About Solutions of Nonlinear Algebraic System with Two Variables

2013; 2(1) : 32-37

Published online February 20, 2013(http:// www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.20130201.15

About solutions of nonlinear algebraic system with two

variables

Rakhshanda Dzhabarzadeh

İnstitute Mathematics and Mechanics of NAN of Azerbaijan, Baku

Email address:

[email protected] (R. Dzhabarzadeh)

To cite this article:

Rakhshanda Dzhabarzadeh. About Solutions of Nonlinear Algebraic System with Two Variables. Pure and Applied Mathematics Journal.

Vol. 2, No. 1, 2013, pp. 32-37. doi: 10.11648/j.pamj.20130201.15

Abstract

:

For nonlinear algebraic system with two variables sufficient conditions of existence of solutions are given. The proof of these statements is received as a corollary of more common reviewing considered in this paper. In particular, in this work the existence of multiple base on eigen and associated vectors of a two parameter system of operators in fi-nite-dimensional spaces is proved. Definitions of the associated vectors, multiple completeness of eigen and associated vectors of two-parameter not selfadjoint systems, nonlinearly depending on spectral parameters are introduced. At the proof of these results we essentially used the notion of the analog of an resultant of two polynomial bundles.

Keywords:

Algebraic, Spectral, Resultant, Nonlinear

1. Introduction

The spectral theory of operators is one of essential direc-tions of a functional analysis. Many partial equadirec-tions and the equations of mathematical physics, connected with physical processes, demanded the new approach for a solu-tion of these problems.

The method of a separation of variables for a solution of the equations with partial derivatives often appears the most

comprehensible as reduces solution of the complicated

equation with many variables to a solution of system of the ordinary differential equations which research is much easier.

So, for example, with help of multiparameter theory problems of a quantum mechanics [7,15,16], theories of a

diffraction[9,10], theories of elastic envelopes, calculation

of nuclear reactors [14], equilibrium processes of diffusion type [13], a Brownian motion [13], boundary value prob-lems for the equations of elliptic-parabolic type, a Cauchy problem for the ultra parabolic equations [11], etc. are solved.

Despite of an urgency and prescription of these re-searches, the spectral theory of multiparameter systems is studied insufficiently. Available outcomes in this area until recently are obtained only for multiparameter selfadjoint systems of the operators linearly depending on spectral parameters.

The founder of researches of spectral problems of the multiparameter selfadjoint systems was F.V. Atkinson [1]. Studied the outcomes which are available for multipara-meter symmetrical differential systems, Atkinson has con-structed the spectral theory of selfadjoint multiparameter systems in finite-dimensional spaces. Further, by means of passage to the limit Atkinson has generalized the received outcomes on a case of multiparameter systems with the selfadjoint compact operators in infinite-dimensional Hil-bert spaces.

In the further, it has appeared possible to build the con-struction of Atkinson in infinite-dimensional spaces. It has allowed construct the spectral theory of multiparameter systems in Hilbert spaces [3, 8] with help a design intro-duced by Atkinson for study of multiparameter systems in finite-dimensional spaces,

But, unfortunately, the technique of research in these works essentially uses only selfadjoint operators entering into multiparameter system of operators. For not selfadjoint multiparameter systems the investigated technique does not allow to solve the simplest problems of the spectral theory

of operators.

The author offers the new approach to research of mul-tiparameter problems.

vectors are introduced.

2. Multiparameter Spectral Theory and

Its Application to Nonlinear

Alge-braic System

For the statement of existence of a solution of

1 1 1

1 1 `1 1

0 1 1

( , )

...

...

0

+ +

= +

+ +

+

+ +

=

m n n

m m m n

a x y

a

a x

a x

a

y

µ

a

y

(1)

2 2 2

2 `2 2

0 1 1

( , )

...

...

0

+ +

=

+

+

+

+

+

+

=

m m n

m m n

b x y

b

b x

b

x

b

y

b

y

we essentially use outcomes of author concerning the multiparameter systems, published in works [4-6] .

In the given article the multiply base property of system of eigen and associated vectors of two parameter system

1 1 1

1 `1 1

m

0 1 m

n

m 1 m n

A( , )x

(A

A

...

A

A

₊

...

A

₊

)x

0

λ µ =

+ λ + + λ

+µ

+ + µ

=

(2)

2 2 2

2 `2 2

m

0 1 m

n

m 1 m n

B( , )y

(B

B

...

B

B

₊

...

B

₊

)y

0

λ µ =

+ λ + + λ

+µ

+ + µ

=

is investigated.

Below we shall reduce these outcomes as they have in-dependent meanings. is established in tensor product

2 1

H

H

⊗

of finite-dimensional spaces

H

₁ andH2

Dimension of space His the product of dimensions of

spaces

H

1 and H2 . In (2) linear operators

)

,...,

1

,

0

(

i

m

1

n

1

A

i

=

+

act in finite-dimensional space

H

1;

and linear operators

B

i

(

i

=

0

,

1

,...,

m

2

+

n

2

)

act in

fi-nite-dimensional space

H

2.

If f1⊗ f2∈H1⊗H2 and g1⊗g2∈H1⊗H2 then

in-ner product of these elements in space H1⊗H2 is defined

by means of the formulae[f1⊗f2,g1⊗g2]H1⊗H2=

( )

f1,g1H1⋅(g1,g2)H2

This definition is spread to other elements of tensor product spaces on linearity.

Let's reduce a series of known positions concerning the spectral theory of multiparameter system.

Definition1. ([1, 3])

(

λ

,

µ

)

∈

C

2 is an eigen value of the

system (2) if there are nonzero elements

x

∈

H y

1

,

∈

H

2

such that (2) is fulfilled. A tensor z=x⊗y is named an

eigenvector of system (2).

Definition2. An operator

A

k

=

A

k

⊗

E

2

+

(accordingly,

k

k E B

B+= 1⊗ ) where

E

₁ (accordingly,E2) is identical

operators in

H

1(accordingly,

H

2), names operator, induced

in

H

=

H

1

⊗

H

2by operators

A

k (accordingly,Bk).

Definition3.[5] A tensor

z

m1,m2 name

(

m

1

,

m

2

)

- the

as-sociated vector to an eigenvector

z

0,0

=

x

⊗

y

if

follow-ing conditions are satisfied

µ

λ

λ

2

µ

1 2 1

)

,

(

!

!

1

0 2 1

r r r r k

r

A

r

r

i

i

∂

∂

∂

+ + ≤

≤

∑

z

k1−r1,k2−r2

=

0

µ λ

µ λ

2 1 2 1 _{( , )}

! ! 1 , 0

2 1

r r r r k r

A r r

i ∂ ∂

∂+ + ≤

≤

∑

₀

2 2 1

1−r,k −r

=

k

z

(3)

.

2

,

1

;

2

,

1

;

=

=

≤

m

i

s

k

s s

(

k1,k2

)

is arrangement from set of the whole nonnegative

numbers on 2 with possible recurring and zero.

Definition4. Systems of elements

{ }

x,1 (k 1,2,...,n)

r

k

k = of

finite-dimensional space H1⊗H2 form

n

-fold base in

this space if any

n

of elements f1,...,fnof space

H

1

⊗

H

2

can be spread out in series ( 1,..., )

1 ,

n i x c

f s

k k ki

i=

∑

=

= with

coefficients, not dependent on an index of vectorsf1,...,fn.

If systems

{ }

, ₁( 1, 2,..., )

r k i _k

x i n

= = are constructed on

{ }

r k k x,1 =₁

with by the certain rules, equalities ( 1,..., )

1 ,

n i x c

f s

k k ki

i=∑ =

=

mean existence of

n

-fold base on system of eigen and

associated vectors of the system (2).

Definition5.[5]. Let be two polynomial bundles

m m

A

A

A

A

(

λ

)

=

0

+

λ

1

+

...

+

λ

,

n n

B

B

B

B

(

λ

)

=

₀

+

λ

₁

+

...

+

λ

,

(4)

depending on the same parameter

λ

and acting,

gener-ally speaking, in various Hilbert spaces

H

1

,

H

2, accordingly.

The operatorRes(A(

λ

),B(

λ

)) is under construction

          

 

          

 

⋅⋅ ⋅ ⋅⋅

⋅

⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅

⋅⋅ ⋅ ⋅⋅

⋅

⋅⋅ ⋅ ⋅⋅

⋅

⋅⋅ ⋅ ⋅⋅

⋅

⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅

⋅⋅ ⋅ ⋅⋅

⋅

⋅⋅ ⋅ ⋅⋅

⋅

⊗ =

+ + + +

+ +

− +

+

+ +

+ +

+ + + +

+ +

− +

+

+ +

+ +

n n m

n

m m m

m

B B B B

B B B

B

B B

B B

A A A A

A A A

A

A A

A A

B A s

2 1 0

1 1

0 2 1 0

2 1 0

1 1

0 2 1 0

0 0 0

0 0

0 0 0

0 0

0 0

0 0

)) ( , ) ( (

Re λ λ (5)

in [2,17] names abstract analog of an Resultant for po-lynomial bundles (4).

In definition of a Resultant (5) of bundles (4) lines with

operators Airepeated

n

of times, and lines with оperators

i

B

repeated exactly

m

of times.

m

,

n

there are the

highest orders of parameter

λ

in bundles of

)

(

λ

A

,(B(λ)_{), accordingly. Thus, the Resultant is an}

oper-ator, acting in space m n

H

H

⊗

)

+

n

m

+

of copies of tensor product spaces

H

1

⊗

H

2

.

Value of

Re

s

(

A

(

λ

)

,

B

(

λ

))

is equal to its formal expansion

when each term of this expansion is tensor product of

op-erators. Let operator

A

_m, or

B

n, is invertible. If the highest

orders of parameter

λ

in bundles of A(

λ

)and

B

(

λ

)

coincide (see [17]) or if the highest orders of parameter

λ

in bundles of A(λ) andB(λ)), generally speaking, can not

coincide (see [2]) bundles (4) have a common point of spectra in the only case when the Resultant of these bundles has nonzero kernel. It is natural, that in case of when bun-dles act in finite-dimensional spaces, this common point of spectra of these bundles is a common eigen value of these bundles.

Theorem. Let operators

A

_i

(

i

=

0

,

1

,...,

m

₁

+

n

₁

)

also

) ,..., 1 , 0

(i m2 n2

Bi = + act in finite-dimensional spaces

1

H

and

2

H

, accordingly, and one of three following conditions is

fulfilled:

a)

max(

m

₁

n

₂

,

m

₂

n

₁

)

=

m

₁

n

₂,

{ }

θ

=

{ }

θ

=

2+ 2

1

,

m n

m

KerB

KerA

;

2 2 1

,

m n

m

B

A

+ are selfadjoint operators everyone in their

space

b)

max(

m

1

n

2

,

m

2

n

1

)

=

m

2

n

1,

{ }

θ

=

{ }

θ

=

1+1

2

,

m n

m

KerA

KerB

operators

B

m₂

A

m₁+n₁are selfadjoint everyone in their

spaces

c)

m

₁

n

₂

=

m

₂

n

₁,

{ }

2 1 1 2 2 1

1 2 2 1 1 2

( mn mn n ( 1)n n mn n mn )

Ker A ⊗B + + − A + ⊗B = θ ,

2 2 1 1 2

1

,

m

,

m n

,

m n

m

B

A

B

A

_{+ +} are the selfadjoint operators, acting

everyone in its finite-dimensional space.

Then themax(m1n2,m2n1)-fold base on system of eigen

and associated vectors of (4) takes place.

Proof. It is fixed one of parameters in (4). For a

deter-minacy, we shall suppose, that it is parameter

λ

and

0 λ

λ= .Then it is received two bundles, each of which

de-pends on one parameter

µ

.

1 `1

1 1 `1 1

0 0 0 1

m n

0 m m 1 m n

A(

, )

A

A

...

A

A

₊

...

A

₊

λ µ =

+ λ

+ +

λ

+ µ

+ + µ

2 2

2 2 `2 2

0 0 0 1

m n

0 m m 1 m n

B(

, )

B

B

...

B

B

₊

...

B

₊

λ µ =

+ λ

+ +

λ

+ µ

+ + µ

(6)

We introduce labels

1 1 0 1 0 0

0

)

...

(

~

m m

A

A

A

A

λ

=

+

λ

+

+

λ

,

2 2 0 1 0 0

0

)

...

(

~

m m

B

B

B

B

λ

=

+

λ

+

+

λ

Then an Resultant of bundles

1 1 ` 1 ` 1

...

)

(

~

)

,

(

_{0 0 1} n _{m n}

m

A

A

A

A

λ

µ

=

λ

+

µ

₊

+

+

µ

₊

2 2 ` 2 2

...

)

(

~

)

,

(

0 0 1 m n

n

m

B

B

B

B

λ

µ

=

λ

+

µ

+

+

+

µ

+ ,

depending on one parameter

µ

looks like:

                          ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⋅⋅ ⋅ ⊗ = = + + + + + + + + + + − + + + + + + + + + + + + + + + + + + + + + − + + + + + + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0 1 1 0 2 1 0 2 1 0 1 1 0 2 1 0 0 0 ) ( ~ 0 0 0 0 ) ( ~ 0 0 0 ) ( ~ ( ) ~ 0 0 0 0 ) ( ~ 0 0 0 ) ( ~ )) , ( , ) , ( ( Re n m m m n m n m m n m m m n m m m n m n m m n m m m B B B B B B B B B B B B A A A A A A A A A A A A B A s λ λ λ λ λ λ µ λ µ λ (7)

In (7) lines with operators

A

_iare repeatedn2of times, and

lines with operators Biare repeated

n

₁ of times. Thus, the

Resultant

Re

s

(

A

(

λ

0

,

µ

),

B

(

λ

0

,

µ

))

is an operator in

space

(

)

1 2

2 1

n n

H

H

⊗

+ that there is a direct sum

n

₁

+

n

₂ of

copies of tensor product spaces

H

₁

⊗

H

₂

.

From outcomes of work [2] follow, that spectra of oper-ators 1 1 ` 1 ` 1 1 1 ... ... ) ,

( 0 0 0 1 0 1 m n

n m m m A A A A A

Aλ µ = +λ + +λ +µ + + +µ + ,

2 2 ` 2 2 2 2 ... ... ) ,

( 0 0 0 1 0 1 m n

n m m m B B B B B

B

λ

µ

= +

λ

+ +

λ

+

µ

₊ + +

µ

₊

intersect if and only if

KerR

(

A

(

λ

0

),

B

(

λ

o

))

≠

{ }

θ

.

As operators A( , ), ( , )

λ µ

B

λ µ

act in finite-dimensional

spaces this common point of their spectra can be only common eigen value of these bundles. The last means, that

the two parameter system (2) has an eigen value(λ0,µ0(λ0)). If

the element 1 2

2

1 ) ( )

~ ,..., ~ , ~

( 1 2 1 2

n n n

n H H

x x x

z= ₊ ∈ ⊗ + belongs to

the kernel of an Resultant Res(A(λ₀,µ),B(λ₀,µ)), then

0

~

~

...

~

~

~

)

(

~

1 1 1 11 2

1

0

+

+

+

=

+ + + + + n n m

m

x

A

x

A

x

A

λ

0

~

~

...

~

~

~

)

(

~

1 3 1 2

0

+

1

+

+

1 1 1+

=

+ + + + + n n m

m

x

A

x

A

x

A

λ

0

~

~

...

~

~

~

)

(

~

2 1 1 1 2 1

2 1 1

0

+

+

+

+

=

+ + + + + + n n n m n m

n

A

x

A

x

x

A

λ

0

~

~

...

~

~

~

)

(

~

2 2 2 2 1 2

1

0

+

+

+

=

+ + + + + n n m m

x

B

x

B

x

B

λ

(8)

0

~

~

...

~

~

~

)

(

~

1 3 1 2

0

+

2

+

+

2 2 2+

=

+ + + + + n n m m

x

B

x

B

x

B

λ

0

~

~

...

~

~

~

)

(

~

2 1 2 2 1 2

1 1 1

0

+

+

+

+

=

+ + + + + + n n n m n m

n

B

x

B

x

x

B

λ

By sequential elimination of elements 1 21 1 2

~_{x H H}

n

2 1 2

~ _{H H}

x ∈ ⊗ from system of equalities (8), we come to

some equation in tensor product spaceH1⊗H2. To the similar

equation we come by representation of value of Resultant ).

, ( , ) ( (

Res A λ₀ B λ₀

By virtue of bulkiness of the received equation

opera-tional coefficients at degrees

λ

i we shall designate

through

C

~

i

(

i

=

1

,

2

,...,

max(

m

1

n

2

,

m

2

n

1

))

.

Let's introduce a label

max(

m

1

n

2

,

m

2

n

1

)

=

r

.Thus, we

come to some equation

0 ~ ~ ... ~ ~ ~ ~ ~ ) ( ~

1 0 1 1 0 1 0 1

0 x =Cx + Cx+ + Cx =

C r

r

λ λ

λ , acting in a tensor

product space

2 1

H

H

H

=

⊗

. As the parameter

λ

is

fixed arbitrarily, we consider a bundle

0

~

~

...

~

~

~

~

~

)

(

~

1 1

1 1 0

1

=

C

x

+

C

x

+

+

C

x

=

x

C

r

r

λ

λ

λ

(9)

in finite-dimensional space

H

=

H

₁

⊗

H

₂.

From outcomes of the spectral theory of operators fol-lows, that the received bundle (9) in finite-dimensional

space has

r

-fold base on system of its eigen and

asso-ciated vectors. Really, at realization of conditions Theorem are satisfied all conditions of the Theorem of Keldysh [12], concerning multiply completeness of system of eigen and associated vectors of polynomial bundle

r r

C C C

C~(

λ

)=~0+

λ

~1+...+

λ

~

The theorem of Keldysh is following: let

T

_nis

selfad-joint completely continuous operator of the finite order (

T

_n

∈

σ

p),KerTn=

{ }

θ , operators ( =0,1,..., −1)

−

n i

T

T n

i n

i

com-pletely continuous operators then

n

-fold completeness of

system of eigen and associated vectors of the operational

bundle n

n

T T

T

T(λ)= 0+λ 1+...+λ ,acting in a Hilbert space,

takes place.

Let's remind, that under an eigen value of a polynomial

bundleT(λ) such the complex number

0

λ

is understood,

that for it exists such nonzero vector

0

x

, that

T

(

λ

₀

)

x

₀

=

x

₀,

m

-th associated vector

k

x

to an eigenvector

0

x

is the

vector, satisfying to conditions

k 0 k 0 k 1

k

0 0 k

d

x T ( ) x T ( ) x d

1 d

. . . T ( ) x ; k 1, . . . , m k ! d

−

= λ + λ

λ

+ + λ =

λ

Let

{ }

xk,0 be a system of eigen and associated vectors of a

bundleT(

λ

),then derivative systems of vectors on system

of eigen and associated vectors of polynomial bundle

)

(

λ

T

are constructed by rules

0

j t

i,0 j

i, j i

i 1,0 0,0 t 0

d

e (x

dt x

t t

x ... x

1! i!

λ

−

=

 

+

 

 

= 

 _{+ +} 

 

 

,... 2 , 1 = i

In case of finite-dimensional space all conditions of the Theorem of Keldysh

are fulfilled ,consequently, system of eigen and

asso-ciated vectors of a bundle r

r_C

C C

C~(λ)=~0+λ~1+...+λ~ forms

r

-fold base in space H1⊗H2.

Really, at realization of a conditions a) of the Theorem

coefficient at

λ

ris an operator

r

C

~

= 1 2 2 2 1

n n m n m B

A ⊗ + . It is easy

to see, that it is selfadjoint operator and ( ⊗ +)=

{ }

θ

1 2 2 2 1

n n m n

m B

A

Ker ;

At realization of conditions b) of the Theorem it is had,

that + ⊗ =

{ }

θ

1 2 2

1 1

n m n

n

m B

KerA , operator 1

2 2

1 1

n m n

n

m B

A + ⊗ is selfadjoint,

and, at last, at realization of conditions c) of the Theorem

the operator 2 1 1 2

1 1 2 ( 1)

n n n n m n m

A ₊ ⊗B + − 1

2 2

1 1

n m n

n m

B

A

+

⊗

is selfadjoint

and 1

2 2

1 1

( n

m n

n

m B

A

Ker ₊ ⊗ 1 2 2 1

{ }

1 2 2

( 1)

n n n n

)

m m n

A

B

₊

θ

+ −

⊗

=

also.

In all these three cases at coefficient at

λ

r_{of bundle}

)

(

~

0

λ

C

is selfadjoint invertible operator.

Remains to prove concurrence of system of eigen and associated vectors of a two-parameter problem (2) and systems of eigen and associated vectors of the polynomial bundle (9) received by the above-stated way in tensor

product spaceH1⊗H2.

Let

x

~

be an eigenvector of the equation (9). Supposing

x

x

~

~

1

=

and using that fact, that (9) there is an expansion of

a Resultant (4), we determine vectors

~

,

2

x

~

x

₃

,...,

~

x

n₁+n₂.

Besides from (8) it is had, that an element

(

~

,

~

,

2 1

x

x

~

x

3

,...,

2 1 2

1

)

(

)

~

2 1

n n n

n

H

H

x

+

∈

⊗

+ enters into the kernel of a Resultant

)) , ( , ) , ( (

Res A

λ

0

µ

B

λ

0

µ

of bundles A(λ0,µ)andB(λ0,µ).

In works [5, 6] the subsequent components of the

ele-ment

(

~

x

1

,

~

x

2

,

,...,

~

3

x

~ )

2 1n

n

x+ entering into the kernel of

Re-sultant

Re

s

(

A

(

λ

0

)

,

B

(

λ

0

))

are for the first time well

enough studied . In particularities, in case of when opera-tional bundles act in finite-dimensional space and depend on the same parameters, the aspect of the first component

1

~

x

of the element

(

~

x

₁

,

~

x

₂

,

~x3,...,

2 1 2

1

)

(

)

~

2 1

n n n

n

H

H

x

₊

∈

⊗

+,

enter-ing into the kernel of a Resultant Res(A(

λ

0,

µ

),B(

λ

0,

µ

)) is

constructed by the following manner :

1

~

_x

_{has the form}

linear combinations of elements

, ... 0 1 1

0 i i

i V U V U V

U⊗ + ₋⊗ + + ⊗ where

U

0

,...,

U

i(accordingly,

i

V

V0,..., ) is a restricted chain of eigen and associated vectors

of the bundle

A

(

λ

₀

,

µ

)

(accordingly, B(λ0,µ)) at the

fixed value of parameter

λ

=

λ

₀. The parameter

µ

, being

some common eigen value

µ

(

λ

₀

)

both of operators

)

,

(

λ

0

µ

A

andB(λ₀,µ).

m

U

U

0

,...,

(accordingly,

s

V

V

0

,...,

) system of eigen and associated vectors of an

operator

A

(

λ

₀

,

µ

)

(accordingly, B(λ0,µ)), ad equating

to their common eigen value

µ

(

λ

0

)

.

,

~

,

~

(

x

1

x

2

,...,

~

3

x

~ )

2 1n

n

x+ coincide with the elements of

deriva-tive systems constructed on the first components

. ... 0 1 1

0 i i

i V U V U V

U ⊗ + − ⊗ + + ⊗ Eigen values

(

λ

0

,

µ

(

λ

0

))

of twoparameter system (2) can be several and all they

have in the first coordinate number

λ

₀.

The form of the first component of element entering the kernel of the resultant of a two polynomial bundles from one parameter acting in finite-dimensional spaces is an-nounced in [2]. For two-parameter systems in Hilbert spaces, at realization of some conditions the aspect of the first coordinate

~

x

₁ of an element (~x1,~x2,

,...,

~

3

x

2 1 2

1 ) ( )

~

2 1

n n n

n H H

x +

+ ∈ ⊗ , entering into the kernel of a Resultant

)) , ( , ) , ( (

ResAλ0 µ Bλ0 µ is announced in [5].

We shall show, that each eigenvector of the equation 0

~ ) ( ~

0 x =

C λ is the first component of an element, entering

into the kernel of Resultant of bundlesA(λ₀,µ)_andB(

λ

0,

µ

).

Really, considering

~

x

=

~

x

₁, we have that the left part of the

equation

C

~

(

λ

0

)

~

x

=

0

represents expansion of a

Resul-tant (7) on a vector

~

x

.

Converting to definition of eigen and associated vectors of two-parameter system, from (9) we discover, that the

element 1

~ ~ _x

x= can be an eigenvector of two-parameter

system of operators (3), ad equating to an eigen value ))

( ,

(λ0 µ λ0 or the associated vector of this system (3) ad

equating to some eigen value

(

λ

0

,

µ

(

λ

0

))

in which

defini-tion there is no derivadefini-tion on the parameter

λ

.

Such associated vectors of two-parameter system would be expedient for naming the associated vectors of

two-parameter system in a direction

µ

. Similarly,

asso-ciated vectors in which definition there is no derivation on

parameter

µ

, we shall name the associated vectors of

two-parameter system in a direction

λ

.

Thus, all eigenvectors of the equation (9) corresponding

to eigen value

0

λ

are either eigen vector or associated

vector on direction

µ

of the twoparameter system

cor-responding to an eigen value(λ0,µ(λ0)).

Remains to prove, that all associated vectors of (9) also be the associated vectors of a two-parameter problem (2).

Let

1

~

_y

_{there is a first associated vector to an}

eigenvec-tor

1

~

x

of the equation (9)

corresponding to its eigen value

0

λ

. Then we have

0 1 0 1 r 0 0 0 1 r

r 1

1 1 0 r 1 1 2 2 1

d C( )y C( )x

d

(C C ... C )y

C x ... r −C x , (max(m n , m n ) r)

λ + λ

λ

= + λ + + λ +

+ + + λ =

ɶ _ɶ ɶ _ɶ ɶ ɶ _{ɶ ɶ} ɶ _ɶ ɶ _ɶ

(10)

and

0

~

~

...

~

~

~

~

~

)

(

~

1 0 1

1 0 1 0 1

0

x

=

C

x

+

C

x

+

+

C

x

=

C

λ

λ

λ

r _r (11)

Equation (11) means that there is nonzero element

2 1 2 1

)

~

,...,

~

,

~

(

_{1 2} n n

n n

H

x

x

x

₊

∈

+ , entering into the kernel of

Re-sultant of bundles

1 1 ` 1 ` 1

1 1

...

...

)

,

(

0 0 0 1 0 1 m n

n m

m m

A

A

A

A

A

A

λ

µ

=

+

λ

+

+

λ

+

µ

₊

+

+

µ

₊ ,

2 2 ` 2 2

2 2

...

...

)

,

(

0 0 0 1 0 1 m n

n m

m m

B

B

B

B

B

B

λ

µ

=

+

λ

+

+

λ

+

µ

+

+

+

µ

+

such that (8) is fulfilled.

It is clear that all eigen vectors of (6) coincide with the 1-st component of element from the KerRe ( (s Aλ µ0, ), (Bλ µ0, )).We

have the completeness of eigen and associated vectors of the

)) , ( , ) , ( (

Res Aλ0 µ Bλ0 µ in space

2 1

)

(

1 2

n n

H

H

⊗

+

If

(

1 2

2 1

)

~

,...,

~

,

~

(

~

2 1

n n n n

H

y

y

y

y

=

₊

∈

+ the 1-st associated

vector to eigen vector 1 2

2 1 )

~ ,..., ~ , ~ ( 1 2

n n n

n H

x x

x ₊ ∈ + of the

opera-torRes(A(λ0,µ),B(λ0,µ))then we have that the following

expressions (12) and equalities (8) are fulfilled

1

1 1 1

0 1 m 1 2

m n n 0 1

A (

)y

A

y

...

d

A

y

A(

)x

0

d

+ +

+

+ +

λ

+

+ +

+

λ

=

λ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

1

1 1 1

0 2 m 1 3

m n n 1 0 2

A ( )y

A

y

...

d

A

y

A( )x

0

d

+ +

+

+ + +

λ

+

+ +

+

λ

=

λ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

2 1 2

1 1 1 2 2

0 n m 1 n 1

m n n n 0 n

A ( )y

A

y

...

d

A

y

A( )x

0

d

+ +

+ +

+ + +

λ

+

+ +

+

λ

=

λ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

(12)

2

2 2 2

0 1 m 1 2

m n n 0 1

B (

)y

B

y

...

d

B

y

B(

)x

0

d

+ +

+

+ +

λ

+

+ +

+

λ

=

λ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

1 2 1

2 2 1 2 1

0 n m 1 n 1

m n n n 0 n

B (

) y

B

y

...

d

B

y

B(

)x

0

d

+ + +

+ +

+

+ +

λ

+

+ +

+

λ

=

λ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

ɶ

_ɶ

1 1 1 1

0 1 m 1 2 m n n

A (

ɶ

+

λ

)x

ɶ

+

A

ɶ

+ ₊

x

ɶ

+ +

... A

ɶ

+ ₊

x

ɶ

=

0

(13)

0

~

~

...

~

~

~

)

(

~

2 1 1 1 2

1

2 1 1 1

0

+

+

+

+

=

+ + +

+ + +

+

n n n m n

m

n

A

x

A

x

x

1 2 2 2 0 1 m 1 2 m n n

B (

+

λ

)x

ɶ

+

B

ɶ

+ ₊

x

ɶ

+ +

... B

ɶ

+ ₊

x

ɶ

=

0

0

~

~

...

~

~

~

)

(

2 1 2 2 1

2

1 2 1

0

+

+

+

+

=

+ + +

+ + +

n n n m n

n

n

B

x

B

x

x

B

λ

The last means, that

1

~

_y

_{there is the associated vector of}

two-parameter system corresponding to eigenvector

1

~

x

with eigen value(

λ

0,

µ

(

λ

0)). From (8) we obtain (11). From

(12) and (13) we obtain (10).By analogy in the image it is possible to prove, that the associated vectors of other orders

of the equations (9) corresponding to an eigen value

λ

₀

also are the associated vectors of two-parameter system (2).

At each eigen value

λ

₀ of the equation (9) corresponding

to

λ

₀ eigen and associate vectors of (9) coincide with eigen

and associated vectors of the two-parameter system (2),

corresponding some eigen values (λ0,µ(λ0))of system (2) .

Eigen vector of the equation (9) is eigenvector of (2) or its

associated vector on direction

µ

.

Associated vectors of the

equation (9),corresponding to eigen value

λ

₀are also

as-sociated vectors of the system (2). Earlier we proved that the system of all eigen and associated vectors of the

equa-tion (9) forms the max(m1n2,m2n1)-fold base of space

.

2 1

H

H

⊗

Then the system of eigen and associated vectors

of two-parameter system (2) also forms max(m1n2,m2n1)

-fold base in spaceH1⊗H2. The Theorem is proved.

In the special case, whenH1 = H2 = R and operators

1 1 2 2

, , ( 1, 2 , ..., ; 1, 2 , ..., )

i k

A B i = m + n k = m + n are

real numbers.

If to consider variables

x

and

y

parameters then the

algebraic system (1) is a special case of multiparameter system (2).

Corollary. Let one of three following conditions is ful-filled:

a)

max(

m

1

n

2

,

m

2

n

1

)

=

m

1

n

2, am1≠0,bm2+n2≠0;

b)max(m1n2,m2n1)=m2n1,

b

m2

≠

0 ,

a

m1+n1

≠

0

c)

m

₁

n

₂

=

m

₂

n

₁ is 2 1 1 2 2 1

1 1 2 ( 1) 1 2 2 0,

n n n n n n m n m m m n

a ₊ b + − a b ₊ ≠

Then the algebraic system (1) has the solutions.

3. Conclusion

In this paper it is proved the existence of solutions of some nonlinear algebraic system .For the proof of this it is essentially used the results of author, obtained for the mul-tiparameter system of operator in the finite dimensional

spaces

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