On the Solution of Equations with Linear-Fractional Shifts
ANNA TARASENKO, OLEKSANDR KARELIN, MANUEL GONZÁLEZ-HERNÁNDEZ,
DARYA KARELINA
Institute of Basic Sciences and Engineering,
Hidalgo State Autonomous University,
Carretera Pachuca-Tulancingo, km.4.5,
Pachuca, Hidalgo, C.P. 42184,
MEXICO
Abstract: - This work represents a continuation of the studies relating to nonlinear equations, carried out by the
authors. Special attention is paid to the operators with linear-fractional shifts that act on the argument of the
unknown function, but also on the unknown function itself. In this work, we study homogeneous equations with
such operators. The main classes of functions for which non-linear equations are considered are Hölder class
real functions. Solutions of the equations have the form of infinite products or the form of infinite continued
fractions; an abstract description of the solutions is also offered. The developed mathematical methods can be
applied to finding the conditions of invertibility of certain operators found in modelling, as well as for the
construction of their inverse operators. Subsequently, we suggest using these results for the modelling of
renewable systems with elements that can be in different states: sick, healthy, immune, or vaccinated. These
results can also be applied to the analysis of balance equations of the model and for finding equilibrium states
of the system.
Key-Words: - Operator with linear-fractional shifts, Non-linear equations, Homogeneous equation, infinite
continued fraction.
Received: August 9, 2022. Revised: February 25, 2023. Accepted: March 21, 2023. Published: April 26, 2023.
I Introduction
Previously, attention to functional operators with
shift was initiated due to the development of the
theory of solvability of boundary value problems
and singular integral equations with Carleman and
non-Carleman shifts, [1], [2], [3].
Now, the interest and motivation for the study of
such operators are growing in connection with the
problems of depletion of natural resources and
research on the possibility of using renewable
resources.
So, in the modelling of systems with renewable
resources, [4], [5], equations that contain functional
operators with shift appear in balance relations. We
consider the simplest functional operator with shift
( ) ( ) ( ) ( ) [ ( )].A x a x x b x x
We found
conditions of invertibility of the operator
A
in
Hölder space with weight and constructed operator
1
A
inverse to it, [6], [7]. In the analysis of
equations of balance relations of such systems, these
inverse operators are used. In this way, due to the
obtained results, equilibrium states of the considered
renewable systems were found, in addition to
formulating the corresponding economic-ecological
problems, [8].
We consider an operator with a more complex
form
( ) ( ) ( ) ( )C x k x x AB x
, where the
operator
B
has the same structure as the operator
A
( ) ( ) ( ) ( )[ ( )]B x c x x d x x
. The operator
C
appears in the investigation of systems with
elements that can be in multiple states, for instance,
be sick, have immunity, or be vaccinated. The
conditions of invertibility of the composition of
operators
AB
can be found as the union of
invertibility conditions for the operator
A
and for
the operator
B
. The inverse operator
1
AB
is
constructed as the superposition of inverse operators
1
A
1
B
.
For the operator
W kI AB
, which arises in
the modelling of systems with renewable resources
with elements that can be in different states, the
conditions of invertibility in the Hölder space with
weight, in the general case, are unknown, as are the
types of the inverse operator. The interest and the
motivation for the study of such operators are
growing.
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Studying the possibility of reducing the operator
W kI AB
to the composition of more simple
functional operators with shift, a system of
nonlinear equations that describes the connections
between the coefficients emerges:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
k x a x c x x x
a x d x b x B c x x x x B x
b x B d x x B x
Substituting
()x
from the first equation of the
system and
()x
from the third equation of the
system to the second equation, we obtain
( ) ( ) / ( ) ( )x G x B x g x
, where
()Gx
and
()gx
are expressed through the known functions
()kx
,
()ax
,
()bx
,
()cx
,
()dx
and
( ) ( )/ ( )x x x
.
Substituting
()x
from the first equation of the
system as well as
()Bx
from the third equation
of the system to the second equation, we obtain a
non-linear equation. As we can observe, the
solubility of non-linear equations plays an especially
important part.
The present work is dedicated to the study of such
nonlinear equations and their various
generalizations.
Special attention is dedicated to operators with a
linear-fractional shift, which acts not only on the
argument of the unknown function
12
34
( ) [ ( )], ( ) ,
a x a
B x x x a x a
but also on the unknown function itself:
12
34
()
( ) .
()
a x a
xa x a
To obtain conditions for the invertibility of
functional operators with a shift in weighted Hölder
spaces and to construct inverse operators, the theory
of functional series was used as the main
mathematical apparatus, [6], [7]. In this work,
homogeneous non-linear equations that contain
operators with the linear-fractional shift are
considered. Other mathematical tools, such as
infinite products and infinite continued fractions,
were used to describe solutions of non-linear
equations with a shift.
2 Equation with Two Shifts in Hölder
Space
Let us remember the definition of a Hölder
space. The function
()x
, which satisfies the
following, conditions on
[0,1]J
,
1 2 1 2
( ) ( )x x C x x
,
12
,x J x J
,
0,1
, is called a Hölder
function with an exponent
and a constant
C
.
The functions of Hölder class form a set. The
norm in is defined by:
, where
( ) max ( ) ,
C
f x f x x J
and
12
12
12
( ) ( )
( ) sup ,
xx
f x f x
f x x J
xx
.
Let us construct a linear-fractional shift
12
34
() a x a
xa x a
with the properties: to be bijective
and to preserve orientation on
J
: if
12
xx
, then
12
( ) ( )xx
for any
11
,,x J x J
, and let c
()x
have only two fixed points:
(0) 0, (1) 1
and
()xx
when
0,1x
.
This implies that
20a
1
34
1
a
aa
that is to say,
1 3 4
a a a
. Additionally, let be a
differentiable function with
and require
that the derivative of the shift be positive:
1 3 4 3 1 14
22
3 4 3 4
()
() ( ) ( )
a a x a a a x aa
dx
dx a x a a x a
,
that is to say
14 0aa
. In this way, we obtain
34
34
() ax
xa x a
, where
34 3 4
a a a
and
34 4 0aa
.
In what follows, by
()Bx
or by
()Bx
we
are going to understand precisely this functional
operator of shift,
( ) [ ( )]B x x
, with the
function
()x
, which has exactly these properties.
At the same time as the shift
()x
, we consider the
linear-fractional shift
12
34
( ) .
b x b
xb x b
Let us search what should be the ratios of
coefficients
()x
y for these shifts to
commute. We write the equality
[ ( )] [ ( )]xx
and carry out some arithmetic operations:
34 12
1 2 34
3 4 3 4
34 1 2
34
34
34
34
,
ax b x b
b b a
a x a b x b
a x b x b
aa
bb
b x b
a x a
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1 34 2 3 4 34 1 2
3 34 4 3 4 3 1 2 4 3 4
,
ba x b a x a a b x b
b a x b a x a a b x b a b x b
1 34 2 3 4 3 1 2 4 3 4
[ ( )][ ( ) ( )]ba x b a x a a b x b a b x b
3 34 4 3 4 34 1 2
[ ( )][ ( )].b a x b a x a a b x b
Furthermore,
2
34 1 3 34 4 3
()a b b a b a x
34 1 4 4 34 2 3 34 4 3 34 2 4 4
2
3 1 4 3 1 34 2 3
3 1 4 3 2 4 3 2 4 4 1 34 2 3
3 2 4 4 2 4
34 2 4 4 3 2 4 4 2 4
[ ( )]
( )( )
[( ) ( )( )]
( ) ,
( ) .
a bb a a b b a b a x a b b a
a b a b b a b a x
a b a b b a a b a b ba b a x
a b a b b a
a b b a a b a b b a
Comparing the coefficients in the corresponding
power functions, we obtain:
34 1 3 34 4 3 3 1 4 3 1 34 2 3
( ) ( )( ), (1)a b b a b a a b a b ba b a
34 1 4 4 34 2 3 34 4 3
[ ( )]a bb a a b b a b a
3 1 4 3 2 4 3 2 4 4 1 34 2 3
[( ) ( )( )], (2)a b a b b a a b a b ba b a
34 2 4 4 3 2 4 4 2 4
( ) .a b b a a b a b b a
(3)
The constants
3 4 1 2 3 4
, , , , ,a a b b b b
, connected by
the relations (1-3), form two shifts that commute
with each other,
34
34
() a a x
xa x a
and
12
34
( ) .
b x b
xb x b
After analysing the system (1), (2), (3), it turns out
that the linear-fractional commutates for
B
are
functional operators with shifts of the same type
3 4 2
34
() b b x b
Bx b x b
, but without the
mandatory requirement
3 4 4 0b b b
, which is
satisfied for
34
34
() a a x
xa x a
:
3 4 4 0a a a
.
Let
20b
and
1 3 4
0, 0, 0b b b
.
The relation (3) is consolidated and transforms to
34 4 3 2 4 4
()a b a b a b
, we obtain
3 4 4 3 2 4 4
()a a b a b a b
and
42
bb
.
Let us turn now to (1). We are going to multiply
this relation by
33
11
aa
:
34 34 34
4
1 3 4 1 3 1 2
3 3 3 3
( ) ( )( )
a a a
a
b b b b b b b
a a a a
and transcribe the obtained equality relative to
the unknown
34
3
a
Za
.
Considering that
4
3
1
aZ
a
42
bb
, we have:
1 3 4 1 3 1 2
( ) ( 1 )( )Zb b Z b b Z b bZ b
and
1 2 1 1 3 2 3 1 1 2 3 2
Z( )bb bb b b b b bb b b
.
We obtain
2
21
b
Zbb
or
34 2
3 2 1
aa b
a b b
as well as
41
3 2 1
1ab
Za b b
.
We now turn the relation (2) into the form
34 2 3 34 3 1 2 34 4 2 3 4 3 1 2 34
a b b a a bb a a b b a a bb a
and multiply it by
33
11
aa
, turning it into
34 34 34
4
1 3 4 1 3 1 2
3 3 3 3
( ) ( )( )
a a a
a
b b b b b b b
a a a a
,
or well,
22
2 3 1 2 2 3 1 2
(Z 1) (Z 1)b b Z bb b b bb Z
and
finally we obtain,
1 2 2 3 1 2 2 3
(2 2 )Zbb b b bb b b
, as well as
1
2
Z
or
34
3
1
2
a
a
and
41
3 2 1
1.
ab
Za b b
We have fundamentally simplified the relations
between the coefficients and we put them together:
42
bb
,
41
3 2 1
ab
a b b
,
4
3
1
2
a
a
,
3 4 4 0a a a
.
Here we recall that the last equality must hold.
So we get
2 2 2 2 2
4
3 4 4 3 4 3 4
3
1
, , .
2
a
a a a a a a a
a
We come to a contradiction:
2
4
2
3
11
24
a
a
.
Now, let
20b
,
1 3 4
0, 0, 0b b b
.
The relation (3) degenerates into
3 34 4 3 3 1 4 3
b a b a a b a b
. From here,
3 4 1
b b b
.
We consider the equation
( ) ( ) ( ) 0x G x B B x
, (4)
with the initial condition and the
condition of concordance
1 (1) (1)G B B
.
The equation will be considered in the space
()HJ
, which means that the coefficient
()Gx
belongs to
()HJ
and the function
()x
is searched for in
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()HJ
. We assume the functions under
consideration to be positive, which corresponds to
the equations that arise in applications.
We write the reductive representation:
( ) ( ) ( )x G x B B x
.
There is a representation of the solution in the form
of an infinite product:
( ) ( ) [ ( ) ( )]x G x B B G x B B x
( ) [ ( ) [ ( ) ( )]]G x B B G x B B G x B B x
1
( ) ( )G x B B B B G x
1 2 2 2
( ) ( ) [ ( )]B B B B B B G x B B x
(5)
The shifts possess multiplicative properties:
[ ( ) ( )] [ ( )] [ ( )]B a x b x B a x B b x
,
[ ( ) ( )] [ ( )] [ ( )]B a x b x B a x B b x
,
hence their composition has the multiplicative
property
[ ( ) ( )] [ ( )] [ ( )]B B a x b x B B a x B B b x
,
which was used in the transformations carried out
above.
We derive the multiplier
( ) [ ( )]
n
B B x
at the
beginning of the representation
( ) ( ) [ ( )] ( ) ( ) ( ) ( )
nn
x B B x G x B B G x B B G x
Note that
1 2 2
( ) [ ]
n n n n
B B B B B B B B B B B B
.
We get
1 2 2
( ) [ ( )]
n n n
x B B B B B B B B B B x
( ) ( ) ( ) ( )
n
G x B B G x B B G x
For some functional operators with fractional
linear shifts
12
34
( ) [ ( )], ( ) b x b
B x x x b x b
and
B
, it is possible to calculate
lim [ ( )]
n
nn
Bx
, where
1 2 2 nn
nB B B B B B B B B
And
lim ( ) ( ) ( ) ( )
n
nG x B B G x B B G x
.
As it was shown, for
B
,
34
34
() a a x
xa x a
,
3 4 4 0a a a
the fractional linear commutates will
be
( ) [ ( )]B x x
,
34
34
() b b x
xb x b
. The
representation (5) of the equation is simplified and
will take the form
2 2 2 2
[ ( )]x B B x G x B B G x B B G x
If the operator
B
is still requested to be
3 4 4 0b b b
, then we obtain that not only is
lim 1
n
nBx
but also
lim 1
n
nBx
.
From here,
0
lim 1
nn
nB B x
.
We obtain
2 2 3 3
[ (1)]x G x B B G x B B G x B B G x
Theorem 1
Solution of equation (4) in the class of functions
()HJ
with the initial condition
0
1 0,
when
34
34
() a a x
xa x a
,
34
34
() b b x
xb x b
,
3 4 3 4
0, 0, 0, 0a a b b
, is represented by
2 2 3 3
0
x G x B B G x B B G x B B G x
The convergence of the infinite product ensures the
existence of a unique solution.
3 Equation with an External Operator
with Shift and an Internal Operator
with Shift
Let us consider the non-linear equation
( ) 0x G x B x
, (6)
where the coefficient
0Gx
and belongs to
()HJ
. The unknown function
()x
is also
searched for in
()HJ
. The operator
()Bx
is
the shift operator, described in section 2. Equation
(6) has two shifts,
is an external nonlinear
operator, described by the formula
( ) [ ( )]xx
, where
()x
is a Hölder class
function.
Let us note that
B
. The operator
()x
transforms the whole function
as a single entity,
while the shift operator
()Bx
transforms only the
argument
x
. Commutativity
BB
has a place. The operator
does not have the multiplicative property
( ) ( ) ( )a b a b
and
( ) ( ) ( )bb
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when
is constant. Let us write out the
concordance condition
1 (1)( )(1)G
, which
follows directly from equation (6). Note that in the
example from [5], the value of
1
is calculated
through the value of
(1)G
.
We are going to write the recurring relation and use
it to express the solution
()x
( ) ( )( ) ( ) ( [ ( ) ( ( ))])G x B x G x B G x B x
2
( ) ( ( ) [ ( ) ( )])G x B G x B G x B x
22
( ) ( ( ) ( ( ) ( )))G x B G x B G x B B x
=
23
( ) ( ( ) ( ( ) ( )))G x B G x B G x B x
23
( ) ( ( ) ( ( ) [ ]))G x B G x B G x B
.
As an example that provides the realization of the
operator
, we are going to take
12
34
()
() ()
b x b
xb x b
. We will assume that the
constants that define the operator
are positive.
The equation (6) obtains the form
12
34
()
( ) ( ) 0
()
b x b
x G x B b x b
.
Let's emphasize again. Here, the internal operator
( ) [ ( )]B x x
acts on the argument; it is defined
by the composite function
. The external
operator is defined by the composite function
12
34
()
( )( ) ()
b x b
xb x b
.
The solution of the equation has the form of an
infinite continued fraction, [9]:
12
34
()
( ) ( ) ()
b B x b
x G x b B x b
2
12
12
2
34
2
12
34
2
34
()
() ()
() ()
() ()
b B x b
b B G x b
b B x b
Gx b B x b
b B G x b
b B x b
Theorem 2
Solution of equation (6) in the class of functions
()HJ
is represented by
2
12
12
2
34
2
12
34
2
34
[]
() []
( ) .
[]
() []
b B b
b B G x b
b B b
Gx b B b
b B G x b
b B b
The convergence of the infinite product ensures the
existence of a unique solution.
In the work, [10], the homogeneous non-linear
equation
1
( ) ( ) 0
()
x G x Bx
(7)
was studied in the Hölder space
()HJ
.
In our notation,
1
() ()
xx
. To simplify, it is
assumed that all the considered functions are
positive. Briefly, we will write the results obtained
in solving this equation, as this material will serve
as a demonstrative example of the method proposed
for the solution of the nonlinear equation (7). We
write the recurrent relation
()
() ()
Gx
xBx
.
From here, it follows that:
2
( ) ( )
( ) [ ( )]
()
()
[ ( )]
G x G x
x B x
B G x
Gx
BBx
2
2
3
()
( ) ( ) ( )
( ) [ ( )] ( ) [ ( )]
B G x
G x G x G x
B
B G x B x B G x B x
2
3
()
()
()
()
[ ( )]
B G x
Gx Gx
B G x BBx
2
4
3
()
()
[ ( )] ( ) ( )
B G x
Gx
Bx
B G x B G x
The condition for the solvability of equation (7) is
the condition for the convergence of the infinite
product according to the Hölder norm to the
function from
()HJ
. The solution of the non-
linear non-homogeneous equation (7) is going to be:
24
35
( ) ( ( )) ( ( ))
( ) (1) ( ) ( ( )) ( ( ))
G x B G x B G x
xG
B G x B G x B G x
The fact that
(1) (1)G
follows from the
equality obtained from the equation when
1x
:
(1)
(1) (1)
G
. Of the two values
(1) (1)G
,
the value
(1) (1)G
chosen corresponds to the
positivity of the solutions
()x
.
4 Conclusions
In this work, certain homogeneous non-linear
equations that contain operators with the linear-
fractional shift are considered. The main method is
the representation of solutions through recurrent
relations. The proposed approach can serve as a
means of constructing inverse operators to operators
with linear-fractional shifts. In certain cases, this
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method is easier. So, in the works, [6], [7], the
construction of inverse operators for linear
functional operators looks cumbersome. The authors
plan to further investigate non-homogeneous non-
linear equations and apply the proposed method to
other types of nonlinear equations, particularly to
equations that contain conjunction operator.
In addition, it is planned to develop a mathematical
model of renewable systems with elements in
various states and subsequently use the obtained
results in the analysis of balance relations obtained
in modelling.
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[8] Karelin A., Tarasenko A., Zolotov V.,
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Oleksandr Karelin, Anna Tarasenko, Manuel
González-Hernández and Darya Karelina
contributed equally to the creation of this article.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors declare that they have no conflicts of
interest in the publication of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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Anna Tarasenko, Oleksandr Karelin,
Manuel González-Hernández, Darya Karelina
E-ISSN: 2224-2880
258
Volume 22, 2023
