Fixed-Point Theory, Variational Inequalities, and Its Approximation AlgorithmsView this Special Issue
Semiconservative Systems of Integral Equations with Two Kernels
The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation , where , are matrix valued functions, , with nonnegative integrable elements, are considered in one semiconservative (singular) case, where the matrix is stochastic one and the matrix 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 kernels (see [1–4]): where the kernel-functions are matrix-valued functions with nonnegative elements; and are the given and sought-for column vectors (vectorfunctions); respectively. Assume that Here is the space of - () order matrix-valued functions, and is the space of column vectors, with components in Lebesgue space . The zero vector or matrix is denoted by . The inequalities between the matrices or vectors, the operation of integration, and some other operations shall be treated componentwise.
Denote by the following -dimensional row vector: Let be an matrix. If then the matrix is a stochastic one (accurate within transpose, see ). If then the matrix is substochastic to a wide extent. We shall call the matrix really substochastic, if and uniform substochastic if there exist such that Let us introduce the following matrices , related with the equation (1.1):
We shall call the kernel conservative, dissipative, or uniform dissipative if the matrix is stochastic, really substochastic, or uniform substochastic, respectively. We shall use analogous names to the kernel .
We shall call (1.1) semiconservative, if one of the kernels , is conservative and the other is dissipative. Without the loss of generality, one can assume that In the uniform semiconservative case of (1.1) we have whereas in the conservative case, both of the kernels are assumed to be conservative.
If , then (1.1) is reduced to the well-known Wiener-Hopf integral equation: Here and are restrictions on of and , respectively.
The theory of the scalar and vector conservative Wiener-Hopf equation (1.10) (where is the conservative one) passed a long way of development. Many (conservative) physical processes in homogeneous half-space are described by such equations. They are of essential interest in the radiative transfer (RT), kinetic theory of Gases (see [6, 7]), in the mathematical theory of stochastic processes, and so forth.
In the RT, the conservative equation (1.10) corresponds to the absence of losses of the radiation inside media (case 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.
Equation (1.1) with two kernels arises in some more general (and more complicated) problems, where the physical processes occur in the infinite media, consisting of two adjacent homogeneous half-spaces (see ). 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 term in (1.1) plays the role of initial sources of radiation. The conservative and semiconservative cases belong to the singular cases of (1.1). In these cases, the unique solvability of (1.1) in the “standard” functional spaces is violated.
A number of results concerning to the scalar conservative equation (1.1) have been obtained by Arabadzhyan . 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 equation (1.1). The main attention will be paid to the uniform semiconservative case (1.9). It will be shown that in certain conditions both the nonhomogeneous equation (1.1) and the corresponding homogeneous equation possess positive locally integrable solutions.
2. Auxiliary Propositions
2.1. Integral Operators
Let . Consider Banach space (B-space) and the corresponding B-space of vector-valued functions (vector columns) . Here is a sign of the transpose. The norm in is defined by Consider the linear topological space of the functions, which are integrable on each finite interval .The space possesses the topology of the componentwise convergence.
The unit operator in each of spaces introduced above is denoted by . Let be the following class of matrix convolution operators on the whole line: if , then The operator acts in the spaces , , and in some other spaces of vector-valued functions.
The class 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 in the B-space . Let be the following matrix: . Taking the (componentwise) modulus in (2.2) and integrating on , we come to the following inequality:
Multiplying this inequality on the left by the vector , we come to the following inequality: From here the estimate follows:
Let us introduce the projectors (projection operators) , acting in the spaces of summable or locally summable functions on by the equalities:
Here is the Heaviside function of the unit jump. In each of the spaces , we have
The projectors are the diagonal matrices of the operators with the diagonal elements .
The operator is an Integral operator:
Here is the diagonal matrix with the diagonal elements .
2.2. On the Invertibility of the Operator in
Let us estimate the norm of in . Assume at first that the kernel functions are arbitrary elements of . Let , . One can obtain the following inequality (similar to (2.3)): Here .
We have , , where
Multiplying (2.11) on the left by the vector , we get Thus, we proved the following.
Lemma 2.1. The following estimate for the norm of the operator in is valid:
If , then the operator is contracting in , hence the operator is invertible, and (1.1) with has a unique solution . If therewith , then .
In accordance with the general theory of the integral equations with two kernels (see [1, 2]), for the invertibility of the operator in , it is necessary the fulfilment of the following conditions of nondegeneration: Here is the unit matrix; the matrices and are the (elementwise) Fourier transforms of and , respectively. For example, , .
In the semiconservative case (1.9), we have: . Hence , that is, the symbol , is degenerated in the point . In the conservative case (where A and B are stochastic matrices), both of the conditions (2.15) are violated. Thus, the operator is noninvertible in in the semiconservative and conservative cases.
3. Semiconservative Nonhomogeneous Equation
In this section, we shall consider the question of the solvability of the uniform semiconservative nonhomogeneous equation (1.1), (1.9) under the following additional assumption: there exists a strong positive vector-column such that . In accordance with Perron-Frobenius theorem (see ), the existence of such vector is secured if the stochastic matrix is an irreducible one.
3.1. One Auxiliary Equation
The following lemma follows from the results :
Lemma A. Equation (1.10), (3.1) possesses the minimal solution which is locally integrable on (see ). The following asymptotics holds This asymptotics admits an adjustment subject to additional assumptions on kernel and free term (see ).
Denote by the following matrix the first moments of matrix-function : with the assumption of componentwise absolute convergence of this integral. Let The number plays a principal role in the classification of the conservative equation (1.10) (see ). If , then If therewith the free term has a finite first moment: , then .
Consider the simple iterations for (1.10):
The sequence possesses the following properties: . It is easy to show that the sequence is monotonic. Indeed we have Using the induction by , we obtain that , which implies the monotonicity of the sequence . The sequence converges monotonically by the topology of to the minimal solution of (1.10):
3.2. One Existence Theorem for (1.1)
This fact is proved using the monotonicity of and the two-sided inequalities (see  Item 2).
Let us introduce the restrictions of the functions on and :
Proof. After the integration of (3.10) over on , we shall have
where , .
Multiplying (3.16) on the left by the vector and taking into account (3.9), we obtain the inequality whence it follows, with due regard for the monotony of sequences , that . We arrive at the following estimate: It follows from B. Levy well-known theorem that the monotonous and bounded by norm sequence converges in : Now compare relations (3.10) for with iterations (3.6), in which ( is determined according to (3.19)). In virtue of , we have the inequality . Hence . According to the Lebesgue theorem, the monotonic sequence converges by the topology : We have obtained that the narrowing of the monotonic iteration sequence to the negative semiaxis is convergent in , and the narrowing of to the positive semiaxis is convergent in . If we denote then in (i.e., in ). Taking limit in (3.10) (see Remark 3.1), we obtain that the vector function satisfies (1.1),(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 of (1.1) could be proved using the fixed point principle of the paper . Anyway, with this method, one cannot obtain the properties and (3.15).
4. The Homogeneous Semiconservative Equation
The homogeneous system (1.1) under the conditions (3.9) will be considered in the present section: Consider at first the corresponding conservative homogeneous system of Wiener-Hopf equations: Let us formulate some results on the existence of positive solutions of the system (4.1) (see ).
Theorem A. Let satisfy the conditions (see (3.9)), and one of the following conditions (a) or (b):
(a)the property of symmetry (here is the sign of transpose): (b)the kernel has a finite first moment (see (3.3)) and that where is determined by (3.4).
Then the equation (4.2) has a positive solution . The vector function is absolutely continuous and monotone increasing. The following asymptotics holds
The convergence of the following integral is necessary and sufficient in order that has a integrable extension on the negative semiaxis If (4.6) holds, then we will have .
It follows from the asymptotics (4.5) that for the fulfilment of the requirement (4.6), it is sufficient that the kernel function has the (componentwise) finite second moment on the negative semiaxis, that is,
Now consider, uniform semiconservative (4.1).
Theorem 4.1. Let the homogeneous equation (4.2) satisfy the conditions (3.9), (4.7) and either of the conditions (4.3) or (4.4). Then there exists a solution , of this equation. The following asymptotics hold:
Proof. In accordance with Theorem A, there exists a solution of (4.1). The inequality (4.6) follows from the condition (4.7) and from the asymptotics (4.5); hence, .
Let us introduce a new sought-for vector function in (4.1) by means of the relation: Substituting (4.9) into (4.1) with due regard for (4.2), we obtain an inhomogeneous equation of the type (1.1) with respect to , in which Because of , we have . In accordance with Theorem 3.2, there exists a (minimal) solution of (1.1) with a free term (4.10) that implies the existence of the strong positive solution of the form (4.9) of the homogeneous equation (4.1).
The asymptotics (4.8) follow immediately from the properties of and , included in Theorem 3.2 and Theorem A. The Theorem is proved.
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