Factorization of a convolution-type integro-differential equation on the positive half line
Sufficient conditions for the existence of solutions are obtained for a class of convolution-type integro-differential equations on the half line. The investigation is based on the three-factor decomposition of the initial integro-differential operator.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509325310558208 |
|---|---|
| author | Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. |
| author_facet | Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. |
| author_sort | Khachatryan, A. Kh. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:49:31Z |
| description | Sufficient conditions for the existence of solutions are obtained for a class of convolution-type integro-differential equations on the half line. The investigation is based on the three-factor decomposition of the initial integro-differential operator. |
| first_indexed | 2026-03-24T02:39:18Z |
| format | Article |
| fulltext |
UDC 517.9
A. Kh. Khachatryan, Kh. A. Khachatryan (Inst. Math. Nat. Acad. Sci., Armenia, Yerevan)
FACTORIZATION OF ONE CONVOLUTION-TYPE
INTEGRO -DIFFERENTIAL EQUATION
ON POSITIVE HALF LINE
FAKTORYZACIQ ODNOHO INTEHRO-DYFERENCIAL|NOHO
RIVNQNNQ TYPU ZHORTKY NA DODATNIJ PIVOSI
Sufficient conditions for the existence of a solution of one class of convolution-type integro-differential
equations on half line are obtained. The investigation is based on three factor decomposition of initial
integro-differential operator.
Otrymano dostatni umovy dlq isnuvannq rozv’qzku odnoho klasu intehro-dyferencial\nyx riv-
nqn\ typu zhortky na pivosi. DoslidΩennq bazugt\sq na rozkladi poçatkovoho intehro-dyfe-
rencial\noho operatora na try mnoΩnyky.
1. Introduction. A number of problems of physical kinetics (see [1 – 3]) are described
by the integro-differential equation
dS
dx
AS x+ ( ) = g x x B K x t dS
dt
dt C K x t S t dt( ) ( ) ( ) ( ) ( )+ − + −
∞ ∞
∫ ∫λ 1
0
2
0
, x R∈ + .
(1.1)
Here, S is an unknown solution from a class of functions absolutely continuous on R
+
and of slow growth in + ∞ , i.e.,
S ∈ � df= f AC R e f x xx∈ ∀ > → → + ∞{ }+ −( ) , ( )s.t. asε ε0 0 ,
where AC R( )+ is the space of functions absolutely continuous on R
+
, A, B, C are
nonpositive parameters, 0 1≤ ⋅ ≤λ( ) , and λ ∈ ∞
+W R1( ) (where W Rp
n( )+ is the So-
bolev space of functions f such that f L Rk
p
( ) ( )∈ + , k = 0, 1, 2, … , n ) . The functions
g and Kj , j = 1, 2, satisfy the following conditions:
0 ≤ g ∈ L R1( )+ (1.2)
and 0 ≤ Kj ∈ L R1( ) such that
K x dxj ( )
−∞
∞
∫ = 1, j = 1, 2. (1.3)
The initial condition to equation (1.1) – (1.3) is joined
S( )0 = s0 ∈ R
+. (1.4)
In the case where
K x1( ) ≡ 0, A = 0, λ( )x = 1, K x2( ) = e ds
s
x s−
∞
∫ 2
1
, (1.5)
the first results of studying equation (1.1) – (1.3) appeared in the works [3 – 5]. Later,
in [6], the equation (1.1) was considered in the more general case where
© A. KH. KHACHATRYAN, KH. A. KHACHATRYAN, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1555
1556 A. KH. KHACHATRYAN, KH. A. KHACHATRYAN
K x1( ) ≡ 0, λ( )x ≡ 1, 0 ≤ K2 ∈ L R1( ), K L2 1
= 1, (1.6)
and, under some additional conditions on functions K2
, g and parameters A, C, the
structural theorems on existence were obtained. Note that in [5, 7], the solvability of
equation (1.1), (1.5) in the space W R1
1( )+ is proved and, by means of the Ambartsu-
mian – Chandrasekhar function, analytical formulae describing the structure of obtain-
ed solution are founded.
In the present work, structural theorems on existence are obtained by putting some
additional conditions on functions λ , K1 and K2 for equation (1.1) – (1.4).
Below, we briefly describe our approach to the investigation. First, we construct
three factor decomposition of the initial integro-differential operator D AI BK D+ − λ
1 –
– CKλ
2 [ where D is a differential operator, I is the unit operator, ( )( )K f xj
λ =
= λ( ) ( ) ( )x K x t f t dtj −
∞
∫0
, j = 1, 2 ] in the form of product of one differential and two
integral operators. Using this factorization, the problem is reduced to the successive
solution of two integtal equations and one first-order simple differential equation. The
former is the Volterra-type integral equation (it can be solved elementary) and the lat-
ter is the integral equation with the kernel λ( ) ( )x w x t− , where w L R( ) ( )⋅ ∈ 1 if A >
> 0 and w x( ) = ρ ρ0 1( ) ( )x x+ if A = 0 ( here, ρ0 1∈L R( ) , ρ1 ∈M R( ) ) .
It should be also noted that above mentioned factorization allow us to construct
a nontrivial solution (from class � ) of the corresponding homogeneous equation for
A = C, i.e.,
dS
dx
AS x+ ( ) = λ( ) ( ) ( ) ( )x B K x t dS
dt
dt A K x t S t dt1
0
2
0
− + −
∞ ∞
∫ ∫ . (1.7)
2. Notations and auxiliary information. Let E
+ be one of the following Banach
spaces: Lp( , )0 ∞ , 1 ≤ p ≤ + ∞ , and L1 ≡ L1(– , )∞ + ∞ . We denote by Ω a class of
the Wiener – Hopf integral operators (see [8]): W ∈ Ω if ( )W f = w x t f t dt( ) ( )−
∞
∫0
,
w ∈ L1 .
It is easy to check that the operator W acts in the space E
+ and the following esti-
mation holds:
W E+ ≤ w x dx( )
−∞
∞
∫ . (2.1)
The kernel w of the operator W is called conservative if
0 ≤ w ∈ L1 , γ df= w x dx( )
−∞
∞
∫ = 1. (2.2)
We also introduce the algebra Ω ±
∈ Ω of lower and upper Volterra-type operators:
V± ∈ Ω
± if
( )( ) ( ) ( )V f x x t f t dt
x
+ += −∫ ν
0
, ( )( ) ( ) ( )V f x t x f t dt
x
− −
∞
= −∫ ν , (2.3)
where x ∈ ( 0, ∞ ) , ν±
+∈L R1( ).
It is easy to see that Ω = Ω Ω+ −� . We denote by Ωλ a class of the following
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
FACTORIZATION OF ONE CONVOLUTION-TYPE INTEGRO-DIFFERENTIAL … 1557
integral operators: Qλ λ∈Ω if
( )( )Q f xλ = λ( ) ( ) ( )x q x t f t dt−
∞
∫
0
, (2.4)
where 0 1≤ ⋅ ≤λ( ) , λ ∈ ∞
+W R1( ) , q L R∈ 1( ) .
It is known that if W ∈ Ω , V±
±∈Ω , then V W− ∈Ω (see [9]). Below, we prove
one generalization of this fact and make essential use of it in the further reasoning.
Lemma 2.1. If Qλ λ∈Ω , then the following possibilities take place:
a) Q Vλ λ
+ ∈Ω , where V+
+∈Ω ,
b) V Q− ∈λ λΩ , where V−
−∈Ω , if and only if there exists a real function r ( t )
on R+ , for which
λ ( )x t+ = λ ( ) ( )x r t , r t t L R( ) ( ) ( )ν−
+∈ 1 .
Proof. Let f E∈ + be an arbitrary function. We have
( )( )Q V f xλ
+ = λ τ τ τ( ) ( ) ( ) ( )x q x t t f d dt
t
0 0
∞
+∫ ∫− −v . (2.5)
Changing the order of integration in (2.5), we obtain
( )( )Q V f xλ
+ = λ τ τ τ
τ
( ) ( ) ( ) ( )x f q x t t dt d
0
∞ ∞
+∫ ∫ − −v =
=
λ τ τ τ( ) ( ) ( ) ( )x f q x z z dz d
0 0
∞ ∞
+∫ ∫ − − v = λ τ τ( ) ( ) ( )x P x t f d
0
∞
∫ − ,
where
P x( ) =
0
∞
+∫ −q x z z dz( ) ( )v . (2.6)
It follows from Fubin’s theorem (see [10]) that P L R∈ 1( ). Now let V−
−∈Ω , Qλ λ∈Ω .
In this case, analogous discussions reduce to the following formula:
( )( )V Q f x−
λ =
0 0
∞ ∞
−∫ ∫ + − +f z x z q x z dz d( ) ( ) ( ) ( )τ λ τ τv =
0
∞
∫ ρ τ τ τ( , ) ( )x f d ,
where
ρ τ( , )x =
0
∞
−∫ + − +v ( ) ( ) ( )z x z q x z dzλ τ .
Let λ ( )x z+ = λ ( ) ( )x r z , where r z z L R( ) ( ) ( )v−
+∈ 1 . Then
ρ τ( , )x =
λ τ( ) ( ) ( ) ( )x z r z q x z dz
0
∞
−∫ − +v df= λ ρ τ( ) ( )x x0 − ,
where ρ0 1∈L R( ).
The inverse statement is proved by analogy.
The lemma is proved.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1558 A. KH. KHACHATRYAN, KH. A. KHACHATRYAN
Let us consider the following homogeneous equation on half line:
B x( ) = λ ( ) ( ) ( )x K x t B t dt
0
∞
∫ − (2.7)
with respect to an unknown function B L R∈ 1
loc( ), where 0 ≤ K ∈ L R1( ), K L1
, 0 ≤
≤ λ ( ⋅ ) ≤ 1 is a measurable function.
Below, we need the following theorem proved in [11]:
Theorem [11]. 1. If 0 ≤ λ ( x ) ≤ 1, 1 1− ∈ +λ( ) ( )x L R , ν( )K df= xK x dx( )
−∞
∞
∫ <
< 0, then equation (2.7) possesses a nontrivial bounded solution B ( x ) ≠ 0 and
B ( x ) = O ( 1 ) as x → + ∞ .
2. If 0 ≤ λ ( x ) ≤ 1, x x L R( ( )) ( )1 1− ∈ +λ , ν( )K = 0, then equation (2.7) pos-
sesses a solution B ( x ) ≥ 0, B ( x ) ≠ 0, and besides,
0
x
B t dt∫ ( ) = O ( x
2
) as x → + ∞ .
3. Factorization problem. We rewrite equation (1.1) in the operator form
D AI BK D CK S+ − −( )1 2
λ λ = g . (3.1)
We consider two possibilities: 1) A > 0, and 2) A = 0.
1. Let A > 0. We consider the following factorization problem: For operators D
and Kj
λ λ∈Ω , j = 1, 2, and for arbitrary α > 0, it is necessary to find operators
Wλ λ∈Ω and Uα ∈ +Ω such that the factorization
D AI BK D CK+ − −1 2
λ λ = I W I U D I−( ) −( ) +( )λ
α α (3.2)
holds as an equality of integral operators acting in W R1
1( )+ .
2. Let A = 0. For operators D and Kj
λ λ∈Ω , j = 1, 2, and for arbitrary α > 0,
it is necessary to find operators Vα ∈ +Ω , Hλ λ∈Ω such that the factorization
D BK D CK− −1 2
λ λ = I H V D I− −( ) +( )λ
α α (3.3)
takes place as an equality of integral operators acting in W R1
1( )+ .
The following lemma holds:
Lemma 3.1. Suppose that A > 0, Kj
λ λ∈Ω , j = 1, 2. Then for each α > 0,
the factorization (3.2) takes place. Kernel functions of the operators Wλ λ∈Ω
and Uα ∈ +Ω have the forms
w x tλ( , ) = λ( ) ( )x w x t− ,
where
w ( x ) = BK x cK t ABK t e dt
x
A x t
1 2 1( ) ( ) ( ) ( )+ −{ }
−∞
− −∫ , (3.4)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
FACTORIZATION OF ONE CONVOLUTION-TYPE INTEGRO-DIFFERENTIAL … 1559
and
u xα( ) = ( ) ( )α θα− −A e xx , where θ( )x =
1 0
1 0
, ,
, .
if x
if x
≥
<
(3.5)
Moreover, if K W R1 1
1∈ ( ) , then w W R∈ 1
1( ) .
Proof. We denote by Γα an inverse operator of the differential operator D I+ α
in the space W R f f1
1 0 0( ) : ( )+ ={ }∩ . It is easy to verify that Γα belongs to Ω+
and has the following form:
( )( )Γα f x =
0
x
x te f t dt∫ − −α( ) ( ) , α > 0. (3.6)
It follows from Lemma 2.1 that Qλ
df= Kj
λ Γα ∈ Ω+ , j = 1, 2, and kernels of the
operators Qj
λ are given by formulae
q x tj
λ( , ) = λ( ) ( )x q x tj − , q xj( ) =
−∞
− −∫
x
j
x tK t e dt( ) ( )α
∈ W R1
1( ), j = 1, 2. (3.7)
We have
D AI BK D CK+ − −1 2
λ λ = D I I AI BK D CK+ − + − −α α λ λ
1 2 = ( )( )I P D I− +α α ,
(3.8)
where
Pα = BK D CK A1 2
λ
α
λ
α ααΓ Γ Γ+ + −( ) . (3.9)
It is easy to see that DΓα = I − α αΓ , hence, Pα = R Uα
λ
α+ , where Rα
λ
∈ Ωλ ,
Rα
λ = BK CK BK1 2 1
λ λ λ
αα+ −( )Γ and Uα ∈ +Ω , the kernel of which is given by (3.5).
We denote by I + Φα the inverse of the operator I U− α in W R1
1( )+ . By means of
simple calculations, it is easy to verify that Φ Ωα ∈ + and, moreover,
( )( )Φα f x = ( ) ( )( )α − ∫ − −A e f t dt
x
A x t
0
, A > 0. (3.10)
Using (3.10), from (3.8) and (3.9) we have
D AI BK D CK+ − −1 2
λ λ = I R I I U D I− +( ) − +α
λ
α α α( ) ( )( )Φ =
= ( )( )( )I W I U D I− − +λ
α α ,
where W λ df= R Rα
λ
α
λ
α+ Φ .
Using Lemma 2.1, we conclude that W λ
∈ Ωλ . It is easy to check that operator
W λ does not depend on α . Actually, let f E∈ + be an arbitrary function. Then
( )( )R f xα
λ
αΦ = ( ) ( ) ( ) ( )( )α λ τ τα
τ− −
∞
− −∫ ∫A x r x t e f d dt
t
A x
0 0
,
where λ α( ) ( )x r x t− is the kernel of the operator Rα
λ . Changing the order of integra-
tion in the last integral and taking into account (3.7), we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1560 A. KH. KHACHATRYAN, KH. A. KHACHATRYAN
( )( )R f xα
λ
αΦ = λ τ τ τα( ) ( ) ( )x Y x f d
0
∞
∫ − ,
where
Y xα ( ) =
−∞
− −
−∞
− −∫ ∫− − −
x
A x
x
xCK A K e d CK A K e d{ ( ) ( )} { ( ) ( )}( ) ( )
2 1 2 1τ α τ τ τ α τ ττ α τ .
Hence, from (3.7) it follows that W λ does not depend on α , its kernel is given by
(3.4). Now we show that operator W λ acts in the space W R1
1( )+ . Really, let f be an
arbitrary function from W R1
1( )+ . We have
( )( )W f xλ = λ( ) ( ) ( )x w x t f t dt
0
∞
∫ − = λ τ τ τ( ) ( ) ( )x w f x d
x
−∞
∫ − .
We denote by ρ( )x the function
ρ( )x = λ τ τ τ( ) ( ) ( )x w f x d
x
−∞
∫ − .
Applying Fubin’s theorem and taking into account that 0 ≤ λ ( x ) ≤ 1, w L R∈ 1( ),
and f W R∈ 1
1( ), we obtain ρ ∈L R1( ). If λ ∈ ∞W R1 ( ), f W R∈ 1
1( ), then from equality
′ρ ( )x = ′ − + + ′ −
−∞ −∞
∫ ∫λ τ τ τ λ τ τ τ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x w f x d x w x f w f x d
x x
x0
it follows that ′ ∈ρ L R1( ). Therefore, ρ ∈W R1
1( ). From (3.4) it follows that if K1 ∈
∈ W R1
1( ) , then w W R∈ 1
1( ).
The lemma is proved.
It is simple to prove the following lemma:
Lemma 3.2. If A = 0, then operator D BK D CK− −1 2
λ λ permits factorization
of type (3.3), where k ernels of operators Vα ∈ +Ω , Hλ λ∈Ω are given,
respectively, by formulae
να( )x = α θαe xx− ( ), h x tλ( , ) = λ( ) ( )x h x t− , (3.11)
where
h x( ) = BK x CK t BK t e dt
x
x t
1 2 1( ) { ( ) ( )} ( )+ −
−∞
− −∫ α α . (3.12)
Further, we essentially use the following lemma that establishes connection betwe-
en first moments of functions w and K j , j = 1, 2:
Lemma 3.3. Suppose that
ν( )K j df= xK x dxj( )
−∞
+∞
∫ < + ∞ , j = 1, 2,
exists. Then ν( )w < + ∞ exists and the following formula holds:
ν( )w =
C AB
A
C
A
K
− +2 2ν( ) for A > 0.
Proof. As ν( )K j < + ∞ , j = 1, 2, then by Fubin’s theorem we have
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
FACTORIZATION OF ONE CONVOLUTION-TYPE INTEGRO-DIFFERENTIAL … 1561
ν( )w =
−∞
+∞
∫ xw x dx( ) =
= B K AB x K t e dt dx c x K t e dt dx
x
A x t
x
A x tν( ) ( ) ( )( ) ( )
1 1 2− +
−∞
∞
−∞
− −
−∞
∞
−∞
− −∫ ∫ ∫ ∫ =
= B K AB K t e xe dx dt C K t e xe dx dtAt
t
Ax At
t
Axν( ) ( ) ( )1 1 2− +
−∞
∞ ∞
−
−∞
∞ ∞
−∫ ∫ ∫ ∫ =
=
C AB
A
C
A
K
− +2 2ν( ).
The lemma is proved.
Remark. If C Kν( )2 ≤ B
C
A
− , A > 0, then ν( )w ≤ 0.
4. Solution of problem (1.1) – (1.4) for A = 0. Let us consider equation (1.1)
when A = 0. Using factorization (3.3), the equation (1.1) (for A = 0 ) we can write
in the following form:
( )( )I H V D I S− − +λ
α α = g. (4.1)
The solution of (4.1) is reduced to successive solution of the following equations:
( )I H V− −λ
α ϕ = g, (4.2)
( )D I S+ α = g. (4.3)
We denote by I + Φ the resolvent of operator I V− α in the space L R1
loc( )+ . It is
easy to check that ( )( )Φ f x = α f t dt
x
( )
0∫ . From representation of operator Φ it
follows that the operator Φ transfers the space L R1( )+ to the space M R( )+ , where
M R( ) is the space of bounded functions on R+ . We represent the operator
I H V− −λ
α in the following form:
I H V− −λ
α = ( )( )I G I V− − α , (4.4)
where G = H Hλ λ+ Φ . It is easy to check that the operator ( )( )Gf x is determined
as
( )( )Gf x = λ( ) ( ) ( )x G x t f t dt
0
0
∞
∫ − ,
where
G x0( ) = B K x C K t dt
x
1 2( ) ( )+
−∞
∫ , K L R1 1∈ ( ),
−∞
∫ ∈
x
K t dt M R2( ) ( ).
Using factorization (4.4), the solution of equation (4.2) is reduced to successive
solution of the following equations:
( )I G− Ψ = g, (4.5)
( )I V− α ϕ = ψ . (4.6)
We rewrite the equation (4.5) in the open form
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1562 A. KH. KHACHATRYAN, KH. A. KHACHATRYAN
ψ( )x = g x G x t t dt( ) ( ) ( )+ −
∞
∫λ ψ
0
0
and consider the following iteration process:
ψ( )( )n x+1 = g x x G x t t dtn( ) ( ) ( ) ( )( )+ −
∞
∫λ ψ
0
0 , ψ( )0 = 0, n = 0, 1, 2 … .
(4.7)
It is easy to see that g x( ) ≤ ψ( )n ↑ by n . We note that if λ ∈ +L R1( ) , then
ψ( ) ( )n L R∈ +
1 , n = 0, 1, 2 … . Really, for n = 0, we have ψ( )1 = g x( ) ∈ L R1( ).
Assume that ψ( )n ∈ ∈ L R1( )+ and prove that ψ( )n+1
∈ L R1( ). Then for arbitrary
r > 0 we have
0
1
r
n x dx∫ +ψ( )( ) ≤
0 0 0
0
∞ ∞ ∞
∫ ∫ ∫+ −g x dx x G x t t dt dxn( ) ( ) ( ) ( )( )λ ψ =
= g x dx B t K x t x dx dt C t K d x dx dtn n
x t
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
0 0
1
0 0 0
2
∞ ∞ ∞ ∞ ∞
−∞
−
∫ ∫ ∫ ∫ ∫ ∫+ − +ψ λ ψ τ τλ ≤
≤
0 0 0 0
∞ ∞ ∞ ∞
∫ ∫ ∫ ∫+ +g x dx B t dt C t dt x dxn n( ) ( ) ( ) ( )( ) ( )ψ ψ λ ⇒ ψ( )n+1 ∈ L R1( )+ .
It is also easy to check that
0
1
∞
+∫ ψ( )( )n x dx ≤ g x dx t G d t dt
t R t
n( ) ( ) ( ) ( )( )
0
0
0
1
∞
∈ −
∞ ∞
+∫ ∫ ∫+ +vrai max
+
λ τ τ τ ψ . (4.8)
Now we suppose that
q0 df= vrai max
+t R t
t G d
∈ −
∞
∫ +λ τ τ τ( ) ( )0 < 1. (4.9)
Then from (4.8), taking into account (4.9), we receive
0
1
∞
+∫ ψ( )( )n x dx ≤ ( ) ( )1 0
1
0
− −
∞
∫q g x dx .
From B. Levi’s theorem (see [10]) it follows that the sequence ψ( )n almost
everywhere in R+ has a limit ψ( )x = lim
n
n x
→∞
ψ( )( ), and besides ψ ∈ +L R1( ).
We prove that ψ( )x is the solution of equation (4.5). Actually, from (4.7) we
have
ψ( )( )n x+1 ≤ g x x G x t t dt( ) ( ) ( ) ( )+ −
∞
∫λ ψ
0
0 , n = 0, 1, 2 … . (4.10)
Passing to the limit in the last inequality, we obtain
ψ( )x ≤ g x x G x t t dt( ) ( ) ( ) ( )+ −
∞
∫λ ψ
0
0 . (4.11)
On the other hand,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
FACTORIZATION OF ONE CONVOLUTION-TYPE INTEGRO-DIFFERENTIAL … 1563
g x x G x t t dtn( ) ( ) ( ) ( )( )+ −
∞
∫λ ψ
0
0 ≤ ψ( )x . (4.12)
From Lebeg’s theorem it follows that
g x x G x t t dt( ) ( ) ( ) ( )+ −
∞
∫λ ψ
0
0 ≤ ψ( )x . (4.13)
Combining inequalities (4.11) and (4.12), we get
ψ( )x = g x x G x t t dt( ) ( ) ( ) ( )+ −
∞
∫λ ψ
0
0 . (4.14)
Now we pass to the solution of the equation (4.6):
ϕ( )x = ψ α ϕα( ) ( )( )x e t dt
x
x t+ ∫ − −
0
. (4.15)
It is obvious that
ϕ( )x = ψ α ψ( ) ( )x t dt
x
+ ∫
0
. (4.16)
Finally solving equation (4.3) and taking into account (1.4), we obtain
S x( ) = s e e t dtx
x
x t
0
0
− − −+ ∫α α ϕ( ) ( ) . (4.17)
Using formula (4.16), we have
S x( ) = s e t dtx
x
0
0
− + ∫α ψ( ) . (4.18)
In its turn, it follows that
0
∞
∫ g x dx( ) ≤ S( )+ ∞ =
0
∞
∫ ψ( )t dt ≤
g x dx
q
( )
0
01
∞
∫
−
. (4.19)
The following theorem holds:
Theorem 4.1. Let 0 ≤ λ ( x ) ≤ 1, λ ∈ +
∞
+L R W R1
1( ) ( )∩ , and let the following
estimation be true :
vrai max
+t R t
t G d
∈ −
∞
∫ +λ τ τ τ( ) ( )0 < 1,
where G x0( ) = BK x C K t dt
x
1 2( ) ( )+
−∞
∫ .
Then problem (1.1) – (1.4) for A = 0 in the class � ( )R+ possesses a
positive solution of the type (4.18) and inequality (4.19) is true.
5. Solution of equation (1.1) – (1.4) for A > 0. In this section, we study
equation (1.1) – (1.4) for A > 0. In this case, we consider the following three
possibilities: 1) A > C ≥ 0, 2) A = C > 0, 3) 0 < A < C.
5.1. Equation (1.1) – (1.4) in case A > C ≥≥≥≥ 0. The following theorem is
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1564 A. KH. KHACHATRYAN, KH. A. KHACHATRYAN
true:
Theorem 5.1. Suppose that a) w x( ) ≥ 0, x R∈ , b) 0 ≤ λ ( x ) ≤ 1, λ &∈ W R∞
+1 ( ) .
Then the problem (1.1) – (1.4) for A > C ≥ 0 in the space W R1
1( )+ has a posi-
tive solution of the type
S x( ) = s e e F t dtx
x
x t
0
0
− − −+ ∫α α( ) ( ) , (5.1)
where α > 0 is the constant, 0 1≤ ∈ +F L R( ).
Proof. Using factorization (3.2), the equation (1.1) may be written in the form
( )( )( )I W I U D I S− − +λ
α α = g . (5.2)
Solution of (5.2) is reduced to successive solution of the following equations:
( )I W F− λ = g , (5.3)
( )I U− α χ = F , (5.4)
( )D I S+ α = χ . (5.5)
We rewrite the equation (5.3) in the open form and consider the iteration
F xn( )( )+1 = g x x w x t F t dtn( ) ( ) ( ) ( )( )+ −
∞
∫λ
0
, F( )0 = 0, n = 0, 1, 2 … . (5.6)
By induction, it is easy to check that
g x( ) ≤ F n( ) ∈ L R1( )+ , n = 1, 2, … , F n( ) ↑ by n . (5.7)
Therefore, we have
0
1
∞
+∫ F x dxn( )( ) ≤
0 0 0
1
∞ ∞ ∞
+∫ ∫ ∫+ −g x dx x w x t F t dt dxn( ) ( ) ( ) ( )( )λ =
=
0 0
1
∞ ∞
+
−∞
∞
∫ ∫ ∫+ +g x dx F t w t t z dz dtn( ) ( ) ( ) ( )( ) λ ≤
0 0
1
∞ ∞
+∫ ∫+g x dx F t dtn( ) ( )( )γ ,
where
γ =
−∞
∞
∫ w x dx( ) =
C
A
< 1. (5.8)
As (5.7) and (5.8) are satisfied, then form B. Levi’s theorem it follows that the se-
quence { }( )( )F xn+ ∞1
0 converges almost everywhere in R+ to an integrable function
F x( ). It is obvious that the function F x( ) is the solution of equation (5.6).
Successively solving equations (5.4) and (5.5), we arrive to result (5.1).
The theorem is proved.
5.2. Equation (1.1) – (1.4) in case A = C > 0. The following theorem holds:
Theorem 5.2. Suppose that the following conditions are satisfied: i) w x( ) ≥ 0,
x R∈ , ii) 0 ≤ λ ( x ) ≤ 1, λ &∈ W R∞
1 ( ), iii) ν( )K j < + ∞ , j = 1, 2, exists and
moreover, ν( )K2 ≤ ( )B A− /1 . Then problem (1.1) – (1.4) for A = C > 0 i n
the class � possesses the solution of the following structure :
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
FACTORIZATION OF ONE CONVOLUTION-TYPE INTEGRO-DIFFERENTIAL … 1565
S x( ) = s e e t dtx
x
A x t
0
0
− − −+ ∫α ϕ( ) ( ) . (5.9)
Here, 0 < α = const, 0 1≤ ∈ +ϕ L Rloc( ),
0
x
t dt∫ ϕ( ) = o f t dt
x
0
∫
( ) for x → + ∞ ,
where f x( ) is the positive increasing function, f ( )0 = 1 and if ν( )K2 < (B –
– 1) / A, then f x( ) = O( )1 for x → + ∞ , and if ν( )K2 = ( )B A− /1 , then
f x( ) = O x( ) for x → + ∞ .
Proof. From the condition A = C > 0 it follows that γ = 1. Together with
(5.3), we consider the following auxiliary equation:
˜ ( )F x = g x w x t F t dt( ) ( ) ˜ ( )+ −
∞
∫
0
, (5.10)
f x( ) =
0
∞
∫ −w x t f t dt( ) ( ) . (5.11)
It was proved in [12, 13] that if ν( )w ≤ 0, 0 1≤ ∈ +g L R( ), then equation (5.10) in
L R1
loc( )+ has positive solution which, almost everywhere in ( 0, + ∞ ) , is the limit of
the following simple iterations:
˜ ( )( )F xn+1 = g x w x t F t dtn( ) ( ) ˜ ( )( )+ −
∞
∫
0
, ˜ ( )F 0 = 0, n = 0, 1, 2, … , (5.12)
and the asymptotic
0
x
F t dt∫ ˜ ( ) = o f t dt
x
0
∫
( ) , x → + ∞ , (5.13)
is true, where f is a positive increasing solution of equation (5.11), f ( )0 = 1.
Mentioned solution f satisfies also the following conditions: f x( ) = O x( ), ( x → ∞ )
for ν( )w = 0 and f x( ) = O( )1 , x → ∞ , for ν( )w < 0. We consider the follo-
wing iteration for equation (5.3) (in the case A = C > 0 ) :
F xn( )( )+1 = g x w x t F t dtn( ) ( ) ( )( )+ −
∞
∫
0
, ˜ ( )F 0 = 0, n = 0, 1, 2, … . (5.14)
It is easy to show that
i) g x( ) ≤ F n( ) ↑ by n , ii) F n( ) ≤ ˜ ( )F n almost everywhere in ( 0, + ∞ ) .
Hence, almost everywhere in R+ , there exists F x( ) = lim ( )( )
n
nF x
→∞
and
0 ≤ g x( ) ≤ F x( ) ≤ ˜ ( )F x . (5.15)
It is obvious that F x( ) is the solution of equation (5.3) for A = C > 0 (the proof of
last fact has analogy with Theorem 4.1). Using (5.13), (5.15) and Lemma 3.3, we
obtain
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1566 A. KH. KHACHATRYAN, KH. A. KHACHATRYAN
0
x
F t dt∫ ( ) = o f t dt
x
0
∫
( ) , x → + ∞ .
Solving equations (5.4) and (5.5), we obtain (5.9).
The theorem is proved.
5.3. Equation (1.1) – (1.4) in case C > A > 0. Doing analogous discussions
as in Theorems 4.1 and 5.1, we get the following theorem:
Theorem 5.3. Let i) w x( ) ≥ 0, x R∈ , ii) 0 ≤ λ ( x ) ≤ 1, λ ( )x &∈ W R∞
+1 ( ) ,
iii) the inequality
vrai max
+t R t
t w d
∈ −
∞
∫ +λ τ τ τ( ) ( ) < 1
takes place. Then problem (1.1) – (1.4) for C > A > 0 in W R1
1( )+ possesses
a solution of the type (4.18).
6. Construction of nontrivial solution of homogeneous equation (1.7). The
factorization (3.2) allows us to construct nontrivial solution of corresponding homoge-
neous equation when A = C > 0. Unfortunately, for other values of parameters A
and C, up to now we were not able to construct a nontrivial solution. It is known only
that, in the case A > C > 0, the homogeneous equation F x( ) = λ ( ) ( )x w x t−
∞
∫0
×
× F t dt( ) in the class � has no nontrivial solutions. It is also known that the homo-
geneous equation, in the case A = C > 0 and ν( )w > 0, in � has no nontrivial
solutions either. On evristic level we conclude that for other values of parameters A
and C nontrivial solutions do not exist.
We consider corresponding homogeneous equation (1.1) – (1.4) for A = C > 0
(see (1.7)).
Using factorization (3.2), we rewrite the equation (1.7) in the form
( )( )( )I W I U D I S− − +λ
α α = 0. (6.1)
The equation is equivalent to the successive solution of the following equations:
( )I W− λ ρ1 = 0, (6.2)
( )I U− α ρ2 = ρ1, (6.3)
( )D I S+ α = ρ2. (6.4)
We write equation (6.2) in the open form: ρ1( )x = λ ρ( ) ( ) ( )x w x t t dt−
∞
∫0 1 .
As A = C > 0, then γ = 1. Using Theorem from [11] (see Sec. 2 of this paper),
Lemma 3.3 and solving equations (6.3) and (6.4), we obtain the following results:
Theorem 6.1. A. Suppose that i) w x( ) ≥ 0, ii) 0 ≤ λ ( x ) ≤ 1, λ ( )x &∈ W R∞
+1 ( ) ,
I x L R− ∈ +λ ( ) ( )1 , iii) ν( )K
B
A2
1< −
.
Then the problem (1.7), (1.3), (1.4) for A = C > 0 in the class � possesses
a nontrivial solution of the type
S x( ) = s e e t dtx
x
A x t
0
0
1
− − −+ ∫α ρ( ) ( ) , (6.5)
where ρ1 ≠ 0 and ρ1( )x = O ( )1 , x → ∞ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
FACTORIZATION OF ONE CONVOLUTION-TYPE INTEGRO-DIFFERENTIAL … 1567
B. L e t i) w x( ) ≥ 0, x R∈ + , ii) 0 ≤ λ ( x ) ≤ 1, λ ( )x &∈ W R∞
+1 ( ) ,
x x( ( ))1 − λ ∈ ∈ L R1( )+ , iii) ν( )K2 ≤
B
A
− 1
. Then the problem (1.7), (1.3), (1.4)
for A = C > 0 in the class � possesses a nontrivial solution of the type (6.5),
where ρ1 ≥ 0, ρ1 ≠ 0, and has the asymptotic behaviour
0 1
x
t dt∫ ρ ( ) = O x( )2 ,
x → + ∞ .
The authors express their gratitude to Professor N. B. Yengibaryan for useful dis-
cussions.
1. Latishev A. V., Yushkanov A. A. A precise solution of the problem of current passage over borders
between crystals in metal (in Russian) // FFT. – 2001. – 43, # 10. – P. 1744 – 1750.
2. Latishev A. V., Yushkanov A. A. Electron plazma in seminfinite metal at presece of alternating
electric field (in Russian) // Zh. Vych. Mat. i Mat. Fiz. – 2001. – 41, # 8. – P. 1229 – 1241.
3. Livsicˆ E. M., Pitaevskii L. P. Physical kinetics (in Russian). – Moscow: Nauka, 1979. – Vol. 10.
4. Khachatryan Kh. A. Integro-differential equations of physical kinetics // J. Contemp. Math. Anal. –
2004. – 39, # 3. – P. 49 – 57.
5. Khachatryan A. Kh., Khachatryan Kh. A. On solvability of some integro-differential equation with
sum-difference kernels // Int. J. Pure and Appl. Math. Sci. (India). – 2005. – 2, # 1. – P. 1 – 13.
6. Khachatryan Kh. A. Ph. D. – Yerevan, 2005. – 115 p. (in Russian).
7. Khachatryan A. Kh., Khachatryan Kh. A. On structure of solution of one integro-differential
equation with completely monotonic kernel // Int. Conf. Harmonic Anal. and Approxim. – Armenia,
Tsakhadzor. – 2005. – P. 42 – 43.
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706 S.
9. Yengibaryan N. B., Arabajian L. G. Some factorization problems for integral operators of
convolution type (in Russian) // Differents. Uravneniya. – 1990. – 26, # 8. – P. 1442 – 1452.
10. Kolmogorov A. N., Fomin V. S. Elements of the theory of functions and functional analysis. –
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Received 17.04.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
|
| id | umjimathkievua-article-3269 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:39:18Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3a/4538a0bda6b7df7d1aa9e44bbe8c9c3a.pdf |
| spelling | umjimathkievua-article-32692020-03-18T19:49:31Z Factorization of a convolution-type integro-differential equation on the positive half line Факторизація одного інтегро-диференціального рівняння типу згортки на додатній півосі Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. Sufficient conditions for the existence of solutions are obtained for a class of convolution-type integro-differential equations on the half line. The investigation is based on the three-factor decomposition of the initial integro-differential operator. Отримано достатні умови для існування розв'язку одного класу інтегро-диференціальних рівнянь типу згортки на півосі. Дослідження базуються на розкладі початкового інтегро-дифе-ренціального оператора на три множники. Institute of Mathematics, NAS of Ukraine 2008-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3269 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 11 (2008); 1555–1567 Український математичний журнал; Том 60 № 11 (2008); 1555–1567 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3269/3284 https://umj.imath.kiev.ua/index.php/umj/article/view/3269/3285 Copyright (c) 2008 Khachatryan A. Kh.; Khachatryan Kh. A. |
| spellingShingle | Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. Factorization of a convolution-type integro-differential equation on the positive half line |
| title | Factorization of a convolution-type integro-differential equation on the positive half line |
| title_alt | Факторизація одного інтегро-диференціального рівняння типу згортки на додатній півосі |
| title_full | Factorization of a convolution-type integro-differential equation on the positive half line |
| title_fullStr | Factorization of a convolution-type integro-differential equation on the positive half line |
| title_full_unstemmed | Factorization of a convolution-type integro-differential equation on the positive half line |
| title_short | Factorization of a convolution-type integro-differential equation on the positive half line |
| title_sort | factorization of a convolution-type integro-differential equation on the positive half line |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3269 |
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