Semiconservative Systems of Integral Equations with Two Kernels - N. B. Yengibaryan; A. G. Barseghyan (International Journal of Mathematics and Mathematical Sciences)
The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation ∫𝑓(𝑥)=𝑔(𝑥)+∞0∫𝑘(𝑥−𝑡)𝑓(𝑡)𝑑𝑡+0−∞𝑇(𝑥−𝑡)𝑓(𝑡)𝑑𝑡, 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 ∫𝐴=∞−∞𝐾(𝑥)𝑑𝑥 is stochastic one and the matrix ∫𝐵=∞−∞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.