon$-approximating set for $K(t,\tau)$;

(b) $\Delta_{m}\in\sigma(k,[0,T]);$

(c) the matrix $Q_{\ast}(\Delta_{m}):R^{nm}\rightarrow R^{nm}$ in (2.22) is
invertible;

(d) the inequality
$\delta_{m}^{\varepsilon}=\mathcal{N}(k,\Delta_{m})\cdot\varepsilon<1$ holds.

Then problem (1.1), (1.2) is well-posed with constant
$\displaystyle{C=\frac{1}{1-\delta_{k}^{\varepsilon}}\cdot\mathcal{N}(k,\Delta_{m})}.$

The conditions of Theorem 4.1 are not only necessary but also sufficient for
the well-posedness of problem (1.1), (1.2).

Theorem 4.3 Problem (1.1), (1.2) is well-posed if and only if there exists
$\varepsilon$-approximating multipoint problem (2.1),(2.2), that is well-posed
with constant $C_{k}$, and the inequality (4.6) holds true.

Proof. The sufficiency of the conditions of Theorem for the well-posedness of
problem (1.1), (1.2) follow from Theorem 4.2.

Let us prove the necessity. Assume that problem (1.1), (1.2) is well-posed
with a constant $C$. Take $\varepsilon>0$ satisfying the inequality
$\varepsilon\cdot C<1/2.$ For chosen $\varepsilon$ take $k\in\mathbb{N}$ and
continuous on $[0,T]$ matrices $\varphi_{j}(t),$ $\psi_{j}(\tau),$
$j=\overline{1,k},$ satisfying inequality (4.1). Let us show that multipoint
problem (2.1), (2.2) with these matrices is well-posed and the constant
$C_{k}$ of well-posedness satisfies inequality (4.4). To this end, we use the
following algorithm.

Step 0. By solving problem (1.1), (1.2), we get the function $x^{(0)}(t).$

Step i. Assuming $x^{(i-1)}(t),$ $i=1,2,\ldots,$ to be known, we solve the
problem

$\frac{dx}{dt}=A(t)x+\int^{T}_{0}K(t,\tau)x(\tau)d\tau+f(t)+\int^{T}_{0}\Bigl{[}\sum\limits_{j=1}^{k}\varphi_{j}(t)\psi_{j}(\tau)-K(t,\tau)\Bigr{]}x^{(i-1)}(\tau)d\tau,\qquad
t\in[0,T],$ $\sum\limits^{m}_{i=0}B_{i}x(t_{i})=d,$

and get the function $x^{(i)}(t).$

It is easy to check that the algorithm converges to $x^{\ast}(t)$ and the
estimate

$\|x^{\ast}\|_{1}\leq\frac{C}{1-C\cdot\varepsilon}\cdot\max(\|f\|_{1},\|d\|),$
$None$

holds, were $x^{\ast}(t)$ is the unique solution to problem (2.1), (2.2).

Since, by assumption, $C\cdot\varepsilon<1/2,$ the well-posedness of the
$\varepsilon$-approximating problem (2.1),(2.2) with constant $C_{k}=2C$
follows from (4.10). Taking into account the choice of $\varepsilon>0$, we get
$q_{k}^{\varepsilon}=C_{k}\cdot\varepsilon<1.$

Theorem 4.3 is proved.

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