Abstract
The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation , where K, T are n × n matrix valued functions, n ≥ 1, 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.
Highlights
Problem StatementConsider the scalar or vector integral equations on the whole line with two kernels see 1– 4: ∞ fx gx K x − t f t dtT x − t f t dt, −∞ < x < ∞, −∞where the kernel-functions K x, T x are matrix-valued functions with nonnegative elements; g and f are the given and sought-for column vectors vectorfunctions ; respectively
Ln×n is the space of n × n- n ≥ 1 order matrix-valued functions, and Ln is the space of column vectors, with components in Lebesgue space L ≡ L1 −∞, ∞
If ςC ≤ ς, 1.5 the matrix C is substochastic to a wide extent
Summary
Consider the scalar or vector integral equations on the whole line with two kernels see 1– 4:. We shall call the kernel K conservative, dissipative, or uniform dissipative if the matrix A is stochastic, really substochastic, or uniform substochastic, respectively. Many conservative physical processes in homogeneous half-space are described by such equations. In the RT, the conservative equation 1.10 corresponds to the absence of losses of the radiation inside media case of pure scattering. 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 7. In each of these half-spaces, the processes may be dissipative or conservative. It will be shown that in certain conditions both the nonhomogeneous equation 1.1 and the corresponding homogeneous equation possess positive locally integrable solutions
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