Nondecreasing solutions of a quadratic Abel equation with supremum in the kernel

We prove an existence theorem for a quadratic Abel integral equation of the second kind with supremum in the kernel. The quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases. We show that the singular quadratic integral equations with supremum has a monotonic solution in C[0,1]. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.