Received 4 October 2014; revised 1 November 2014; accepted 7 November 2014

ABSTRACT

In this paper, we consider functional operators with shift in weighted Hölder 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.

Keywords:

Functional Operator with Shift, Hölder Space, Conditions of Invertibility, Renewable Resources

1. Introduction

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 [1] -[3] . Conditions of invertibility for functional operators with shift in weighted Lebesgue spaces were obtained [1] .

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 [4] [5] , and the theory of linear functional operators with shift is the adequate mathematical instrument for the investigation of such systems.

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.

2. Auxiliary Lemmas

We introduce [6] weighted Hölder spaces, in which we consider functional operators with shift.

A function that satisfies the following condition on,

is called Hölder’s function with exponent and constant on.

Let be a power function which has zeros at the endpoints,:

The functions that become Hölder functions and turn into zero at the points, , after being multiplied by, form a Banach space: The norm in the space is defined by

where

and

We denote by the set of all bounded linear operators mapping the Banach space into the Banach space. The norm of an operator we will denote by.

Let be a bijective orientation-preserving displacement on J: if, then for any; and let have only two fixed points: In

addition, let be a differentiable function, let and let belong to.

Without loss of generality, we assume that for any fixed, is fulfilled; we have and.

We will use the following notation,

Lemma 1.

An essential point is that is independent from.

Proof. This lemma follows from the properties of shift. W

Lemma 2. If the following condition is fulfilled,

(1)

then the following inequalities are correct in some half-neighborhoods of the endpoints:

(2)

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

From Lemma 1 and Lemma 2 it follows that for a positive integer exists such that for any at most values of will be outside of.

The following lemmas hold.

Lemma 3. Operator is bounded in space,

Operator is bounded in space,

Lemma 4. For, the inequality is correct.

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

3. Conditions of Invertibility for Operator A in Weighted Hölder Spaces

In weighted Hölder space, the operators: where, and where, are invertible simultaneously when. It is obvious that and.

If a certain natural number exists such that where then operator is invertible in space and

This statement in weighted Lebesgue spaces was proved in [1] . The proof is completely transferred without change to the weighted Hölder space as the applied algebraic operations do not depend on the specific properties of the spaces.

Analogously, if and a certain natural number exists such that where then the operator is invertible in space and its inverse operator is

It is obvious that

We will use the following notation

Theorem 1. From conditions (1) it follows that such exists for which

Proof. In order to prove

(3)

we estimate every summand separately, starting with the first

(4)

We took into account Lemma 3 and Lemma 4.

From (2) of Lemma 2, it follows that the first factor on the right side of inequality (4): tends to zero when.

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

(5)

From Lemma 1 and (2) of Lemma 2, it follows that only values of may be outside of the set

, where inequality holds. Here the number is from Lemma 1.

From (2) of Lemma 2 and the identity

it follows that some number exists such that is fulfilled for all fixed, with the possible exception of values of.

All expressions with from (5) tend to zero when.

Thus, such exists that which means that operator is invertible in space

We will now formulate and prove conditions of invertibility for operator in the space of Hölder class functions with weight. In [4] these conditions were only formulated but not proved.

Theorem 2. Operator acting in Banach space, is invertible if the following condition is fulfilled:

where function is defined by:

Proof. We consider the following case:

In space, operators and where, are invertible simultaneously.

Thus, such exists that

which means that operator is invertible in space

The case can be considered analogously.

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 [4] . The balance relation describing the state of cyclic equilibrium is the equation for the unknown distribution function which is sought in space. In [5] , a reproductive summand has been added for a more accurate description of the process of reproduction; this term has been expressed by integrals with degenerate kernels.

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 [4] , we shall obtain two equations with two unknowns, and:

(6)

where and are the densities of the distributions of the first and second resources by their respective individual parameters such as weight or length and

,

are the terms of reproduction and interaction process respectively.

Let be the space in which our model is considered. Suppose that for

in the conditions of invertibility of Theorem 2 are fulfilled. Thus, the inverse operators of and exist, and. We apply these inverse operators to the left side of Equation (6) and obtain Fredholm equations of the second type with degenerate kernels. Using a known method of solving such equations, we can find densities and of the cyclic equilibrium of the system.

References