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Semiconservative systems of integral equations with two kernels. (English) Zbl 1223.45004
Summary: The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation \(f(x) = g(x) + \int_0^{\infty} k(x - t) f(t) dt + \int_{-\infty}^0 T(x - t)f(t)dt\), where \(K, T\) are \(n \times n\) matrix valued functions, \(n \geq 1\), with nonnegative integrable elements, are considered in one semiconservative (singular) case, where the matrix \(A = \int_{-\infty}^{\infty} K(x)dx\) is stochastic one and the matrix \(B = \int^{\infty}_{-\infty} T(x)dx\) is substochastic one. It is shown that in certain conditions the nonhomogeneous equation simultaneously with the corresponding homogeneous one possesses positive solutions.
MSC:
45F15 Systems of singular linear integral equations
45M20 Positive solutions of integral equations
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