International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 917951, 11 pages
http://dx.doi.org/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

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

Academic Editor: Giuseppe Marino

Copyright © 2011 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.

Abstract

The solvability and the properties of solutions of nonhomogeneous and homogeneous vector integral equation ∫ 𝑓 ( 𝑥 ) = 𝑔 ( 𝑥 ) + ∞ 0 ∫ 𝑘 ( 𝑥 − 𝑡 ) 𝑓 ( 𝑡 ) 𝑑 𝑡 + 0 − ∞ 𝑇 ( 𝑥 − 𝑡 ) 𝑓 ( 𝑡 ) 𝑑 𝑡 , 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 ) 𝑑 𝑥 is stochastic one and the matrix ∫ 𝐵 = ∞ − ∞ 𝑇 ( 𝑥 ) 𝑑 𝑥 is substochastic one. It is shown that in certain conditions the nonhomogeneous equation simultaneously with the corresponding homogeneous one possesses positive solutions.