On the quadratic integral equations and their applications

The class of quadratic integral equations contains, as a special case, numerous integral equations encountered in the theory of radiative transfer, the queuing theory, the kinetic theory of gases and the theory of neutron transport. As a pursuit of this, in the following pages, sufficient conditions are given for the existence of positive continuous solutions to some possibly singular quadratic integral equations. Meanwhile, we prove the existence of maximal and minimal solutions of our problems. The method used here depends on both Schauder and Schauder-Tychonoff fixed point principles. Unlike all previous contributions of the same type, no assumptions in terms of the measure of noncompactness were imposed on the nonlinearity of the given functions. As far as we know, the approach presented in this paper, in particular, the discussion of the existence of maximal and minimal solutions to the quadratic integral equations was never applied in the field of the quadratic integral equations and so is new. (C) 2011 Elsevier Ltd. All rights reserved.