On the Iterated Method for the Solution of Functional Equations with
Shift Whose Fixed Points are Located at the Ends of a Contour
ANNA TARASENKO, OLEKSANDR KARELIN, MANUEL GONZALEZ-HERNANDEZ,
JOSELITO MEDINA-MARIN
Institute of Basic Sciences and Engineering,
Hidalgo State Autonomous University,
Carretera Pachuca-Tulancingo, km.4.5,
Pachuca, Hidalgo, C.P. 42184
MEXICO
Abstract: - In this paper, we offer an approach for solving functional equations containing a shift operator and
its iterations. With the help of an algorithm, the initial equation is reduced to the first iterated equation, then,
applying the same algorithm, we obtain the second iterated equation. Continuing this process, we obtain the
-th iterated equation and the limit iterated equation. We prove a theorem on the equivalence of the initial
equation and the iterated equations. Based on the analysis of the solvability of the limit equation, we find a
solution to the initial equation. Equations of this type appear when modeling renewable systems with elements
in different states, such as being sick, healthy with immunity, and without immunity. The obtained results
represent appropriate mathematical tools for the study of such systems.
Key-Words: - Functional operator with shift, Inverse operator, Iterated method, Hölder functions, Hölder
functions with weight, Renewable systems.
Received: June 7, 2024. Revised: October 19, 2024. Accepted: November 9, 2024. Published: December 15, 2024.
1 Introduction
When modeling systems with renewable resources
[1], functional equations with shift arise
. To solve equations with the operator ,
, the results on the
invertibility of the operator in Hölder
space with weight were used.
More complex models have been developed to
study renewable systems with elements in three
states. The balance relations of the model included
not only the shift operator , but also its second-
order compositions . To solve the
equation:
iterative methods of solution were developed and
applied.
In this work, we generalized the iterative
method proposed in [1] and obtained formulas for
solving the equation with compositions of the
operator of the -th order:
.
Here, operators are considered in the class of
Hölder functions and in the spaces of Hölder
functions with weight, [2], [3].
The statements and theorems proved in this
work will serve as an effective mathematical
apparatus in the study of renewable systems with
elements in different states.
2 Iteration Equations. Connection
between the Initial Values of the
Sought Function and the Free
Term
From [3], we recall the definitions of the space of
Hölder class functions with weight
, where
.
Functions that satisfy the following
condition: ,
are called Hölder functions and form the class
When , then such functions are called
Lipschitz functions.
Functions that become Hölder functions and
turn into zero at points after being
multiplied by , form a Banach space. Functions
of this space are called Hölder functions with weight
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.90
Anna Tarasenko, Oleksandr Karelin,
Manuel Gonzalez-Hernandez, Joselito Medina-Marin
E-ISSN: 2224-2880
874
Volume 23, 2024
and have the notation
. The norm in
the space
is defined by:
= ,
where
and
,
.
Let be a bijective orientation-preserving
shift on : if , then and
has only two fixed points , and
, when , . In addition, let
be a differentiable function with
and
.
From the properties of non-Carleman shift,
an important property of the functional operator
follows:
. (1)
We consider the equation:
(2)
or, in operator form,
,
we write
(3)
We will consider equation (2) in two cases:
I. The first case of the problem in space :
(4)
Coefficients belong
to, the known free term on the right side q
belongs to , the unknown function is sought
in space .
II. The second case of the problem in space
:
(5)
Coefficients belong
to, the known free term on the right side
belongs to
, the unknown function is
sought in space
.
We will now proceed to describe the process of
constructing iterated equations. We write the
original equation (3) in recurrent form:
. (6)
Substituting the expression for the unknown
function into the right side of the same equation (4),
we get an equation after the first iteration:
,
.
We denote the obtained equation as the first
iterated equation. Using the same algorithm, we
construct the second iterated equation and move
operator to the end of it, in front of.
Here, we have indicated the results of the first
and the second iteration. Continuing the iterative
process, at the step , we obtain -th iterated
equation:
, (7)
where
(8)
Now, in the first case of the problem, space
, we establish connections between the initial
values of the free term and the initial values of the
unknown function of the original equation (2).
These values will be useful in the construction of
the solution of the considered equation. The
following simple statement holds:
Lemma 1. There are relationships
where and
where .
Proof. It is easy to see that from original equation
(2) we obtain:
and
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Anna Tarasenko, Oleksandr Karelin,
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Volume 23, 2024
This follows from the fact that constants do not
change under the action of the shift and
is a fixed point for non-Carleman
shift .
3 Equivalence of the Original
Equation and the Iterated
Equations
That the solution to the original equation is a
solution to the iterated equations directly follows
from the description of the algorithm for
constructing the iterated equations. We formulate
this affirmation as a theorem due to its importance
to what follows in this article and we will provide its
proof.
Theorem 1.
The initial equation and iterated equations are
equivalent to each other.
Proof. At first, we will prove that if is a solution
to the original equation, then this function is a
solution to all iterated equations. This follows
directly from the algorithm of construction of
iterated equations. So, it is obvious that all solutions
of original equation (6):
are included in the set of solutions of the first
iterated equation
. Continuing our reasoning in this way,
we come to the conclusion of the first part of the
proof. Now we will show that the iterated equations
do not have any other solutions except for the
solutions of the original equation.
Let be the solution of the first iterated
equation: ,
then the function will be
the solution of the second iterated equation:
,
= Now, we add and subtract the term
and perform some
transformations,
Here the fact that is a solution of the first iterated
equation was used.
Counting we come to initial
equation . The theorem
has been proved. Note that we did not separate cases
1 and 2 since differences in statements do not affect
the proof of the theorem.
We will continue the analysis of the solvability
of original equation (2) using its representation in
the form of -th iterated equation (7), (8). We
represent the operator , obtained after
iterations, in another way:
We introduce an operator:
that will play an important role and
.
Note that = (0).
Operator
can be written out in detail:
:
The relation holds.
The n-th iterated equation is represented as:
.
4 Solution to the Initial Equation
Passing to the limit, directing to infinity we
have:
.
We start with the first statement in space :
To calculate the value of the unknown function
at the end points by Lemma 1, we assume
.
We can introduce operators and
when their limits exist in
operator norm and functional norm of the Hölder
space, respectively. Taking into account the linearity
of the operators, we have a unique solution to the
original equation:
and .
Here,
(9)
and
. (10)
Paying attention to formula for the initial values
from Lemma 1 and to the limit operators (9, 10), the
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.90
Anna Tarasenko, Oleksandr Karelin,
Manuel Gonzalez-Hernandez, Joselito Medina-Marin
E-ISSN: 2224-2880
876
Volume 23, 2024
solution of the initial equation can be written as
follows:
. (11)
Theorem 2.
Let the sequence of operators {converge in
operator norm to the operator acting in , the
functional sequence converge to some
function from space and
.
Then the equation in space
,
where coefficients and free term belong to
and unknown function is searched from has a
unique solution that is determined by the formula:
Now, consider the second statement in space
.
Let us remember that we study the equation (5)
with the free term belonging to
the
unknown function belonging to
and
the coefficients taken from
We translate the equation from
into
We multiply the left and the right sides of
(5) by the weight function and obtain
Here, according to the definition of the Hölder
space with weight
, functions
(x)= and
belong to the space of Hölder and vanish at
the end points, ==0,
We write down equivalent equations for the
unknown function ,
and, finally, we get the equation:
(12)
where
.
Here, the following was used:
,
Let's introduce the operator:
and write equation (12) in the form:
=S .
To simply the proof of the following statements,
we will add to the requirements imposed on the shift
one more: has a second derivative and (1)
, (1) . In what follows we will assume that
has this property.
We take the weight function from the definition
of space
: ,
.
Lemma 2.
Functions
,
, … ,
belong to.
Proof. First, we prove that belongs to the
Lipschitz class :
.
We use Lagrange´s Theorem. If function
is continuous on a closed segment and
differentiable on the open interval ), then the
following relation holds:
;
(13)
in more detail:
.
The function
is bounded by
some constant . We calculate this constant.
The derivative of the function is equal to
and
The derivative has no singularities at . In
order to assert the continuity of the derivative we
consider the limit when tends to zero:
.
The continuous function is defined on
segment and therefore reaches its extreme
values on this segment, which we will denote by
and
. We came to an estimate:
|
.
Coming back to (13), we conclude that function
is a Lipschitz function.
Note that and that the
product of a Hölder function from and a
Lipschitz function from gives us a Hölder
function from.
The following equality holds:
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Anna Tarasenko, Oleksandr Karelin,
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=
and
belongs to .
We move on to:
.
The requirements imposed on the shift and the
belonging of the shift to Hölder class with the
exponent imply that () belongs .
Presenting
as:
,
we, similarly, have that
belongs to .
From Lemma 2 it follows that, the coefficients
belong to the space ,
since according to the statement of the problem, the
coefficients are taken from
the space .
To facilitate understanding of the ideas and the
algorithms for constructing operators used in the
article, the authors propose a list of sources that
provide classical definitions of operator theory and
functional analysis [4], [5] [6], [7], as well as
specific features of Hölder spaces of weighted
functions and operators acting in them, and indicate
some applications [8], [9].
We introduce the operators:
;
the limit operator
, if it exists in
operator norm and acts on ; operator:
+ and
operator as the sum of functional series
when it converges to some function from
Equation (12) is a special case of the original
equation (4), with the right-hand side vanishing at
the point . We write down its
solution using the formula (11): ,
where the term
disappears because
. We get a theorem:
Theorem 3.
Let the sequence of operators
converge in
operator norm to operator acting in and
the functional series converge in
norm of space to some function
from the space and, moreover, let
Then, the equation:
,
where the coefficients belong to , free term
belongs to
and unknown function is
sought from
has a unique solution:
where
4 Conclusion
The equation considered in this paper is a linear one.
The authors plan to generalize the proposed method
and apply it to the study of some nonlinear
equations that arise when modeling systems with
renewable resources. In order to achieve this, there
is a need to develop the theory of continued
fractions and infinite products [10], which
complicates the construction of solutions to
nonlinear equations. The first advances in this
direction have already been made and the results
obtained will serve as new tools for studying
renewable systems with elements in different states;
for example, infected, not infected, with immunity,
and without immunity. Another direction of
research is the introduction of several shifts and
their iterations into the equation under
consideration, which will allow taking into account
more complex relationships and interactions
between the elements of the system.
References:
[1] Karelin, O., Tarasenko, A. and M. Gonzalez-
Hernandez, Study of the Equilibrium of
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Applying Operators with Shift, IEEE
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in Sciences and Industry (MCSI), Athens,
Greece, 2023, pp. 27-32,
https://doi.org/10.1109/MCSI60294.2023.000
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[2] Gakhov, F. D., Boundary value problems,
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[3] Litvinchuk, G. S., Solvability theory of
boundary value problems and singular integral
equations with shift, Kluwer Acad. Publ,
2000. https://doi.org/10.1007/978-94-011-
4363-9.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.90
Anna Tarasenko, Oleksandr Karelin,
Manuel Gonzalez-Hernandez, Joselito Medina-Marin
E-ISSN: 2224-2880
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Volume 23, 2024
[4] Yosida, K., Functional analysis, Springer,
1995. https://doi.org/10.1007/978-3-642-
61859-8.
[5] Nair, M. T., Functional analysis: A first
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https://doi.org/10.1007/978-3-030-51945-2.
[7] Kravchenko, V. G. and G. S. Litvinchuk,
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[8] Duduchava, R. V., Unidimensional Singular
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[9] Duduchava, R. V., Convolution integral
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[10] Khinchin, A. Y., Continued fractions,
University of Chicago Press, 1992. ISBN:
978-0486696300
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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 have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.90
Anna Tarasenko, Oleksandr Karelin,
Manuel Gonzalez-Hernandez, Joselito Medina-Marin
E-ISSN: 2224-2880
879
Volume 23, 2024
