Semiconservative Systems of Integral Equations with Two Kernels

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Volume 2011, Article ID 917951,11pages doi:10.1155/2011/917951

Research Article

Semiconservative Systems of Integral

Equations with Two Kernels

N. B. Yengibaryan and A. G. Barseghyan

Institute of Mathematics of National Academy of Sciences of Armenia, 24b Marshal Baghramian Avenue, 0019 Yerevan, Armenia

Correspondence should be addressed to N. B. Yengibaryan,[email protected]

Received 3 February 2011; Revised 23 April 2011; Accepted 30 May 2011

Academic Editor: Giuseppe Marino

Copyrightq2011 N. B. Yengibaryan and A. G. Barseghyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equationfx gx ₀∞kx−tftdt0

−∞Tx−tftdt, whereK, T aren×n

matrix valued functions,n ≥ 1, with nonnegative integrable elements, are considered in one semiconservative singular case, where the matrix A _−∞∞ Kxdx is stochastic one and the matrix B _−∞∞ Txdx is substochastic one. It is shown that in certain conditions the nonhomogeneous equation simultaneously with the corresponding homogeneous one possesses positive solutions.

1. Introduction: Problem Statement

Consider the scalar or vector integral equations on the whole line with two kernelssee1–

4:

fx gx

_∞

0

Kx−tftdt

₀

−∞Tx−tftdt , −∞< x <∞,

1.1

where the kernel-functionsKx, Tx are matrix-valued functions with nonnegative ele-ments;g andf are the given and sought-for column vectorsvectorfunctions; respectively. Assume that

K, T ∈Ln×n, g∈Ln, K, T, g ≥O. 1.2

HereLn×n is the space ofn×n-n ≥ 1order matrix-valued functions, andLn _{is the space}

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matrix is denoted by O. The inequalities between the matrices or vectors, the operation of integration, and some other operations shall be treated componentwise.

Denote byςthe followingn-dimensional row vector:

ς 1,1, . . . ,1. 1.3

LetC≥Obe ann×nmatrix. If

ςCς, 1.4

then the matrixCis a stochastic oneaccurate within transpose, see5. If

ςC≤ς, 1.5

then the matrix C is substochastic to a wide extent. We shall call the matrix C really substochastic, ifςC≤ς, ςC /ς and uniform substochastic if there existμ∈0,1such that

ςC≤μς, 0≤μ <1. 1.6

Let us introduce the followingn×nmatricesA, B≥O, related with the equation1.1:

A

_∞

−∞Kxdx, B

_∞

−∞Txdx. 1.7

We shall call the kernelKconservative, dissipative, or uniform dissipative if the matrix

A is stochastic, really substochastic, or uniform substochastic, respectively. We shall use analogous names to the kernelT.

We shall call1.1semiconservative, if one of the kernelsK,T is conservative and the other is dissipative. Without the loss of generality, one can assume that

ςAς, ςB≤ς, ςB /ς. 1.8

In the uniform semiconservative case of1.1we have

ςAς, ςB≤μς, 0≤μ <1, 1.9

whereas in the conservative case, both of the kernelsK, T are assumed to be conservative. IfTO, then1.1is reduced to the well-known Wiener-Hopf integral equation:

ϕx hx

_∞

0

Kx−tϕtdt, x >0, 1.10

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The theory of the scalar and vector conservative Wiener-Hopf equation1.10 where

Kis the conservative onepassed a long way of development. Manyconservativephysical processes in homogeneous half-space are described by such equations. They are of essential interest in the radiative transferRT, kinetic theory of Gasessee6,7, in the mathematical theory of stochastic processes, and so forth.

In the RT, the conservative equation1.10corresponds to the absence of losses of the radiation inside mediacase of pure scattering. However, such losses occur as a result of an exit of radiation from media. In case of the dissipative one, there are losses inside media as well.

Equation1.1with two kernels arises in some more generaland more complicated problems, where the physical processes occur in the infinite media, consisting of two adjacent homogeneous half-spaces see 7. In each of these half-spaces, the processes may be dissipative or conservative. Another area of applications is connected with the RT in the atmosphere-ocean system.

In the theory of RT, the free termgin1.1plays the role of initial sources of radiation. The conservative and semiconservative cases belong to the singular cases of1.1. In these cases, the unique solvability of1.1in the “standard” functional spacesLn

p 1 ≤ p ≤ ∞is

violated.

A number of results concerning to the scalar conservative equation1.1have been obtained by Arabadzhyan3. The systems of conservative or semiconservative equations with two kernels have not ever been treated.

The present paper is devoted to the solvability and the properties of the solutions of the nonhomogeneous and homogeneous vector equation1.1. The main attention will be paid to the uniform semiconservative case1.9. It will be shown that in certain conditions both the nonhomogeneous equation1.1and the corresponding homogeneous equation possess positive locally integrable solutions.

2. Auxiliary Propositions

2.1. Integral Operators

Let a, b ⊂ −∞,∞. Consider Banach space B-space La, b ≡ L1a, b and the

corresponding B-spaceLn_{a, b}_{of vector-valued functions}_{vector columns}_f _f

1, . . . , fnT.

HereT is a sign of the transpose. The norm inLn_{a, b}_{is defined by}

_fn

k1

_fk

La,bς

b

a

_f_x_dx. _2.1

Consider the linear topological space LLoc0,∞ of the functions, which are integrable on each finite interval 0, r, r < ∞.The space Ln

Loc0,∞ possesses the topology of the

componentwise convergence.

The unit operator in each of spaces introduced above is denoted byI. LetΩn be the

following class of matrix convolution operators on the whole line: ifU ∈Ωn, then

ϕx Uf x

_∞

−∞Ux−tftdt, U∈L

n×n_. _2.2

The operatorU ∈ Ωn acts in the spacesLn,Lnp 1 ≤ p ≤ ∞, and in some other spaces of

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The classΩn is an algebra where the kernel function of the operator product is the

convolution of the kernel functions of the factors.

Let us estimate the norm of operator U ∈ Ωn in the B-space Ln. Let C ≥ O be the

followingn×nmatrix:C ckm _−∞∞ |Ux|dx. Taking thecomponentwisemodulus in

2.2and integrating on−∞,∞, we come to the following inequality:

_∞

−∞

_ϕ_x_dx_≤_C∞

−∞

_f_t_dt. _2.3

Multiplying this inequality on the left by the vector ς, we come to the following inequality:

_Uf_≤_γ_f_, _where_γ _max k

n

m1

ckm. 2.4

From here the estimate follows:

_U_≤_γ. _2.5

Let us introduce the projectors projection operators P±, acting in the spaces of

summable or locally summable functions on−∞,∞by the equalities:

Pfx fxϑx, P₋fx fxϑ−x. 2.6

Hereϑis the Heaviside function of the unit jump. In each of the spacesLp1≤p≤ ∞, we have

P±1. 2.7

Denote byK, T∈Ωnthe following operators, whose kernel functionsK, T participate

in1.1:

Kfx

_∞

−∞Kx−tftdt,

Tfx

_∞

−∞Tx−tftdt. 2.8

Equation1.1admits the following operator entry

fgWf, 2.9

whereWKPTP−.

The projectors P± are the diagonal matrices of the operators with the diagonal

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The operatorWis an Integral operator:

Wfx

_∞

−∞Wx, tftdt, whereWx, t Kx−tϑt Tx−tϑ−t. 2.10

Hereϑxis the diagonal matrix with the diagonal elementsϑx.

2.2. On the Invertibility of the Operator

I

−

W

in

L

n

Let us estimate the norm of W in Ln_{. Assume at first that the kernel functions} _{K, T} _are

arbitrary elements ofLn×n. Let ϕ Wf ,f ∈ Ln_{. One can obtain the following inequality}

similar to2.3:

_∞

−∞

ϕxdx≤A

_∞

0

ftdtB

₀

−∞

ftdt. 2.11

HereA akm

_∞

−∞|Kx|dx, B bkm

_∞

−∞|Tx|dx.

We haveςA≤λς,ςB≤μς, where

λmax

m n

k1

akm, μmax

m n

k1

bkm. 2.12

Multiplying2.11on the left by the vectorς, we get

ϕ≤λς

_∞

0

ftdtμς

₀

−∞

ftdt≤maxλ, μf. 2.13

Thus, we proved the following.

Lemma 2.1. The following estimate for the norm of the operatorWinLn_{is valid:}

W≤qmaxλ, μ. 2.14

Ifq <1, then the operatorWis contracting inLn_{, hence the operator}_I₋_W_{is invertible,}

and1.1withg∈Ln_{has a unique solution}_f_∈_Ln_{. If therewith}_{K, T, g}_≥_O_{, then}_f_≥_O_.

In accordance with the general theory of the integral equations with two kernelssee

1,2, for the invertibility of the operator I−W inLn_{, it is necessary the fulfilment of the}

following conditions of nondegeneration:

detJ−Ks/0, detJ−Ts/0, −∞< s <∞. 2.15

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In the semiconservative case1.9, we have:ςJ−K0 ςJ−A O. Hence detJ− A 0, that is, the symbolJ−Ks, is degenerated in the points0. In the conservative case

where A and B are stochastic matrices, both of the conditions2.15are violated. Thus, the operatorI−Wis noninvertible inLn_{in the semiconservative and conservative cases.}

3. Semiconservative Nonhomogeneous Equation

In this section, we shall consider the question of the solvability of the uniform semiconserva-tive nonhomogeneous equation1.1,1.9under the following additional assumption: there exists a strong positive vector-columnηsuch thatAηη, η > O. In accordance with Perron-Frobenius theoremsee8, the existence of such vectorηis secured if the stochastic matrix

Ais an irreducible one.

3.1. One Auxiliary Equation

At the outset, consider the auxiliary conservative Wiener-Hopf equation1.10, where

O≤h∈Ln0,∞, x >0, ςAς, Aηη 3.1

with the conservative kernelKparticipating in1.1. The following lemma follows from the results9:

Lemma A. Equation1.10,3.1possesses the minimal solutionϕ≥Owhich is locally integrable on0,∞(see [9]). The following asymptotics holds

x

0

ϕtdtox2, x−→ ∞. 3.2

This asymptotics admits an adjustment subject to additional assumptions on kernelKand free termh

(see [9]).

Denote byνthe following matrix the first moments of matrix-functionK:

ν

_∞

−∞xKxdx, 3.3

with the assumption of componentwise absolute convergence of this integral. Let

σςνη, −∞< σ <∞. 3.4

The numberσplays a principal role in the classification of the conservative equation1.10

see9. Ifσ <0, then

_x

0

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If therewith the free termhhas a finite first moment:₀∞thtdt <∞, thenϕ∈Ln₀_,_∞_.

Consider the simple iterations for1.10:

ϕm1x hx

_∞

0

Kx−tϕmtdt, ϕ0O, m0,1,2, . . . . 3.6

The sequenceϕmpossesses the following properties:O≤ ϕm ∈Ln₀_,_∞_{. It is easy}

to show that the sequenceϕmis monotonic. Indeed we have

ϕm1x−ϕmx

_∞

0

Kx−tϕmt−ϕm−1tdt, ϕ0O, m0,1,2, . . . . 3.7

Using the induction by m, we obtain that ϕm1x − ϕmx ≥ O, which implies the monotonicity of the sequence ϕm. The sequence ϕm converges monotonically by the topology ofLn_Loc0,∞to the minimal solutionϕof1.10:

ϕm↑ϕ inLn_Loc0,∞. 3.8

3.2. One Existence Theorem for

1.1

Consider now1.1under conditions

ςAς, Aηη, ςB≤μς, 0≤μ <1. 3.9

Let us consider the following iterations for1.1:

fm1x gx

_∞

0

Kx−tfmtdt

₀

−∞Tx−tf

m_t_dt, _3.10

f0O, m0,1,2, . . . . 3.11

We have

fm∈Ln, m0,1, . . . , O≤fmx↑ bym. 3.12

Letf≥Obe any positive solution of1.1,3.9:

fx gx

_∞

0

Kx−tftdt

0

−∞Tx−tftdt.

3.13

It is easy to verify by induction thatfm≤f, for eachm≥0. Hence, if the sequencefm → f

converges by the topology ofLn

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Remark 3.1. If the sequencefm → f converges by the topology ofLn_Loc−∞,∞, then one can take the limit in3.10, andf≥Owill be the minimal positive solution of1.1.

This fact is proved using the monotonicity offmand the two-sided inequalitiessee

10Item 2.

Let us introduce the restrictions of the functionsfmon0,∞and−∞,0:

ωm fm_0,∞, ψmfm_−∞,0 ∈Ln−∞,0. 3.14

Theorem 3.2. Let the conditions 3.9 hold. Then 1.1 has the minimal positive solution f ∈

Ln

Loc−∞,∞withf|−∞,0∈Ln−∞,0and

x

0

ftdtox2, x−→∞. 3.15

If∃σ <0, then₀xftdtox, x → ∞.

Proof. After the integration of3.10overxon−∞,∞, we shall have

am1bm1γAamBbm, 3.16

wheream₀∞ωmxdx, bm_−∞0 ψmxdx,γ_−∞∞ gxdx.

Multiplying3.16on the left by the vectorςand taking into account3.9, we obtain the inequality

ςam1ςbm1≤ςγςamμςbm, 3.17

whence it follows, with due regard for the monotony of sequences am, bm, that 1 −

μςbm≤ςγ. We arrive at the following estimate:

ςbmψm≤1−μ−1ςγ. 3.18

It follows from B. Levy well-known theorem that the monotonous and bounded by norm sequenceψmconverges inLn_−∞,₀_:

O≤ ψm ↑ψ∈Ln−∞,0. 3.19

Now compare relations3.10forx >0 with iterations3.6, in whichhx gx _−∞0 Tx−

tψtdt, x > 0 ψ is determined according to3.19. In virtue ofψm ≤ ψ, we have the

inequalityωmx≤ϕmx, x >0, m0,1, . . .. Henceωmx≤ϕx, x >0, m0,1, . . .. According to the Lebesgue theorem, the monotonic sequenceωmconverges by the topology

Ln

Loc0,∞:

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We have obtained that the narrowing of the monotonic iteration sequencefmto the negative semiaxis is convergent inLn_−∞,₀_{, and the narrowing of} _fm _{to the positive semiaxis is}

convergent inLn_Loc0,∞. If we denotefx ωx, x>0_{ψx, x<0},thenfm → f inLn_Loc−∞,∞ i.e.,

in Ln_{−∞, r}_, _{for all}_{r <} _∞_{. Taking limit in} _3.10 _see_{Remark 3.1}_{, we obtain that the}

vector functionf satisfies1.1,3.9, and thereby, it is its minimal solution. The Theorem is proved.

Observe that, under the assumptions of Theorem 3.2, the existence of the locally integrable solution of1.1could be proved using the fixed point principle of the paper10. Anyway, with this method, one cannot obtain the propertiesf|_−∞,0∈Ln_−∞,₀_and_3.15_.

4. The Homogeneous Semiconservative Equation

The homogeneous system1.1under the conditions3.9will be considered in the present section:

Gx

_∞

0

Kx−tGtdt

₀

−∞Tx−tGtdt.

4.1

Consider at first the corresponding conservative homogeneous system of Wiener-Hopf equations:

Sx

_∞

0

Kx−tStdt. 4.2

Let us formulate some results on the existence of positive solutions of the system4.1 see

9.

Theorem A. LetK satisfy the conditionsς A ς, Aη η(see 3.9), and one of the following

conditions (a) or (b):

athe property of symmetry (hereT is the sign of transpose):

K−x KTx, 4.3

bthe kernelKhas a finite first momentν(see3.3) and that

σ≤0, 4.4

whereσis determined by3.4.

Then the equation4.2has a positive solutionSx> O. The vector functionSis absolutely continuous and monotone increasing. The following asymptotics holds

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Let usin conditions of the Theorem Acontinue the vector functionSto all the real axis in accordance with the equality4.2. Then the equality4.2takes place on the whole real axis.

The convergence of the following integral is necessary and suﬃcient in order thatS

has a integrable extension on the negative semiaxis

_∞

0

Stdt

_−t

−∞Kxdx <∞.

4.6

If4.6holds, then we will haveS∈Ln

Loc−∞,∞, Sx>O.

It follows from the asymptotics4.5that for the fulfilment of the requirement4.6, it is suﬃcient that the kernel functionKhas thecomponentwisefinite second moment on the negative semiaxis, that is,

_∞

0

K−xx2dx <∞. 4.7

Now consider, uniform semiconservative4.1.

Theorem 4.1. Let the homogeneous equation4.2satisfy the conditions3.9,4.7and either of the conditions4.3or4.4. Then there exists a solutionG > O,G∈Ln_Loc−∞,∞of this equation. The following asymptotics hold:

x

−∞GtdtO

x2, x−→ ∞. 4.8

Proof. In accordance with Theorem A, there exists a solutionS > O of4.1. The inequality

4.6follows from the condition4.7and from the asymptotics4.5; hence,S∈Ln_−∞,₀_.

Let us introduce a new sought-for vector functionf ≥ O in 4.1by means of the relation:

Gx fx Sx, −∞< x <∞. 4.9

Substituting4.9into4.1with due regard for4.2, we obtain an inhomogeneous equation of the type1.1with respect tof, in which

gx

0

−∞Tx−tStdt, x∈R.

4.10

Because ofS ∈ Ln_−∞,₀_{, we have} _g _∈ _Ln_{. In accordance with}_{Theorem 3.2}_{, there exists a}

minimalsolution of 1.1with a free term4.10that implies the existence of the strong positive solution of the form4.9of the homogeneous equation4.1.

The asymptotics4.8follow immediately from the properties offandS, included in

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It is remarkable that under the conditions ofTheorem 4.1, both the nonhomogeneous equation 1.1 with g ∈ Ln _{and the homogeneous equation} _4.1 _{simultaneously have}

positive solutions.

References

1 I. C. Gohberg and I. A. Feldman, Convolution Equations and Projection Methods for Their Solution, vol. 41 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1974.

2 S. Pr ¨ossdorf, Einige Klassen Singul¨arer Gleichungen, Akademie-Verlag, Berlin, Germany, 1974.

3 L. G. Arabadzhyan, “On a conservative integral equation with two kernels,” Mathematical Notes, vol. 62, no. 3, pp. 271–277, 1997.

4 A. G. Barsegyan, “Transfer equation in adjacent half-spaces,” Contemporary Mathematical Analysis, vol. 41, no. 6, pp. 8–19, 2006.

5 W. Feller, An Introduction of Probability Theory and Its Applications, vol. 2, John Wiley & Sons, New York, NY, USA, 2nd edition, 1971.

6 C. Cercignani, Theory and Application of the Boltzmann Equation, Elsevier, New York, NY, USA, 1975. 7 B. Davison, Neutron Transport Theory , Oxford Clarendon Press, Oxford, UK, 1957.

8 P. Lancaster, Theory of Matrices, Academic Press, London, UK, 1969.

9 N. B. Engibaryan, “Conservative systems of integral convolution equations on the half-line and the whole line,” Sbornik: Mathematics, vol. 193, no. 6, pp. 847–867, 2002.

10 N. B. Engibaryan, “On the fixed points of monotonic operators in the critical case,” Izvestiya RAN,

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