Well-posedness criteria for one family of boundary value problems

Abstract

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.

Full text

Bulletin of the Karaganda University. Mathematics Series, No. 4(112), 2023, pp. 5–20
MATHEMATICS
DOI 10.31489/2023M4/5-20
UDC 517.925
P.B. Abdimanapova1, S.M. Temesheva2,∗
1Almaty Technological University, Almaty, Kazakhstan;
2Al-Farabi Kazakh National University,
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
(E-mail: peryzat74@mail.ru, temeshevasvetlana@gmail.com)
Well-posedness criteria for one family of boundary value problems
This paper considers a family of linear two-point boundary value problems for systems of ordinary differential
equations. The questions of existence of its solutions are investigated and methods of finding approximate
solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value
problems for systems of ordinary differential equations are established. The uniqueness of the solution
of the problem under consideration is proved. Algorithms for finding an approximate solution based on
modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their
convergence is proved. According to the scheme of the parameterization method, the problem is transformed
into an equivalent family of multipoint boundary value problems for systems of differential equations. By
introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra
integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed
algorithm are established, which also ensure the existence of a unique solution of the family of boundary
value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of
linear boundary value problems for the system of ordinary differential equations are obtained.
Keywords: Family of linear boundary value problems, multipoint boundary value problem, existence of
solution, singular solution, well-posedness, necessary and sufficient condition.
Introduction
Problem statement and research methods
This paper is devoted to the study of a family of linear boundary value problems for differential
equations ∂v
∂t =A(x, t)v+f(x, t),(x, t)∈[0, ω]×(0, T ),(1)
B1(x)v(x, 0) + B2(x)v(x, T ) = d(x), x ∈[0, ω],(2)
where (n×n)-matrix A(x, t)en-vector-function f(x, t)are continuous on [0, ω]×[0, T ],B1(x),B2(x)
and n-vector-function d(x)are continuous on [0, ω],xis a parameter of the family (x∈[0, ω]);
kA(x, t)k ≤ a0,kv(x, t)k= max
i=1,n
kvi(x, t)k.
∗Corresponding author.
E-mail: temeshevasvetlana@gmail.com
Mathematics series. No. 4(112)/2023 5

P.B. Abdimanapova, S.M. Temesheva
In the present paper problem (1), (2) is investigated by the parameterization method [1].
The originallity of the parameterization method lies in the simple idea of introducing parameters at
some points of the set on which the boundary value problem is considered, which subsequently allows
us to construct an algorithm for finding a solution, obtain sufficient solvability conditions, establish
solvability criteria for linear and nonlinear two-point boundary value problems, multipoint boundary
value problems, boundary value problems with impulse influence, singular boundary value problems,
nonlocal boundary value problems for differential equations, loaded differential equations, integro-
differential Fredholm equations, differential equations with delayed argument, partial differential equa-
tions and others. These results are presented in the works of Dzhumabaev and his students (Assanova
[2], Temesheva [3–7], Orumbayeva [8–10], Uteshova [11, 12], Iskakova [13, 14], Imanchiyev [15, 16],
Bakirova [17], Kadirbayeva [18], Tleulessova [19], Abildayeva [20], Abdimanapova [21]).
Dzhumabaev and Assanova [22] studied a nonlocal boundary value problem for systems of linear
hyperbolic equations with mixed derivative. A special substitution allowed to reduce this problem to
an equivalent boundary value problem, which can be considered as a family of two-point boundary
value problems for systems of ordinary differential equations, where the spatial variable servers as a
parameter of the family.
This approach can also be used to study the linear nonlocal boundary value problem for a system
of partial differential equations (m= 1,2, . . .)
∂m+1u
∂t∂xm=A(x, t)∂mu
∂xm+f(x, t), u ∈Rn,(x, t)∈[0, ω]×(0, T ),
∂ku
∂xkx=0
=ψk(t), t ∈[0, T ], k = 0,1, . . . , m −1,∂0u
∂x0= 0,
B1(x)∂mu(x, t)
∂xmt=0
+B2(x)∂mu(x, t)
∂xmt=T
=d(x).
This fact motivated us to investigate problem (1), (2).
In this paper problem (1), (2) is investigated by the parameterization method with a modified
algorithm. Sufficient conditions for the existence of a unique solution are obtained. The well-posedness
criteria for problem (1), (2) are established.
Notation
•Nis a natural number;
•νis a natural number;
•Ωr= [0, ω]×[(r−1)h, rh),h=T /N,r= 1, N ;
•C([0, ω],Rn)is the space of continuous functions d: [0, ω]→Rnwith the norm kdk0=
max
x∈[0,ω]kd(x)k;
•C([0, ω]×[0, T ],Rn)is the space of continuous functions v: [0, ω]×[0, T ]→Rnwith the norm
kvk1= max
(x,t)∈[0,ω]×[0,T ]kv(x, t)k;
•the index rtakes on the values 1,2, . . . , N ;
•the index stakes on the values 1,2, . . . , N + 1;
•C([0, ω]×[0, T ],Ωr,RnN )is the space of systems of functions v(x, [t]) = (v1(x, t), v2(x, t), . . . , vN(x, t))
with the norm kvk2= max
r=1,N
sup
(x,t)∈Ωr
kvr(x, t)k, where the function vr: Ωr→Rnis continuous
and has a finite limit at t→rh −0uniformly with respect to x∈[0, ω]for all r;
•C([0, ω],Rn(N+1))is the space of functions λ(x)=(λ1(x), λ2(x), . . . , λN+1(x)) with the norm
kλk3= max
s=1,N+1
max
x∈[0,ω]kλs(x)k, where λs: [0, ω]→Rnare continuous for all s;
6 Bulletin of the Karaganda University

Well-posedness criteria for one ...
•C([0, T ],Rn)is the space of continuous functions v: [0, T ]→Rnwith the norm kvk4=
max
t∈[0,T ]kv(t)k;
•Iis the identity matrix of size n;
•Ois the zero matrix of size n×n;
•Oh1iis the first column of the matrix O.
1 Solvability of a family problems (1),(2)
Definition 1. v∗(x, t)∈C([0, ω]×[0, T ],Rn), continuously differentiable with respect to tand
satisfying equation (1) and boundary conditions (2) for each fixed x∈[0, ω], is called a solution of the
problem (1), (2).
Problem (1), (2) is investigated by the parameterization method [1]. For a fixed N, we make the
partition [0, ω]×[0,T ) =
N
S
r=1
Ωr.
According to the scheme of the parameterization method, the problem (1), (2) is transformed into
the equivalent family of multipoint boundary value problems with parameter for systems of differential
equations
∂evr
∂t =A(x, t)(evr+λr(x)) + f(x, t),(3)
evr(x, (r−1)h)=0,(4)
B1(x)λ1(x) + B2(x)λN+1(x) = d(x),(5)
λr(x) + lim
t→rh−0evr(x, t)−λr+1(x) = 0, r = 1, N, (6)
where (x, t)∈Ωr,x∈[0, ω],λr(x) = v(x, (r−1)h),λN+1(x) = lim
t→T−0v(x, t),evr(x, t) = v(x, t)−
λr(x),r= 1, N . A solution of problem (3)–(6) is a pair (λ∗(x),ev∗(x, [t])) λ∗(x)∈C([0, ω],Rn(N+1)),
ev∗(x, [t]) ∈C([0, ω]×[0, T ],Ωr,RnN )such that for each ris continuous and continuously differentiable
with respect to ton Ωrfunction ev∗
r(x, t)at λr(x) = λ∗
r(x)satisfies equation (3), condition (4), and
λ∗
1(x),λ∗
N+1(x),λ∗
r(x),lim
t→rh−0ev∗
r(x, t), satisfy (5), (6).
If the family of pairs (λ∗(x),ev∗(x, [t])) is a solution of the family of problems (3)–(6), then the
family of functions
v∗(x, t) = λ∗
r(x) + ev∗
r(x, t)for (x, t)∈Ωr, r = 1, N ,
λ∗
N+1(x)for x∈[0, ω], t =T
is a solution to the family of boundary value problems (1), (2).
If the family of systems of functions bv(x, [t]) = (bv1(x, t),bv2(x, t),...,bvN(x, t)) is a solution to
problem (1)-(2), then the solution to problem (3)–(6) is the pair (b
λ(x),b
ev(x, [t])) with elements b
λ(x) =
(b
λ1(x),b
λ2(x),...,b
λN+1(x)),b
λr(x) = bvr(x, (r−1)h),r= 1, N ,b
λN+1(x) = lim
t→T−0bvN(x, t),x∈[0, ω],
b
ev(x, [t]) = (b
ev1(x, t),b
ev2(x, t),...,b
ev2(x, t)),b
evr=bvr(x, t)−bvr(x, (r−1)h),(x, t)∈Ωr,r= 1, N .
In problem (3)–(6), the initial conditions (4) appeared for elements of the family of systems of
functions ev(x, [t]). For a known λr(x), the Cauchy problem (3), (4) on Ωris equivalent to the family
of Volterra integral equations of the second kind:
evr(x, t) =
t
Z
(r−1)h
A(x, τ )evr(x, τ)dτ +
t
Z
(r−1)h
A(x, τ )dτ ·λr(x) +
t
Z
(r−1)h
f(x, τ )dτ. (7)
Mathematics series. No. 4(112)/2023 7

P.B. Abdimanapova, S.M. Temesheva
In (7), replacing evr(x, τ )by the right hand side of (7) and repeating this process νtimes, we obtain
the following representation of the function evr(x, t):
evr(x, t) = Dν,r(x, t)·λr(x) + Fν,r(x, t) + Gν,r(x, t, ev),(8)
where
Dν,r(x, t) =
t
Z
(r−1)h
A(x, τ1)dτ1+
t
Z
(r−1)h
A(x, τ1)
τ1
Z
(r−1)h
A(x, τ2)dτ2dτ1+. . . +
+
t
Z
(r−1)h
A(x, τ1)
τ1
Z
(r−1)h
A(x, τ2). . .
τν−1
Z
(r−1)h
A(x, τν)dτν. . . dτ2dτ1,
Fν,r(x, t) =
t
Z
(r−1)h
f(x, τ1)dτ1+
t
Z
(r−1)h
A(x, τ1)
τ1
Z
(r−1)h
f(x, τ2)dτ2dτ1+. . . +
+
t
Z
(r−1)h
A(x, τ1). . .
τν−2
Z
(r−1)h
A(x, τν−1)
τν−1
Z
(r−1)h
f(x, τν)dτνdτν−1. . . dτ1,
Gν,r(t, x, ev) =
t
Z
(r−1)h
A(x, τ1). . .
τν−1
Z
(r−1)h
A(x, τν)evr(x, τν)dτν. . . dτ1,
t∈[(r−1)h, rh),r= 1, N .
Determining from (8) the limits
lim
t→rh−0evr(x, t) = Dν,r (x, rh)·λr(x) + Fν,r (x, rh) + Gν,r(rh, x, ev), x ∈[0, ω], r = 1, N ,
substituting them into (5), (6) and multiplying (5) by h > 0, we obtain the family of systems of linear
algebraic equations with respect to λr(x),x∈[0, ω]:
hB1(x)λ1(x) + hB2(x)λN+1(x) = hd(x),(9)
(I+Dν,r(x, rh))λr(x)−λr+1(x) = −Fν,r(x, rh)−Gν,r (rh, x, ev), r = 1, N . (10)
We write system (9), (10) in the form:
Qν(h, x)λ(x) = −Fν(h, x)−Gν(h, x, ev), λ(x)∈C([0, ω],Rn(N+1)),
where
Qν(h, x) =








hB1(x)O O . . . O hB2(x)
I+Dν,1(x, h)−I O . . . O O
O I +Dν,2(x, 2h)−I . . . O O
. . . . . . . . . . . . . . . . . .
O O O . . . −I O
O O O . . . I +Dν,N (x, N h)−I








,
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Well-posedness criteria for one ...
Fν(h, x)=(−hd(x), Fν,1(h, x), Fν,2(2h, x), . . . , Fν,N (N h, x)),
Gν(h, x, ev) = (Oh1i, Gν,1(h, x, ev), Gν,2(2h, x, ev), . . . , Gν,N (N h, x, ev)).
As can be seen, the process of finding a solution to problem (1), (2) is reduced to solving a family of
systems of linear algebraic equations (10) for some ev(x, [t]) and solving the family of Cauchy problems
(3), (4) on Ωrwhen λr(x),r= 1, N is found.
Let us describe the algorithm for finding a solution to problem (3)–(6). Let the matrix Qν(h, x)be
reversible for all x∈[0, ω].
Step 0. (a)The family of parameters λ(1)(x)is found from the equation Qν(h, x)λ(x) = −Fν(h, x).
(b)We determine the components of the system of functions ev(0)(x, [t]) by solving the Cauchy
problems (3), (4) on Ωrat λr(x) = λ(0)
r(x),r= 1, N .
(c)On [0, ω]×[0, T ]we define the function
v(0)(x, t) = (λ(0)
r(x) + ev(0)
r(x, t)for (x, t)∈Ωr, r = 1, N ,
λ(0)
N+1(x)for x∈[0, ω], t =T.
Step 1. (a)The family of parameters λ(1)(x)is found from the equation Qν(h, x)λ(x) = −Fν(h, x)−
Gν(h, x, ev(0)).
(b)We determine the components of the system of functions ev(1)(x, [t]) by solving the Cauchy
problems (3), (4) on Ωrat λr(x) = λ(1)
r(x),r= 1,(N+ 1).
(c)On [0, ω]×[0, T ]we define the function
v(1)(x, t) = (λ(1)
r(x) + ev(1)
r(x, t)for (x, t)∈Ωr, r = 1, N ,
λ(1)
N+1(x)for x∈[0, ω], t =T.
At the k-th step, we find the pair (λ(k)(x),ev(k)(x, [t])),k= 0,1,2, . . .. On ¯
Ωwe define the piecewise
continuous function
v(k)(x, t) = (λ(k)
r(x) + ev(k)
r(x, t)for (x, t)∈Ωr, r = 1, N ,
λ(k)
N+1(x)for x∈[0, ω], t =T.
Condition 1. For some h > 0 : N h =T,νand for any x∈[0, ω]the matrix Qν(h, x) : Rn(N+1) →
Rn(N+1) is invertible and the following inequalities are satisfied:
k(Qν(h, x))−1k ≤ γν(h, x)≤γν(h),
qν(h) = γν(h)nea0h−
ν
X
j=0
(a0h)j
j!o<1.(11)
The following statement establishes sufficient conditions for the feasibility and convergence of the
proposed algorithm. It should be noted that this statement ensures the existence of a unique solution
of the family of boundary value problems with parameters (3)–(6).
Theorem 1. Let Condition 1 be met. Then the sequence of pairs (λ(k)(x),ev(k)(x, [t])) converges to
the unique solution (λ∗(x),ev∗(x, [t])) of problem (3)–(6) and the following estimates hold true:
kλ∗−λ(k)k36qν(h)
1−qν(h)kλ(k)−λ(k−1)k3,(12)
kev∗
r(x, t)−ev(k)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ∗
r(x)−λ(k)
r(x)k,(13)
where k= 1,2, . . .,(x, t)∈Ωr,r= 1, N .
Mathematics series. No. 4(112)/2023 9

P.B. Abdimanapova, S.M. Temesheva
Proof. The continuity of the matrices A(x, t)and B1(x),B2(x)on [0, ω]×[0, T ]and [0, ω], respectively,
implies the continuity of the matrix Qν(h, x) : Rn(N+1) →Rn(N+1) on [0, ω]. Let us fix ex,bx∈[0, ω].
The matrix (Qν(h, x))−1:Rn(N+1) →Rn(N+1) is continuous for all x∈[0, ω], since the inequality
k(Qν(h, ex))−1−(Qν(h, bx))−1k ≤ γ2
ν(h))kQν(h, bx)−Qν(h, ex)kholds.
The solution of problem (3)-(6) is found by the algorithm. Solving the equation Qν(h, x)λ(x) =
−Fν(h, x),we find λ(0)(x). Since the matrix (Qν(h, x))−1and the vector Fν(h, x)are continuous for
all x∈[0, ω], we have λ(0)(x)∈C([0, ω],Rn(N+1))and
kλ(0)k3≤γν(h)hmax n1,
ν−1
X
j=0
(a0h)j
j!omax{kdk0,kfk1}.
For any rand x∈[0, ω], we find the function ev(0)
r(x, t)from the Cauchy problem (3), (4) with
λr(x) = λ(0)
r(x):
∂evr
∂t =A(x, t)evr+A(x, t)λ(0)
r(x) + f(x, t),evr(x, (r−1)h)=0, r = 1, N .
Then for ev(0)
r(x, t)we have the estimate
kev(0)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ(0)
r(x)k+ (t−(r−1)h)ea0(t−(r−1)h)kfk1,
whence it follows that
kev(0)k2≤ea0h−1kλ(0)k3+hea0hkfk1.
Then, following the algorithm, we solve the equation Qν(h, x)λ(x) = −Fν(h, x)−Gν(h, x, ev(0) )and
find λ(1)(x). We have
kλ(1) −λ(0)k3=k − (Qν(h, x))−1·Gν(h, x, ev(0) )k ≤ γν(h) max
r=1,N
kGν,r(rh, x, ev(0))k≤
≤γν(h) max
r=1,N nrh
Z
(r−1)h
a0. . .
τν−1
Z
(r−1)h
a0kev(0)
r(x, τν)kdτν. . . dτ1o≤γν(h)(a0h)ν
ν!kev(0)k2.
We define the components of the system of functions ev(1) (x, [t]) = (ev(1)
1(x, t),ev(1)
2(x, t),..., ev(1)
N(x, t))
by solving the Cauchy problem (3), (4) with λr(x) = λ(1)
r(x):
∂evr
∂t =A(x, t)evr+A(x, t)λ(1)
r(x) + f(x, t),evr(x, (r−1)h)=0, r = 1, N .
The difference (ev(1)
r(x, t)−ev(0)
r(x, t)) is estimated as follows:
kev(1)
r(x, t)−ev(0)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ(1)
r(x)−λ(0)
r(x)k.
We assume that the pair (λ(k−1)(x),ev(k−1)(x, [t])) is determined and for all (x, t)∈Ωrthe following
inequalities hold:
kλ(k−1) −λ(k−2)k3≤qν(h)kλ(k−2) −λ(k−3)k3,
kev(k−1)
r(x, t)−ev(k−2)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ(k−1)
r(x)−λ(k−2)
r(x)k.(14)
At the k-th step of the algorithm, solving the equation Qν(h, x)λ(x) = −Fν(h, x)−Gν(h, x, ev(k−1)),
we find λ(k)(x). Taking into account (14), we establish that
kλ(k)−λ(k−1)k3≤qν(h)kλ(k−1) −λ(k−2)k3, k = 2,3,.... (15)
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Well-posedness criteria for one ...
We define the components of the system of functions ev(k)(x, [t]) = (ev(k)
1(x, t),ev(k)
2(x, t),...,ev(k)
N(x, t))
by solving the Cauchy problem (3), (4) with λr(x) = λ(k)
r(x):
∂evr
∂t =A(x, t)evr+A(x, t)λ(k)
r(x) + f(x, t),evr(x, (r−1)h) = 0, r = 1, N .
For all (x, t)∈Ωr,r= 1, N (k= 1,2,3, . . .)we estimate the difference (ev(k)
r(x, t)−ev(k−1)
r(x, t)):
kev(k)
r(x, t)−ev(k−1)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ(k)
r(x)−λ(k−1)
r(x)k.(16)
By the condition of Theorem, qν(h)<1, so it follows from (15), (16) that the pair (λ(k)(x),ev(k)(x, [t])),
k= 0,1,2, . . ., converges to (λ∗(x),ev∗(x, [t])), the solution of problem (3)–(6) in C([0, ω],Rn(N+1))×
C([0, ω]×[0, T ],Ωr,RnN ).
It is not difficult to establish the validity of the inequalities:
kλ(k+`)−λ(k)k36qν(h)
1−qν(h)kλ(k)−λ(k−1)k3,(17)
kλ(k)−λ(0)k361−qk
ν(h)
1−qν(h)γν(h)(a0h)ν
ν!kev(0)k2,
kev(k+`)
r(x, t)−ev(k)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ(k+`)
r(x)−λ(k)
r(x)k,(18)
kev(k)
r(x, t)−ev(0)
r(x, t)k ≤ ea0(t−(r−1)h)−1kλ(k)
r(x)−λ(0)
r(x)k,
(x, t)∈Ωr,r= 1, N ,k= 1,2, . . .. In the inequalities (17), (18), letting `→ ∞, we establish the
validity of the estimates (12), (13).
Let us show the uniqueness of the solution of problem (3)–(6). Let v∗(x, t)and bv(x, t)be two
solutions of problem (1), (2). Then the pairs (λ∗(x),ev∗(x, [t])) and (b
λ(x),b
ev(x, [t])) are solutions to the
boundary value problem (3)–(6), here
λ∗(x)∈C([0, ω],Rn(N+1)), λ∗
s(x) = v∗(x, (s−1)h), s = 1, N + 1,
ev∗
r(x, [t]) ∈C([0, ω]×[0, T ],Ωr,RnN ),
ev∗
r(x, t) = v∗(x, t)−v∗(x, (r−1)h),(x, t)∈Ωr, r = 1, N ,
b
λ(x)∈C([0, ω],Rn(N+1)),b
λs(x) = bv(x, (s−1)h), s = 1, N + 1,
b
ev(x, [t]) ∈C([0, ω]×[0, T ],Ωr,RnN ),
b
evr(x, t) = bv(x, t)−bv(x, (r−1)h),(x, t)∈Ωr, r = 1, N .
Under our assumptions, the following equations hold:
ev∗
r(x, t) =
t
Z
(r−1)h
A(x, τ )ev∗
r(x, τ )dτ +
t
Z
(r−1)h
A(x, τ )dτ ·λ∗
r(x) +
t
Z
(r−1)h
f(x, τ )dτ,
b
evr(x, t) =
t
Z
(r−1)h
A(x, τ )b
evr(x, τ )dτ +
t
Z
(r−1)h
A(x, τ )dτ ·b
λr(x) +
t
Z
(r−1)h
f(x, τ )dτ,
Q−1
ν(h, x)λ∗(x) = −Fν(h, x) + Gν(h, x, ev∗),
Mathematics series. No. 4(112)/2023 11

P.B. Abdimanapova, S.M. Temesheva
Q−1
ν(h, x)b
λ(x) = −Fν(h, x) + Gν(h, x, b
ev).
Then the following inequalities are true
kev∗−b
evk2≤ea0h−1· kλ∗−b
λk3,(19)
kλ∗−b
λk3≤qν(h)kλ∗−b
λk3.
Hence, by virtue of inequality (11), λ∗(x) = b
λ(x). Then from (19) we obtain that v∗(x, t) = bv(x, t)
for (x, t)∈[0, ω]×[0, T ]. Theorem 1 is proved.
Since problem (1), (2) and problem (3)–(6) are equivalent, the following statement holds true.
Corollary 1. Let Condition 1 be met. Then the sequence v(k)(x, t) (k= 0,1,2, . . .)converges to the
unique solution v∗(x, t)of problem (1), (2) and the following estimates are true:
kv∗−v(0)k1≤γν(h)ea0h
1−qν(h)·(a0h)ν
ν!ea0h−1max
s=1,N+1
max
x∈[0,ω]kv(0)(x, (s−1)h)k+hea0hkfk1.
2 Well-posedness criteria for the family of problems (1),(2)
Definition 2. The boundary value problem (1), (2) is called well-posed if for any f(x, t)∈C([0, ω]×
[0, T ],Rn),d(x)∈C([0, ω],Rn)it has a unique solution v(x, t)and
kvk1≤Kmax nkdk1,kfk1o,
where Kis a constant, independent of f(x, t)and d(x). The number Kis called the well-posedness
constant of problem (1), (2).
Let us consider the equation
1
hQ∗(h, x)λ(x) = −F∗(h, A, f, d, x), λ(x)∈C([0, ω],Rn(N+1)),
where Q∗(h, x) = lim
ν→∞ Qν(h, x),F∗(h, A, f, d, x) = lim
ν→∞
1
hFν(h, x).
Theorem 2. The boundary value problem (1), (2) is well-posed for all x∈[0, ω]if and only if there
exists h0∈(0, T ]such that for any h∈(0, h0] : Nh =Tthere is a number ν=ν(h), such that the
matrix Qν(h, x) : Rn(N+1) →Rn(N+1) is invertible and the following inequalities hold:
k(Qν(h, x))−1k ≤ γν(h),(20)
qν(h) = γν(h)nea0h−
ν
X
j=0
(a0h)j
j!o<1.(21)
Proof. The sufficiency of the conditions of Theorem 2 for the well-posedness of problem (1), (2)
follows from Corollary 1.
Necessity. Let problem (1), (2) be well-posed with a constant K. Problem (1), (2) for every fixed
bx∈[0, ω]is a linear two-point boundary value problem for the ordinary differential equation:
dbv
dt =b
A(t)bv+b
f(t), t ∈(0, T ),bv∈Rn,(22)
b
B1bv(0) + b
B2bv(T) = b
d. (23)
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Well-posedness criteria for one ...
Here bv(t) = v(bx, t),b
A(t) = A(bx, t),b
f(t) = f(bx, t),b
B1=B1(bx),b
B2=B2(bx),b
d=d(bx).
Since for f(x, t) = b
f(t),d(x) = b
dwe have:
kbv∗k4= max
t∈[0,T ]kv∗(bx, t)k ≤ max
(x,t)∈[0,ω]×[0,T ]kv∗(x, t)k ≤ Kmax{kdk0, f k1}=Kmax{k b
dk,kb
fk4},
then the correct solvability of problem (1), (2) follows from the correct solvability of problem (22), (23)
with constant Kfor every fixed bx∈[0, ω].
For any ε > 0there is h0∈(0, T ], satisfying the inequality
1
a0h0
(ea0h0−1−a0h0)≤ε
(2 + ε)(1 + ε).
Then, by Theorem 3 [1; p. 42], we obtain the following estimate for all h∈(0, h0] : N h =T:
k(Q∗(h, bx))−1k6(1 + ε)K
h.
In view of the arbitrariness of bx∈[0, ω], we obtain
k(Q∗(h, x))−1k6(1 + ε)K
h,∀x∈[0, ω].
Let us choose ν1such that:
2(1 + ε)K
hnea0h−
ν1
X
j=0
(a0h)j
j!o<1.
For any ν, we have there is the inequality
kQ∗(h, x)−Qν(h, x)k6
∞
X
j=ν+1
(a0h)j
j!=nea0h−
ν
X
j=0
(a0h)j
j!o.
Then it follows from the theorem on small perturbations of boundedly invertible operators that for all
ν≥ν1the matrix Qν(h, x) : Rn(N+1) →Rn(N+1) is invertible and
k(Qν(h, x))−1k6k(Q∗(h, x))−1k
1− k(Q∗(h, x))−1k·kQ∗(h, x)−Qν(h, x)k<2(1 + ε)K
h.
Thus, for all ν≥ν1,h∈(0, h0] : Nh =Tand x∈[0, ω], taking γν(h) = 2(1 + ε)K
h, we obtain
that the inequalities (20), (21). Theorem 2 is proved.
Theorem 3. The boundary value problem (1), (2) is well-posed if and only if for any νthere exists
h=h(ν) : Nh =T, such that the matrix Qν(h, x) : Rn(N+1) →Rn(N+1) is invertible for all x∈[0, ω]
and the inequalities (20), (21) are true.
Proof. Sufficiency. The well-posedness of problem (1), (2) under the conditions of Theorem follows
from Corollary 1.
Necessity. Let the problem (1), (2) be well-posed with constant K. Reasoning as in the proof of
Theorem 2, for a given ε > 0we find h0=h0(ε)such that for all h∈(0, h0] : N h =Tand x∈[0, ω]
the matrix Q∗(h, x) : Rn(N+1) →Rn(N+1) is invertible and
k(Q∗(h, x))−1k6(1 + ε)K
h.
Mathematics series. No. 4(112)/2023 13

P.B. Abdimanapova, S.M. Temesheva
We choose h1∈(0, h0]such that the relation is satisfied:
2(1 + ε)K
h1nea0h1−
ν
X
j=0
(a0h1)j
j!o<1.(24)
Since k(Q∗(h, x))−1k · kQ∗(h, x)−Qν(h, x)k<0.5,then, by virtue of (24), by the small perturbation
theorem of boundedly reversible operators, for all h∈(0, h1] : N h =Tand x∈[0, ω]the inequality
holds k(Qν(h, x))−1k<2(1 + ε)K
h.
Taking γν(h) = 2(1 + ε)K
h, by virtue of choosing h∈(0, h1] : Nh =T, we obtain the fulfillment of
inequalities (20) and (21). Theorem 3 is proved.
Theorem 4. Let for some νthere exist h0=h0(ν)such that for all h∈(0, h0] : Nh =Tand
x∈[0, ω]the matrix Qν(h, x) : Rn(N+1) →Rn(N+1) is invertible and
k(Qν(h, x))−1k ≤ γ
h,
where γis a constant, independent of hand x. Then problem (1), (2) is well-posed with constant
K=γ.
Proof. For any ε > 0there is h0∈(0, T ]satisfying the inequality
1
a0h0
(ea0h0−1−a0h0)≤ε
(2 + ε)(1 + ε).
We choose h1∈(0, h0] : N h1=Tsuch that the following inequality is satisfied:
γ
h1nea0h1−
ν
X
j=0
(a0h1)j
j!o<1.
Then qν(h)≤qν(h1)<1for all h∈(0, h1] : Nh =Tand, by Corollary 1, the problem (1), (2) has a
unique solution v∗(x, t)and
max
(x,t)∈[0,ω]×[0,T ]kv∗(x, t)k ≤ ea0h γ
1−qν(h)·(a0h)ν
ν!·ea0h−1
h+ 1×
×γmax n1,
ν−1
X
j=0
(a0h)j
j!o+γ
1−qν(h)
(a0h)ν
ν!ea0hmax{kdk0,kfk1}+hea0hkfk1.
Letting h→0in the above inequality, we obtain that
max
(x,t)∈[0,ω]×[0,T ]kv∗(x, t)k ≤ γmax{kdk0,kfk1}.
Theorem 4 is proved.
Theorem 5. Let problem (1), (2) be well-posed with constant K. Then for any νand ε > 0there
exists h0=h0(ν, ε)such that for all h∈(0, h0] : Nh =Tand x∈[0, ω]the matrix Qν(h, x) :
Rn(N+1) →Rn(N+1) is invertible and
k(Qν(h, x))−1k ≤ (1 + ε)K
h.
14 Bulletin of the Karaganda University

Well-posedness criteria for one ...
Proof. For a given ε > 0, find h0=h0(ε)such that for all h∈(0, h0] : Nh =Tand x∈[0, ω]the
matrix Q∗(h, x) : Rn(N+1) →Rn(N+1) is invertible and the following estimate holds true:
k(Q∗(h, x))−1k6(2 + ε)K
2h.
Let us choose h1∈(0, h0]satisfying the inequality:
(2 + ε)K
h1nea0h1−
ν
X
j=0
(a0h1)j
j!o<ε
1 + ε.
Since k(Q∗(h, x))−1k · kQ∗(h, x)−Qν(h, x)k<1
2·ε
1 + εthen, the theorem on small perturbations of
boundedly invertible operators, for all h∈(0, h1] : Nh =Tand x∈[0, ω]the following estimate holds
k(Qν(h, x))−1k<(1 + ε)K
h=γν(h)and, based on (24),
qν(h) = γν(h)nea0h−
ν
X
j=0
(a0h)j
j!o<ε
2 + ε<1.
Then, according Corollary 1, there exists a unique solution v∗(x, t)of problem (1), (2) and the
following estimate holds:
max
(x,t)∈[0,ω]×[0,T ]kv∗(x, t)k ≤ ea0h(1 + ε)K
1−qν(h)·(a0h)ν
ν!·ea0h−1
h+ 1(1 + ε)K×
×max n1,
ν−1
X
j=0
(a0h)j
j!o+(1 + ε)K
1−qν(h)
(a0h)ν
ν!ea0hmax{kdk0,kfk1}+hea0hkfk1.
Letting h→0, we obtain the estimate max
(x,t)∈[0,ω]×[0,T ]kv∗(x, t)k ≤ (1 + ε)Kmax{kdk0,kfk1}.
Theorem 5 is proved.
Conclusion
The paper proposes a modified algorithm of the parameterization method: an additional parameter
is introduced and at the last point of the segment on which the boundary value problem is considered.
This is the difference between the proposed modified algorithm and the classical algorithm of the
parameterization method. This modification allows us to simplify the structure of the linear operator
equation with respect to the introduced parameters. Sufficient conditions for the existence of a single
solution of the problem (1), (2) and criteria of correct solvability of the family of linear boundary
value problems for the system of ordinary differential equations are obtained. Note that the idea of
the methodology used in this paper has wide prospects of development for the study of problems of
solutions of linear and nonlinear boundary value problems.
Acknowledgments
This research is funded by the Science Committee of the Ministry of Science and Higher Education
of the Republic of Kazakhstan (Grant No. AP19675193)
Mathematics series. No. 4(112)/2023 15

P.B. Abdimanapova, S.M. Temesheva
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П.Б. Абдиманапова1, С.M. Темешева2
1Алматы технологиялық университетi, Алматы, Қазақстан;
2Әл-Фараби атындағы Қазақ ұлттық университетi;
Математика және математикалық модельдеу институты, Aлматы, Қазақстан
Шеттiк есептiң бiр үйiрiнiң қисынды шешiмдiлiк
критерийлерi туралы
Мақалада дифференциалдық теңдеулер жүйелерi үшiн сызықтық екi нүктелi шеттiк есептер үй-
iрi карастырылған. Оның шешiмдерiнiң бар болу сұрақтары зерттелiп, жуық шешiмдi табу әдiстерi
ұсынылған. Жәй дифференциалдық теңдеулер жүйесi үшiн сызықтық екi нүктелi шеттiк есептер үй-
iрiнiң жеткiлiктi шарттары анықталған. Қарастырылған есептiң шешiмiнiң жалғыздығы дәлелдендi.
Д.С. Жұмабаевтың параметрлеу әдiсiнiң алгоритмдерiнiң бiр модификациясы негiзiнде зерттелетiн
есептiң жуық шешiмiн табу алгоритмдерi берiлген және олардың жинақтылығы дәлелденген. Па-
раметрлеу әдiсiнiң схемасы бойынша есеп дифференциалдық теңдеулер жүйелерi үшiн көп нүктелi
шеттiк есептерiнiң эквиваленттi үйiрiне түрлендiрiлген. Жаңа белгiсiз функцияларды енгiзу арқы-
лы бiз зерттелетiн есептi баламалы есепке, екiншi тектi Вольтерра интегралдық теңдеуiне келтiремiз.
Параметрметрлi шеттiк есептер үйiрiнiң жалғыз шешiмiнiң бар болуын қамтамасыз ететiн ұсынылған
Mathematics series. No. 4(112)/2023 17

P.B. Abdimanapova, S.M. Temesheva
алгоритмнiң орындылығы мен жинақтылығының жеткiлiктi шарттары анықталды. Жәй дифферен-
циалдық теңдеулер жүйесi үшiн сызықтық шеттiк есептер үйiрiнiң қисынды шешiмдiлiгiнiң қажеттi
және жеткiлiктi шарттары алынды.
Кiлт сөздер: сызықтық шеттiк есептер үйiрi, көпнүктелi шеттiк есеп, шешiмнiң бар болуы, жалғыз
шешiм, қисынды шешiмдiлiк, қажеттi және жеткiлiктi шарт.
П.Б. Абдиманапова1, С.M. Темешева2
1Алматинский технологический университет, Алматы, Казахстан;
2Казахский национальный университет имени аль-Фараби;
Институт математики и математического моделирования, Алматы, Казахстан
О критериях корректной разрешимости одного семейства
краевых задач
В статье рассмотрено семейство линейных двухточечных краевых задач для систем дифференциаль-
ных уравнений. Исследованы вопросы существования его решений и предложены методы нахожде-
ния приближенных решений. Установлены достаточные условия существования семейства линейных
двухточечных краевых задач для системы обыкновенных дифференциальных уравнений. Доказана
единственность решения рассматриваемой задачи. Даны алгоритмы нахождения приближенного ре-
шения исследуемой задачи, основанные на одной модификации алгоритмов метода параметризации
Д.С. Джумабаева, и доказана их сходимость. По схеме метода параметризации задача будет преоб-
разована в эквивалентное семейство многоточечных краевых задач для систем дифференциальных
уравнений. Введя новые неизвестные функции, сведем исследуемую задачу к эквивалентной задаче,
интегральному уравнению Вольтерра второго рода. Установлены достаточные условия осуществимо-
сти, сходимости предложенного алгоритма, вместе с тем обеспечивающие существование единственно-
го решения семейства краевых задач с параметрами. Получены необходимые и достаточные условия
корректной разрешимости семейства линейных краевых задач для системы обыкновенных диффе-
ренциальных уравнений.
Ключевые слова: семейство линейных краевых задач, многоточечная краевая задача, существование
решения, единственное решение, корректная разрешимость, необходимое и достаточное условие.
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