On Invertibility of Functional Operators with Shift in Weighted Hölder Spaces

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.


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.

Auxiliary Lemmas
We introduce [6] is called Hölder's function with exponent µ and constant C on J .Let ρ be a power function which has zeros at the endpoints The functions that become Hölder functions and turn into zero at the points 0 x = , 1 x = , after being multiplied by ( ) , form a Banach space: The norm in the space , sup .
We denote by ( ) the set of all bounded linear operators mapping the Banach space H into the Banach space H .The norm of an operator ( ) we will denote by ( ) be a bijective orientation-preserving displacement on J: if 1 2 x x < , then ( ) ( ) , x J x J ∈ ∈ ; and let ( ) have only two fixed points: ( ) ( ) ( ) and let ( ) Without loss of generality, we assume that for any fixed ( ) ( ) We will use the following notation, ( ) ( ) , , , .
Proof.This lemma follows from the properties of shift ( ) then the following inequalities are correct in some ε half-neighborhoods of the endpoints 0, Proof.This lemma follows from (1) and from the properties of ( ) From Lemma 1 and Lemma 2 it follows that for 0 ε > a positive integer 0 n exists such that for any .
We shall take advantage of these lemmas in the proof of invertibility conditions in Section 3.

Conditions of Invertibility for Operator A in Weighted Hölder Spaces
In weighted Hölder space , and where u b a = , are invertible simultaneously when 0 a ≠ .It is obvious that A aU = and If a certain natural number n exists such that ( ) 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 0 b ≠ and a certain natural number n exists such that ( ) and its inverse operator is ( ) ( ) From conditions (1) it follows that such n exists for which In order to prove we estimate every summand separately, starting with the first .
We took into account From Lemma 1 and (2) of Lemma 2, it follows that only 0 n values of ( ) Here the number 0 n is from Lemma 1.
From (2) of Lemma 2 and the identity it follows that some number 1 0 ε > exists such that ( ) ( ) is fulfilled for all fixed 1 2 , x x , We will now formulate and prove conditions of invertibility for operator A in the space of Hölder class functions with weight.In [4] these conditions were only formulated but not proved.where function β σ is defined by: ( ) ( )

From ( 2 )
of Lemma 2, it follows that the first factor on the right side of inequality (when n → ∞ .Now, we estimate the second summand of (3).The following estimate holds tend to zero when n → ∞ .Thus, such n exists that

Theorem 2 .
Operator A acting in Banach space weighted Hölder spaces µ ρ , in which we consider functional operators with shift.A function ( ) x ϕ that satisfies the following condition on