Advances in Pure Mathematics, 2014, 4, 595-600 Published Online November 2014 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2014.411068

How to cite this paper: Tarasenko, A. and Karelin, O. (2014) On Invertibility of Functional Operators with Shift in Weighted Hölder Spaces. Advances in Pure Mathematics, 4, 595-600. http://dx.doi.org/10.4236/apm.2014.411068

On Invertibility of Functional Operators with Shift in Weighted Hölder Spaces Anna Tarasenko1, Oleksandr Karelin2 1Mathematical Research Center, Hidalgo State University, Pachuca, Mexico 2Advanced Research Center on Industrial Engineering, Hidalgo State University, Pachuca, Mexico Email: anataras@uaeh.edu.mx, karelin@uaeh.edu.mx Received 4 October 2014; revised 1 November 2014; accepted 7 November 2014

Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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 ( )0 ,H Jµ ρ , in which we consider functional operators with shift.

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A. Tarasenko, O. Karelin

596

A function ( )xϕ that satisfies the following condition on [ ]0,1J = , ( ) ( ) ( )1 2 1 2 1 2, , , 0,1 ,x x C x x x J x J

µϕ ϕ µ− ≤ − ∈ ∈ ∈

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 0x = , 1x = :

( ) ( ) ( )0 1 0 10 1 , 1 , 1 .x x xµ µρ µ µ µ µ µ µ= − − < < + < < +

The functions that become Hölder functions and turn into zero at the points 0x = , 1x = , after being multiplied by ( )xρ , form a Banach space: ( ) [ ]0 , , 0,1 .H J Jµ ρ = The norm in the space ( )0 ,H Jµ ρ is defined by

( ) ( ) ( ) ( ) ( )0 , ,H J H Jf x x f xµ µρ ρ=

where

( ) ( ) ( ) ( ) ( ) ( ) ( ) ,H J Cx f x x f x x f xµ µρ ρ ρ= +

and

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 2 1 2

1 1 2 2

, , 1 2

max , sup .C x x J x x

x f x x f xx f x x f x x f x

x x µµρ ρ

ρ ρ ρ∈ ≠

−= =

−

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

Let ( )xβ be a bijective orientation-preserving displacement on J: if 1 2x x< , then ( ) ( )1 2x xβ β< for any 1 2,x J x J∈ ∈ ; and let ( )xβ have only two fixed points: ( ) ( ) ( )0 0, 1 1, , when 0, 1.x x x xβ β β= = ≠ ≠ ≠ In

addition, let ( )xβ be a differentiable function, let ( )d 0,d

xxβ ≠ and let ( )d

dx

xβ belong to ( )H Jµ .

Without loss of generality, we assume that for any fixed ( )0,1x∈ , ( ) ( )lim 0, lim 1m mm mx xβ β−→+∞ →+∞= = is fulfilled; we have ( )0 1β ′ ≤ and ( )1 1β ′ ≥ .

We will use the following notation,

( ) ( )0 1 ,, , 1 ,sr

r sr s x x xµ µ µ µ ρ= − = − = − ( ) ( ) ( ), 1 ,x x x xµµ

µ µ µρ ρ= = −

( ) ( ) ( ) ( ) ( ) ( ) ( )1 2, ; , ; , ; 1 21 2

, , , .r s j r s j j jx x

x x x x x xx xµ µ µ

β βρ ρ β ρ ρ β

− = = = −

Lemma 1. ( )( )( )( )( )( )0 1 2 1 2 0 2 1, 0 , , ,x x J n x J n n n n n n nβ ε∀ ∈ ∀ > ∃ ∈ ∀ ∈ ∃ ∈ < = −N N ( ) [ ] [ ] [ ]1 20, 1 ,1 , \ , .n x n n nβ ε ε ∈ − ∈ N

An essential point is that 0 2 1n n n= − is independent from x . Proof. This lemma follows from the properties of shift ( )xβ . Lemma 2. If the following condition is fulfilled,

( ) ( ) ( )( ) ( )( )( )

0 10 10; 0 1, 1 1,0 1

b ba x

a aµ µ µ µ

β β− + − +′ ′≠ < < (1)

then the following inequalities are correct in some ε half-neighborhoods of the endpoints 0, 1x x= = :

( ) ( )( ) [ ] [ ],

, ;1

1, 0, 1 ,1 .r sr s

xu x q x

xρ

ε ερ

≤ < ∈ − (2)

Proof. This lemma follows from (1) and from the properties of ( )xβ , ( )a x , ( )b x . From Lemma 1 and Lemma 2 it follows that for 0ε > a positive integer 0n exists such that for any

A. Tarasenko, O. Karelin

597

[ ]0,1x∈ at most 0n values of ( )n xβ will be outside of [ ] [ ]0, 1 ,1ε ε− . The following lemmas hold. Lemma 3. Operator Bβ is bounded in space ( )H Jµ , ( )( ) .CH JB µ

µβ β ′≤

Operator Bβ is bounded in space ( )0 ,H Jµ ρ , ( )( ) [ ] ( ) ( )( )0 ,

.H J H J

H J

B Bµ µ

µ

β βρ

ρρ β

≤

Lemma 4. For ( )0 ,H Jµϕ ρ∈ , the inequality ( )0 ,,

2 H JC

µ

µρ

µ µ

ρϕ ϕρ

≤ 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 ( )0 ,H Jµ ρ , the operators: ,A aI bBβ= − where ,a H b Hµ µ∈ ∈ , and ,U I uBβ= −

where u b a= , are invertible simultaneously when 0a ≠ . It is obvious that A aU= and 1 1 1A U Ia

− −= .

If a certain natural number n exists such that ( )( )( )0

1

0 ,

1,n

nj

j H J

u x Bµ

β

ρ

−

=

A. Tarasenko, O. Karelin

598

we estimate every summand separately, starting with the first

( )01 1 1

, ; , ;, , , ,

=0 =0 =0, ; 1 , ; 1

2 .n n n

r s j r s jnj j r s j H JC

j j jr s j r s jC C C

u B u uµ

µβ µ µ µ µ ρ

ρ ρρ ϕ ρ ρ ϕ ρ ϕ

ρ ρ

− − −

+ +

≤ ≤

∏ ∏ ∏ (4)

We took into account ( ) ( )1

, ;,, ,

0, ; , ; 1

, ,n

r s jr s nr s r sC Cjr s n r s j

Bβρρ

ρ ϕ ρ ϕρ ρ

−

= +

= =∏ Lemma 3 and Lemma 4.

From (2) of Lemma 2, it follows that the first factor on the right side of inequality (4): 1

, ;,

0 , ; 1

nr s j

jj r s j C

uµ µρ

ρρ

−

= +∏

tends to zero when n →∞ . Now, we estimate the second summand of (3). The following estimate holds

( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

1 2 1 2

1 2

1

0

2 2; 2 1

;1 1 1 1 2 1 20 0; ; 1 1

33; 3 1

; 2 2 1 2 1 20 0; ; 2 1

sup sup ,

sup ,

nn

jj

n njn n

j n n j jx x x xj jn jC

jnnin n

j i k kx x j i j kn iC

u B

xu x u x u u x x x

x

xu u x u x x x

x

β

µ

µ µµµ µ

µ µ

µ µµµ

µ µ

ρ ϕ

ρρ ϕρ ρ ϕ β

ρ ρ

ρρ ϕρ β

ρ ρ

−

=

− −+

+<

A. Tarasenko, O. Karelin

599

Proof. We consider the following case: ( )( ) ( ) ( )

0, ,

, 0,1.ia x x J

a i i b i iµ µ

β− +

≠ ∈

′> ⋅ =

In space ( )0 ,H Jµ ρ , operators aI bBβ− and ,U I uBβ= − where u b a= , are invertible simultaneously. Thus, such n exists that

( )( )0

0

1

,0 ,

,n

nj H J

j H J

u Bµ

µ

β ρρ

ϕ ϕ−

=

A. Tarasenko, O. Karelin

600

Kluwer Academic Publishers, Dordrecht, Boston, London. http://dx.doi.org/10.1007/978-94-011-4363-9 [3] Kravchenko, V.G. and Litvinchuk, G.S. (1994) Introduction to the Theory of Singular Integral Operators with Shift.

Kluwer Academic Publishers, Dordrecht, Boston, London. http://dx.doi.org/10.1007/978-94-011-1180-5 [4] Tarasenko, A., Karelin, A., Lechuga, G.P. and Hernández, M.G. (2010) Modelling Systems with Renewable Resources

Based on Functional Operators with Shift. Applied Mathematics and Computation, 216, 1938-1944. http://dx.doi.org/10.1016/j.amc.2010.03.023

[5] Karelin, O., Tarasenko, A. and Hernández, M.G. (2013) Application of Functional Operators with Shift to the Study of Renewable Systems When the Reproductive Processed Is Described by Integrals with Degenerate Kernels. Applied Mathematics (AM), 4, 1376-1380. http://dx.doi.org/10.4236/am.2013.410186

[6] Duduchava, R.V. (1973) Unidimensional Singular Integral Operator Algebras in Spaces of Holder Functions with Weight. Proceedings of A. Razmadze Mathematical Institute, 43, 19-52. (In Russian)

http://dx.doi.org/10.1007/978-94-011-4363-9http://dx.doi.org/10.1007/978-94-011-1180-5http://dx.doi.org/10.1016/j.amc.2010.03.023http://dx.doi.org/10.4236/am.2013.410186

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On Invertibility of Functional Operators with Shift in Weighted Hölder SpacesAbstractKeywords1. Introduction2. Auxiliary Lemmas 3. Conditions of Invertibility for Operator A in Weighted Hölder Spaces References