EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 287-302
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Asymptotic solutions of scalar integro-differential
equations with partial derivatives and with fast

oscillating coefficients

Burkhan T. Kalimbetov1,∗, Alisher N. Temirbekov1, Abdimukhan S. Tolep1

1 Institut Natural Sciences, K.A.Yasawi International Kazakh-Turkish University, Turkestan,
Kazakhstan

Abstract. In the paper, ideas of the Lomov regularization method are generalized to the Cauchy
problem for a singularly perturbed partial integro-differential equation in the case when the integral
term contains a rapidly varying kernel. Regularization of the problem is carried out, the normal
and unique solvability of general iterative problems is proved.

2020 Mathematics Subject Classifications: 35C20, 35F10, 45K05

Key Words and Phrases: Singularly perturbed, partial integro differential equation, regular-
ization of an integral, solvability of iterative problems

1. Introduction

In the paper, we consider the Cauchy problem for the integro-differential equation with
partial derivatives:

Lεy(x, t, ε) ≡ ε ∂y∂x = a(x)y +
x∫
x0

K(x, t, s)y(s, t, ε)ds+ h(x, t)+

+εg(x)cosβ(x)
ε y, y(x0, t, ε) = y0(t) ( (x, t) ∈ [x0, X]× [0, T ] ),

(1)

where β′(x) > 0, g(x), a(x) is a scalar functions, y0(t) constant, ε > 0 is a small parameter.
The problem of constructing a regularized asymptotic solution [1] of the problem (1) is
posed. Earlier, in [2], [3], [4], [5], [6], [7], systems for ordinary integro-differential equations
were mainly considered. In this paper we consider an partial integro-differential equa-
tions. Construction of asymptotic solutions for singularly perturbed integro-differential
equations with partial derivatives in the case when integral operators change rapidly was
first investigated in the works [8], [9], [10]. Construction of asymptotical solutions for

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3664

Email addresses: burkhan.kalimbetov@ayu.edu.kz (B.T. Kalimbetov),
alisher.temirbekov@ayu.edu.kz (A.N. Temirbekov), abdimuhan.tolep@ayu.edu.kz (A.S. Tolep)

http://www.ejpam.com 287 c© 2020 EJPAM All rights reserved.



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 288

ordinary integro-differential equations with fast oscillating coefficients from the position
of the regularization method are considered in [11].

Denote by λ1(x) = −a(x), β′(x) is a frequency of fast oscillating cosine. In the fol-
lowing, functions λ2(x) = −iβ′(x), λ3(x) = +iβ′(x) will be called the spectrum of a fast
oscillating coefficient.

We assume that the conditions are fulfilled:
(i) K(x, t, s) ∈ C∞{x0 < x < s < X, 0 < t < T}, h(x, t) ∈ C∞([x0, X]× [0, T ]), a(x),

g(x), β(x) ∈ C∞[x0, X],
(ii) λ1(x) 6= λj(x), j = 2, 3, λi(x) 6= 0, (∀x ∈ [x0, X]), i = 1, 2, 3;
(iii) Reλ1(x) ≤ 0, (∀x ∈ [x0, X]);
(iv) for ∀x ∈ [x0, X] and n2 6= n3 inequalities

n2λ2(x) + n3λ3(x) 6= λ1(x),
λ1(x) + n2λ2(x) + n3λ3(x) 6= λ1(x), (∀x ∈ [x0, X])

for all multi-indices n = (n2, n3) with |n| ≡ n2 + n3 ≥ 1 (n2 and n3 are non-negative
integers) are holds.

We will develop an algorithm for constructing a regularized [1] asymptotic solution of
problem (1).

2. Regularization of the problem

Denote by σj = σj(ε) independent of magnitude σ1 = e−
i
ε
β(t0), σ2 = e+ i

ε
β(t0), and

rewrite system (1) as

ε ∂y∂x = a(x)y + εg(x)
2

e− i
ε

t∫
t0

β′(θ)dθ

σ1+ +e
+ i

ε

t∫
t0

β′(θ)dθ

σ2

 y+

+
x∫
x0

K(x, t, s)y(s, t, ε)ds+ h(x, t), y(x0, t, ε) = y0.

(2)

Introduce the regularized variables:

τj =
1

ε

x∫
x0

λj(θ)dθ ≡
ψj(x)

ε
, j = 1, 3

and instead of problem (2), consider the problem

ε ∂ỹ∂x +
3∑
j=1

λj(x) ∂ỹ∂τj − a(x)ỹ −
x∫
x0

K(x, t, s)ỹ(s, t, ψ(s)
ε , ε)ds−

−εg(x)
2 (eτ2σ1 + eτ3σ2)ỹ = h(x, t), ỹ(x0, t, 0, ε) = y0,

(3)

for the function ỹ = ỹ(x, t, τ, ε) where is indicated: ψ = (ψ1, ψ2, ψ3). It is clear that if

ỹ = ỹ(x, t, τ, ε) is a solution of the problem (3), then the function is ỹ = ỹ(x, t, ψ(x)
ε , ε) an



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 289

exact solution to problem (2), therefore, problem (3) is extended with respect to problem
(2). However, it cannot be considered fully regularized, since it does not regularize the
integral

Jỹ =

x∫
x0

K(x, t, s)ỹ(s, t, ψ(s, ε), ε)ds.

Definition. A class Mε is said to be asymptotically invariant (with ε → +0) with
respect to an operator P0 if the following conditions are fulfilled:

1) Mε ⊂ D(P0) for each fixed ε > 0;
2) the image P0µ(x, t, ε) of any element µ(x, t, ε) ∈Mε decomposes in a power series

P0µ(x, t, ε) =

∞∑
n=0

εnµn(x, t, ε)(ε→ +0, µn(x, t, ε) ∈Mε, n = 0, 1, ...),

convergent asymptotically for ε→ +0) (uniformly with ∈ [t0, T ]).
From this definition it can be seen that the class Mε depends on the space U, in which

the operator P0 is defined. In our case P0 = J. For the space U we take the space of vector
functions y (x, t, τ) , represented by sums

y(x, t, τ, σ) =
3∑
i=1

yi(x, t, σ)eτi +
∗∑

2≤|m|≤Ny

ym(x, t, σ)e(m,τ)+

+y0(x, t, σ) +
∗∑

1≤|m|≤Ny

ye1+m(x, t, σ)e(e1+m,τ), yi(x, t, σ),

ym(x, t, σ), ye1+m(x, t, σ) ∈ C∞ ([x0, X]× [0, T ]) ,
1 ≤ |m| ≡ m2 +m3 ≤ Ny, i = 0, 3, m = (0,m2,m3).

(4)

where is denoted: (m,λ(x)) ≡ m2λ2(x) +m3λ3(x), (e1 +m,λ(x)) ≡ λ1(x) +m2λ2(x) +
m3λ3(x); an asterisk ∗ above the sum sign indicates that the summation for |m| ≥ 1 it
occurs only over multi-indices m = (0,m2,m3) with m2 6= m3, e1 = (1, 0, 0) , σ = (σ1, σ2) .

Note that here the degree Ny of the polynomial y (x, t, τ) , relative to the exponentials
eτj depends on the element y. In addition, the elements of space U depend on bounded
in ε > 0 terms of constants σ1 = σ1 (ε) and σ2 = σ2 (ε) and which do not affect the
development of the algorithm described below, therefore, in the record of element (4) of
this space U , we omit the dependence on σ = (σ1, σ2) for brevity. We show that the class
Mε = U |τ=ψ(t)/ε is asymptotically invariant with respect to the operator J .

The image of the integral operator J on an arbitrary element y (x, t, τ) , of the space
U has the form

Jy(x, t, τ) =

x∫
x0

K(x, t, s)y0(s, t)ds+
3∑
i=1

x∫
x0

K(x, t, s)yi(s, t)e

1
ε

s∫
x0

λi(θ)dθ

ds+

+
∗∑

2≤|m|≤Ny

x∫
x0

K(x, t, s)ym(s, t)e

1
ε

s∫
x0

(m,λ(θ))dθ

ds+



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 290

+
∗∑

1≤|m|≤Ny

x∫
x0

K(x, t, s)ye1+m(s, t)e

1
ε

s∫
x0

(e1+m,λ(θ))dθ

ds.

Apply the operation of integration by parts to the first term.

Ji(x, t, ε) =

x∫
x0

K(x, t, s)yi(s, t)e

1
ε

s∫
x0

λi(θ)dθ

ds = ε

x∫
x0

K(x, t, s)yi(s, t)

λi(s)
de

1
ε

s∫
x0

λi(θ)dθ

=

= ε

K(x, t, s)yi(s, t)

λi(s)
e

1
ε

s∫
x0

λi(θ)dθ

∣∣∣∣∣∣
s=x

s=x0

−
x∫

x0

(
∂

∂s

K(x, t, s)yi(s, t)

λi(θ)

)
e

1
ε

s∫
x0

λi(θ)dθ

ds

 =

= ε

K(x, t, x)yi(x, t)

λi(x)
e

1
ε

x∫
x0

λi(θ)dθ

− K(x, t, x0)yi(x0, t)

λi(x0)

−
−ε

x∫
x0

(
∂

∂s

K(x, t, s)yi(s, t)

λi(s)

)
e

1
ε

s∫
x0

λi(θ)dθ

ds .

Continuing this process, we obtain the series

Ji(x, t, ε) =
∞∑
ν=0

(−1)νεν+1

(Iνi (K(x, t, s)yi(s, t)))s=x e

1
ε

x∫
x0

λi(θ))dθ

−

− (Iνi (K(x, t, s)yi(s, t)))s=x0
]
,

where I0
i = 1

λi(s)
· , Iνi = 1

λi(s)
Iν−1
i (ν ≥ 1, i = 1, 3).

Applying the integration operation in parts to integrals

Jm(x, t, ε) =

x∫
x0

K(x, t, s)ym(s, t)e

1
ε

s∫
x0

(m,λ(θ))dθ

ds,

Je1+m(x, t, ε) =

x∫
x0

K(x, t, s)ye1+m(s, t)e

1
ε

s∫
x0

(e1+m,λ(θ))dθ

ds,

we note that for all multi-indices m = (0,m2,m3) , m2 6= m3, inequalities

(m,λ(x)) ≡ m2λ2(x) +m3λ3(x) 6= 0 ∀x ∈ [x0, X] , m2 +m3 ≥ 2

are satisfied. In addition, for the same multi-indices we have

(e1 +m,λ(x)) 6= 0∀x ∈ [x0, X] , m2 6= m3, |m| = m2 +m3 ≥ 1.



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 291

Indeed, if (e1 +m,λ(x)) = 0 for some x ∈ [x0, X] and m2 6= m3, m2 + m3 ≥ 1, then
m2λ2(x) +m3λ3(x) = −λ1(x)), m2 +m3 ≥ 1, which contradicts condition (iv). Therefore,
integration by parts in integrals Jm (t, ε) , Je1+m (t, ε) is possible. Performing it, we will
have:

Jm(x, t, ε) =

x∫
t0

K(x, t, s)ym(s, t)e

1
ε

s∫
x0

(m,λ(θ))dθ

ds = ε

x∫
x0

K(x, t, s)ym(s, t)

(m,λ(s))
de

1
ε

s∫
x0

(m,λ(θ))dθ

=

= ε

K(x, t, x)ym(x, t)

(m,λ(x))
e

1
ε

x∫
x0

(m,λ(θ))dθ

− K(x, t, x0)ym(x0, t)

(m,λ(x0))

−
−ε

x∫
x0

(
∂

∂s

K(x, t, s)ym(s, t)

(m,λ(s))

)
e

1
ε

s∫
x0

(m,λ(θ))dθ

ds =

=
∞∑
ν=0

(−1)νεν+1

(Iνm (K(x, t, s)ym(s, t)))s=t e

1
ε

x∫
x0

(m,λ(θ))dθ

−

− (Iνm (K(x, t, s)ym(s, t)))s=t0
]
,

where I0
m = 1

(m,λ(s)) · , I
ν
m = 1

(m,λ(s))
∂
∂s I

ν−1
m (ν ≥ 1, |m| ≥ 2),

Je1+m(x, t, ε) =

x∫
x0

K(x, t, s)ye1+m(s, t)e

1
ε

s∫
x0

(e1+m,λ(θ))dθ

ds =

= ε

s∫
x0

K(x, t, s)ye1+m(s, t)

(e1 +m,λ(s))
de

1
ε

s∫
x0

(e1+m,λ(θ))dθ

=

= ε

K(x, t, x)ye1+m (x, t)

(e1 +m,λ(x))
e

1
ε

x∫
x0

(e1+m,λ(θ))dθ

− K(x, t, x0)ye1+m (x0, t)

(e1 +m,λ(x0))

−
−ε

x∫
x0

(
∂

∂s

K (t, s) ye1+m(s, t)

(e1 +m,λ (s))

)
e

1
ε

s∫
x0

(e1+m,λ(θ))dθ

ds =

=
∞∑
ν=0

(−1)νεν+1

(Iνe1+m

(
K(x, t, s)ye1+m(s, t)

))
s=t

e

1
ε

x∫
x0

(e1+m,λ(θ))dθ

−

−
(
Iνe1+m

(
K(x, t, s)ye1+m(s, t)

))
s=t0

]
,

where I0
e1+m = 1

(e1+m,λ(s)) · , I
ν
e1+m = 1

(e1+m,λ(s))
∂
∂s I

ν−1
e1+m (ν ≥ 1, |m| ≥ 1,



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 292

Therefore, the image of the operator J on the element (5) of the space U is represented
as a series

Jy(x, t, τ) =

x∫
x0

K(x, t, s)y0(s, t)ds+

+

3∑
i=1

∞∑
ν=0

(−1)νεν+1

(Iνi (K(x, t, s)yi(s, t)))s=t e

1
ε

x∫
x0

λi(θ))dθ

−

− (Iνi (K(x, t, s)yi(s, t)))s=t0
]

+

+
∗∑

2≤|m|≤NY

∞∑
ν=0

(−1)νεν+1

(Iνm (K(x, t, s)ym(s, t)))s=t e

1
ε

x∫
x0

(m,λ(θ))dθ

−

− (Iνm (K(x, t, s)ym(s, t)))s=t0
]

+

+
∑

1≤|m|≤NY

∞∑
ν=0

(−1)νεν+1[
(
Iνe1+m

(
K(x, t, s)ye1+m(s, t)

))
s=t
×

×e
1
ε

x∫
x0

(e1+m,λ(θ))dθ

−
(
Iνe1+m

(
K(x, t, s)ye1+m(s, t)

))
s=t0

]
.

It is easy to show (see, for example, [12], pp. 291-294) that this series converges
asymptotically for ε → +0 (uniformly in (x, t) ∈ [x0, X] × [0, T ]). This means that the
class Mε is asymptotically invariant (for ε→ +0) with respect to the operator J .

We introduce operators Rν : U → U, acting on each element y(x, t, τ) ∈ U of the form
(5) according to the law:

R0y(x, t, τ) =

x∫
x0

K(x, t, s)y0(s, t)ds, (60)

R1y(x, t, τ) =
3∑

i=1

[(
I0
i (K(x, t, s)yi(s, t))

)
s=x

eτi − (
(
I0
i (K(x, t, s)yi(s, t))

)
s=x0

]+

+
∗∑

1≤|m|≤Ny

[
(
I0
m (K(x, t, s)ym(s, t))

)
s=x

e(m,τ) −
(
I0
m (K(x, t, s)ym(s, t))

)
s=x0

]+

+
∗∑

1≤|m|≤Ny

[(
I0
e1+m

(
K(x, t, s)y

e1+m
(s, t)

))
s=x

e(e1+m,τ)− (61)

−
(
I0
e1+m

(
K(x, t, s)y

e1+m
(s, t)

))
s=x0

]
,



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 293

Now let ỹ(x, t, τ, ε) be an arbitrary continuous function on (x, t, τ) ∈ [x0, X]× [0, T ]×
{τ : Re τj , j = 1, 3}, with asymptotic expansion

ỹ(x, t, τ, ε) =

∞∑
k=0

εkyk(x, t, τ), yk(x, t, τ) ∈ U, (7)

converging as ε→ +0 (uniformly in (x, t, τ) ∈ [x0, X]× [0, T ]×{τ : Re τj , j = 1, 3}). Then
the image Jỹ (x, t, τ, ε) of this function is decomposed into an asymptotic series

Jỹ(x, t, τ, ε) =

∞∑
k=0

εkJyk(x, t, τ) =

∞∑
r=0

εr
r∑
s=0

Rr−sys(x, t, τ)|τ=ψ(t)/ε.

This equality is the basis for introducing an extension of an operator J on series of the
form (7):

J̃ ỹ ≡ J̃

( ∞∑
k=0

εkyk(x, t, τ)

)
=
∞∑
r=0

εr

(
r∑

k=0

Rr−kyk(x, t, τ)

)
.

Although the operator J̃ is formally defined, its utility is obvious, since in practice it is
usual to construct the N -th approximation of the asymptotic solution of the problem (2),
in which impose only N -th partial sums of the series (7), which have not a formal, but a
true meaning. Now you can write a problem that is completely regularized with respect
to the original problem (2):

Lεỹ(x, t, τ, ε) ≡ ε ∂ỹ∂x +
3∑
j=1

λj(x) ∂ỹ∂τj − a(x)ỹ − J̃ ỹ − εg(x)
2 (eτ2σ1 + eτ3σ2)ỹ =

= h(x, t), ỹ(x0, t, 0, ε) = y0, ((x, t) ∈ [x0, X]× [0, T ]) .

(8)

3. Solvability of iterative problems

Substituting the series (7) into (8) and equating the coefficients of the same powers of
ε, we obtain the following iterative problems:

Ly0 ≡
3∑
j=1

λj(x)∂y0∂τj
− a(x)y0 −R0y0 = h(x, t), y0(x0, t, 0) = y0; (90)

Ly1 = −∂y0

∂x
+
g(x)

2
(eτ2σ1 + eτ3σ2)y0 +R1y0, y1(x0, t, 0) = 0; (91)

Ly2 = −∂y1

∂x
+
g(x)

2
(eτ2σ1 + eτ3σ2)y1 +R1y1 +R2y0, y2(x0, t, 0) = 0; (92)

............................................................

Lyk = −∂yk−1

∂x + g(x)
2 (eτ2σ1 + eτ3σ2)yk−1 +Rky0 +R1yk−1, yk(x0, t, 0) = 0, k ≥ 1. (9k)



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 294

Each iterative problem (9k) has the form

Ly ≡
3∑
j=1

λj(x)
∂y

∂τj
− a(x)y −R0y = H(x, t, τ), y(x0, t, 0) = y∗, (10)

where H(x, t, τ) ∈ U, is the known vector function of space U, y∗ is the known constant
vector of the complex space C, and the operator R0 has the form (see (60))

R0y(x, t, τ) ≡ R0

y0(x, t) +
3∑
i=1

yi(x, t)e
τi+

∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+

+

∗∑
1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ)

 ∆
=

x∫
x0

K(x, t, s)y0(s, t)ds.

We introduce scalar (for each x ∈ [x0, X]) product in space U :

< u,w >≡< u0(x, t) +

3∑
i=1

ui(x, t)e
τi +

∗∑
2≤|m|≤Ny

um(x, t)e(m,τ)+

+

∗∑
1≤|m|≤Ny

ue1+m(x, t)e(e1+m,τ), w0(x, t) +

3∑
i=1

wi(x, t)e
τi+

+
∗∑

2≤|m|≤Nw

wm(x, t)e(m,τ) +
∗∑

1≤|m|≤Nw

we1+m(x, t)e(e1+m,τ) >
∆
=

∆
= (u0(x, t), w0(x, t)) +

3∑
i=1

(ui(x, t), wi(x, t)) +

∗∑
2≤|m|≤min(Ny ,Nw)

(um(x, t), wm(x, t)) +

+

∗∑
1≤|m|≤min(Ny ,Nw)

(
ue1+m(x, t), we1+m(x, t)

)
,

where we denote by (∗ , ∗) the usual scalar product in the complex space C. Let us prove
the following statement.

Theorem 1. Let conditions (i)-(ii), (iv) be fulfilled and the right-hand side H(x, t, τ)
of system (10) belongs to the space U . Then the system (10) is solvable in U, if and only
if

H1(x, t, τ) ≡ 0, ∀x ∈ [x0, X] . (11)

Proof. We will determine the solution of system (10) as an element (5) of the space
U :

y(x, t, τ) = y0(x, t) +
3∑
i=1

yi(x, t)e
τi +

∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 295

+
∗∑

1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ) ≡ y0(x, t) +
3∑
i=1

yi(x, t)e
τi+ (12)

+
∗∑

2≤|m|≤Ny

ym(x, t)e(m,τ) +
∗∑

2≤|m1|≤Ny

ym
1
(x, t)e(mk,τ),

where for convenience introduced multi-indices m1 = e1 + m ≡ (1,m2,m3) , m2 and m3

are non-negative integer numbers. Substituting (12) into system (10), we will have

3∑
i=1

[λi(x)− a(x)] yi(x, t)e
τi +

∗∑
2≤|m|≤Ny

[(m,λ(x))− a(x)] ym(x, t)e(m,τ)+

+
∗∑

2≤|m1|≤Ny

[(
m1, λ(x)

)
− a(x)

]
ym

1
(x, t)e(m1,τ)−

−a(x)y0(x, t)−
x∫

x0

K(x, t, s)y0(s, t)ds = H0(x, t)+

+

3∑
i=1

Hi(x, t)e
τi +

∗∑
2≤|m|≤Ny

Hm(x, t)e(m,τ) +

∗∑
2≤|m1|≤Ny

Hm1
(x, t)e(m1,τ).

Equating here the free terms and coefficients separately for identical exponents, we obtain
the following systems of equations:

−a(x)y0(x, t)−
x∫

x0

K(x, t, s)y0(s, t)ds =H0(x, t), (13)

[λi(x)− a(x)] yi(x, t) = Hi(x, t), i = 1, 4, (13i)

[(m,λ(x))− a(x)] ym(x, t) = Hm(x, t), m2 6= m3, 2 ≤ |m| ≤ Ny, (13m)[(
m1, λ(x)

)
− a(x)

]
zm

1
(x, t) = Hm1

(x, t),m2 6= m3,2 ≤
∣∣m1

∣∣ ≤ Ny. (14)

The equation (13) can be written as

y0(x, t) =

x∫
x0

(
−a−1(x)K(x, t, s)

)
y0(s, t)ds− a−1(x)H0(x, t). (130)

Due to the smoothness of the kernel −a−1(x)K(x, t, s) and heterogeneity −a−1(x)H0(x, t),
this Volterra integral equation has a unique solution z0(x, t) ∈ C∞ ([x0, X]× [0, T ]) . The
equations (132) and (133) also have unique solutions

zi(x, t) = [λ1(x)− a(x)]−1Hi(x, t) ∈ C∞ ([x0, X]× [0, T ]) , i = 2, 3.



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 296

Equation (131) are solvable in space C∞ ([x0, X]× [0, T ]) if and only if there are identities

H1(x, t) ≡ 0 ∀x ∈ [x0, X] ,

It is not difficult to see that these identities coincide with identities (11).
Further, since (m,λ(x)) ≡ m2λ2(x) + m3λ3(x) 6= λ1(x), |m| = m2 + m3 ≥ 2 (see

condition (iv)) the absence of resonance), the equation system (13m) has a unique solution

zm(x, t) = [(m,λ(x))− a(x)]−1Hm(x, t), 2 ≤ |m| ≤ Ny ∈ C∞ ([x0, X]× [0, T ]) .

We now consider equation (14). Let
(
m1, λ(x)

)
= λ1(x),

∣∣m1
∣∣ ≥ 2. Then

λ1(x) +m2λ2(x) +m3λ3(x) = λ1(x)⇔

⇔ m2λ2(x) +m3λ3(x) = 0⇔ m2 6= m3, m2 +m3 ≥ 1,

which cannot be (see definition of class U). Unique solution of equation (18) for
∣∣m1

∣∣ ≥ 2
in the class C∞ ([x0, X]× [0, T ]) :

zm
1
(x, t) =

[(
m1, λ(x)

)
− a(x)

]−1
Hm1

(x, t), 2 ≤
∣∣m1

∣∣ ≤ Ny.

Thus, condition (11) is necessary and sufficient for the solvability of equation (10) in
the space U . The theorem is proved.

Remark. If identity (11) holds, then under conditions (i)-(ii) and (iv), equation (10)
has the following solution in the space U :

y(x, t, τ) = y0(x, t) + α1(x, t)eτ1 +

3∑
i=2

[λi(x)− a(x)]−1Hi(x, t)e
τi+

+

∗∑
2≤|m|≤Ny

[(m,λ(x))− a(x)]−1Hm(x, t)e(m,τ)+ (14)

+

∗∑
1≤|m|≤Ny

[(e1 +m,λ(x))− a(x)]−1He1+m(x, t)e(e1+m,τ),

where α1(x, t) ∈ C∞ ([x0, X]× [0, T ]) are arbitrary function, y0(x, t) is the solution of an
integral equation (130), m ≡ (0,m2,m3) ,m2 6= m3, |m| = m2 +m3 ≥ 1.

4. The unique solvability of the general iterative problem in the space
U . Residual term theorem

Let us proceed to the description of the conditions for the unique solvability of equation
(10) in space U . Along with problem (10), we consider the equatiom

Ly(x.t, τ) = −∂y
∂x

+
g(x)

2
(eτ2σ1 + eτ3σ2) y +Q(x, t, τ), (15)



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 297

where y = y(x, t, τ) is the solution (14) of the equation (10), Q(x, t, τ) ∈ U is the well-
known function of the space U. The right part of this equation:

G(x, t, τ) ≡ −∂y
∂x

+
g(x)

2
(eτ2σ1 + eτ3σ2) y +Q(x, t, τ) =

= − ∂

∂x

y0(x, t) +
3∑
i=1

yi(x, t)e
τi +

∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+

+

∗∑
1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ)

+

+
g(x)

2
(eτ2σ1 + eτ3σ2)

y0(x, t) +
3∑
i=1

yi(x, t)e
τi +

∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+

+

∗∑
1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ)

+Q(x, t, τ),

may not belong to space U , if y = y(x, t, τ) ∈ U. Indeed, taking into account the form (14)
of the function y = y(x, t, τ) ∈ U, we will have

Z(x, t, τ) ≡ G(x, t, τ) +
∂y

∂x
− g(x)

2
(eτ2σ1 + eτ3σ2)

[
y0(x, t) +

3∑
i=1

yi(x, t)e
τi+

+
∗∑

2≤|m|≤Ny

ym(x, t)e(m,τ) +
∗∑

1≤|m|≤Ny

ze1+m(x, t)e(e1+m,τ)

 =

=
g(x)

2
y0(x, t) (eτ2σ1 + eτ3σ2) +

3∑
i=2

g(x)

2
yi(x, t)

(
eτi+τ2σ1 + eτi+τ3σ2

)
+

+
g(x)

2
y1(x, t)

(
eτ1+τ2σ1 + eτ1+τ3σ2

)
+
g(x)

2
(eτ2σ1 + eτ3σ2)

 ∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+

+

∗∑
1≤|m|≤Ny

ze1+m(x, t)e(e1+m,τ)

+Q(x, t, τ).

Here are terms with exponents

eτ3+τ2 = e(m,τ)|m=(0,1,1),

eτ2+(m,τ) (if m2 + 1 = m3) , eτ3+(m,τ) (if m3 + 1 = m2) , (∗)



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 298

eτ2+(e1+m,τ) (if m2 + 1 = m3)m3 + 1 = m2,

do not belong to space U, since in multi-index m = (0,m2,m3) of the space U must be
m2 6= m3, m2 +m3 ≥ 1. Then, according to the well-known theory (see, [1] , p. 234), we
embed these terms in the space U according to the following rule (see (∗)):

êτ2+τ3 = e0 = 1, ̂eτ2+(m,τ) = e0 = 1 (m2 + 1 = m3,m2 6= m3) ,

̂eτ3+(m,τ) = e0 = 1 (m3 + 1 = m2,m2 6= m3) ,

̂eτ2+(e1+m,τ) = eτ1 (m2 + 1 = m3,m2 6= m3) . (∗∗)

In Z(x, t, τ) need of embedding only the terms

M(x, t, τ) ≡
3∑
i=2

g(x)

2
yi(x, t)

(
eτi+τ2σ1 + eτi+τ3σ2

)
+
g(x)

2
y1(x, t)

(
eτ1+τ2σ1 + eτ1+τ3σ2

)
,

S(x, t, τ) ≡ g(x)

2
(eτ2σ1 + eτ3σ2) [

∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+
∗∑

1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ)].

We describe this embedding in more detail, taking into account formulas (∗∗) :

M(x, t, τ) ≡ g(x)

2
y1(x, t)

(
eτ1+τ2σ1 + eτ1+τ3σ2

)
+

3∑
i=2

g(x)

2
yi(x, t)

(
eτi+τ2σ1 + eτi+τ3σ2

)
=

=
g(x)

2

[
y1(x, t)eτ1+τ2σ1 + y1(x, t)eτ1+τ3σ2 + y2(x, t)e2τ2σ1 + y2(x, t)σ2+

+y3(x, t)σ1 + y3(x, t)e2τ3σ2

]
⇒

⇒ M̂(x, t, τ) =
g(x)

2

[
y1(x, t)eτ1+τ2σ1 + y1(x, t)eτ1+τ3σ2 + y2(x, t)e2τ2σ1+

+y2(x, t)σ2 + y3(x, t)σ1 + y3(x, t)e2τ3σ2

]
,

(note that in M̂(x, t, τ) there are no members containing eτ1 , measurement exponents
|m| = 1) :

S(x, t, τ) ≡ g(x)

2
(eτ2σ1 + eτ3σ2)

 ∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)+

∗∑
1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ)

 =

=
g(x)

2

 ∗∑
2≤|m|≤Ny

ym(x, t)
(
eτ2+(m,τ)σ1 + eτ3+(m,τ)σ2

)
+

+

∗∑
1≤|m|≤Ny

ye1+m(x, t)
(
e(e1+m,τ)+τ2σ1 + e(e1+m,τ)+τ3σ2

)
⇒



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 299

⇒ Ŝ(x, t, τ) =
g(x)

2
[

∑
2 ≤ |m| ≤ Ny,
m2 + 1 = m3

ym(x, t)σ1 +
∑

2 ≤ |m| ≤ Ny,
m3 + 1 = m2

ym(x, t)σ2+

+
∗∑

2 ≤ |m| ≤ Ny,
m2 + 1 6= m3,m3 + 1 6= m2

ym(x, t)e(m,τ)+

+


∑

1 ≤ |m| ≤ Ny,
m2 + 1 = m3

ye1+m(x, t)σ1 +
∑

1 ≤ |m| ≤ Ny,
m3 + 1 = m2

ye1+m(x, t)σ2

 e
τ1+

+
∗∑

1 ≤ |m| ≤ Ny,
m2 + 1 6= m3,m3 + 1 6= m2

ye1+m(x, t)e(e1+m,τ),

After embedding, the right-hand side of system (15) will look like

Ĝ(x, t, τ) = − ∂

∂x

y0(x, t) +
3∑
i=1

yi(x, t)e
τi+

∗∑
2≤|m|≤Ny

ym(x, t)e(m,τ)

−
− ∂

∂x

 ∗∑
1≤|m|≤Ny

ye1+m(x, t)e(e1+m,τ)

+ M̂(x, t, τ) + Ŝ(x, t, τ) +Q(x, t, τ),

moreover, in Ŝ(x, t, τ) the coefficients at eτ1 do not depend on z1(x, t). As indicated in [1],
the embedding G(x, t, τ) → Ĝ(x, t, τ) will not affect the accuracy of the construction of
asymptotic solutions of problem (2), since G(x, t, τ)→ Ĝ(x, t, τ).

Theorem 2. Let conditions (i)-(ii), (iv) be fulfilled and the right-hand side H(x, t, τ) ∈
U of equation (10) satisfy condition (11). Then problem (10) under additional conditions

Ĝ(x, t, τ) ≡ 0 ∀t ∈ [x0, X] , (16)

where Q(x, t, τ) is the known vector function of space U , is uniquely solvable in U .
Proof. Since the right-hand side of equation (10) satisfies condition (11), this equation

has a solution in space U in the form (14), where α1(x, t) ∈ C∞ ([x0, X]× [0, T ]) are
arbitrary function so far. Submit (14) to the initial condition y (x0, t, 0) = y∗. We get
α1(x0, t) = y∗, where denoted

y∗ = y∗ + a−1(x0)H0(x0, t)−
3∑
i=2

[λi(x0)− a(x0)]−1Hi(x0, t)−



B.T. Kalimbetov, A.N. Temirbekov, A.S. Tolep / Eur. J. Pure Appl. Math, 13 (2) (2020), 287-302 300

−
∗∑

2≤|m|≤Ny

[(m,λ(x0))− a(x0)]−1Hm(x0, t)−

−
∗∑

1≤|mk|≤Ny

[(
mk, λ(x0)

)
− a(x0)

]−1
Hmk

(x0, t).

where do we find the values α1(x0, t) = y∗. Then condition (16) takes the form

− ∂

∂x
α1(x, t)eτ1+

+


∑

1 ≤ |m| ≤ Ny,
m2 + 1 = m3

ye1+m(x, t)σ1 +
∑

1 ≤ |m| ≤ Ny,
m3 + 1 = m2

ye1+m(x, t)σ2

 e
τ1+

+Q1(x, t)eτ1 ≡ 0 ∀(x, t) ∈ [x0, X]× [0, T ], .

We obtain linear ordinary differential equations with respect to the function α1(x, t),
involved in the solution (14) of equation (10). Attaching to them the initial conditions
α1 (t0) = y∗ computed earlier, we find uniquely the function α1(x0, t) = y∗ and, therefore,
we construct solution (14) in the space in a unique way. The theorem 2 is proved.

Applying Theorems 1 and 2 to iterative problems (9k) (in this case, the right-hand
sides H(k)(x, t, τ) of these problems are embedded in the space U , i.e. H(k)(x, t, τ) we
replace with Ĥ(k)(x, t, τ) ∈ U), we find uniquely their solutions in space U and construct
series (7). Justasin [1], we prove the following statement.

Theorem 3. Suppose that conditions (i)-(ii), (iv) are satisfied for problem (2). Then,
when ε ∈ (0, ε0](ε0 > 0 is sufficiently small), problem (2) has a unique solution y(x, t, ε) ∈
C1 ([x0, X]× [0, T ]) , in this case, the estimate

||y(x, t, ε)− yεN (x, t)||C[x0,X]×[0,T ] ≤ cNεN+1,

holds true, where zεN (x, t) is the restriction (for τ = ψ(t)
ε ) of the N - partial sum of

series (7) (with coefficients yk(x, t, τ) ∈ U, satisfying the iteration problems (9k)), and the
constant cN > 0 does not depend on ε ∈ (0, ε0].

Acknowledgements

This work is supported by the grant AP05133858 ”Contrast structures in singularly
perturbed equations and their applications in the theory of phase transitions” Ministry of
Education and Science of the Republic of Kazakhstan.



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