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In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems with renewable resources.

The interest towards the study of functional operators with shift was stipulated by the development of solvability theory and Fredholm theory for some classes of linear operators, in particular, singular integral operators with Carleman and non-Carleman shift [

Our study of functional operators with shift in Hölder spaces with weight has an additional motivation: on modeling systems with renewable resources, equations with shift arise [

In Section 2, some auxiliary lemmas are proposed. These are to be used in the proof of invertibility conditions. In Section 3, conditions of invertibility for functional operators with shift in Hölder spaces with power wight are obtained. We provide the main idea and the scheme of proof of the conditions of invertibility. At the end of the article, an application to modeling systems with renewable resources is specified.

We introduce [

A function

is called Hölder’s function with exponent

Let

The functions that become Hölder functions and turn into zero at the points

where

and

We denote by

Let

addition, let

Without loss of generality, we assume that for any fixed

We will use the following notation,

Lemma 1.

An essential point is that

Proof. This lemma follows from the properties of shift

Lemma 2. If the following condition is fulfilled,

then the following inequalities are correct in some

Proof. This lemma follows from (1) and from the properties of

From Lemma 1 and Lemma 2 it follows that for

The following lemmas hold.

Lemma 3. Operator

Operator

Lemma 4. For

We shall take advantage of these lemmas in the proof of invertibility conditions in Section 3.

In weighted Hölder space

If a certain natural number

This statement in weighted Lebesgue spaces was proved in [

Analogously, if

It is obvious that

We will use the following notation

Theorem 1. From conditions (1) it follows that such

Proof. In order to prove

we estimate every summand separately, starting with the first

We took into account

From (2) of Lemma 2, it follows that the first factor on the right side of inequality (4):

Now, we estimate the second summand of (3). The following estimate holds

From Lemma 1 and (2) of Lemma 2, it follows that only

From (2) of Lemma 2 and the identity

it follows that some number

All expressions with

Thus, such

We will now formulate and prove conditions of invertibility for operator

Theorem 2. Operator

where function

Proof. We consider the following case:

In space

Thus, such

which means that operator

The case

Now, we will focus on the application of these results to the modeling of systems with renewable resources. For the study of such systems, cyclic models were elaborated based on functional operators with shift [

If we model the behavior of a system with two resources, taking into account the interaction between them, by integrals with degenerate kernels and following the principles of modeling from [

where

are the terms of reproduction and interaction process respectively.

Let

in