On a class of integral equations of Urysohn type with strong non-linearity

We study a class of homogeneous and non-homogeneous integral equations of Urysohn type with strong non-linearity on the positive semi-axis. It is assumed that some non-linear integral operator of Wiener–Hopf–Hammerstein type is a local minorant of the corresponding Urysohn operator. Using special methods of the linear theory of convolution-type integral equations, we construct positive solutions for these classes of Urysohn equations. We also study the asymptotic behaviour of these solutions at infinity. As an auxiliary fact in the course of the proof of these assertions, we construct a one-parameter family of positive solutions for non-linear integral equations of Wiener–Hopf–Hammerstein type whose operator is a minorant for the original Urysohn operator. We give particular examples of non-linear integral equations for which all the hypotheses of the main theorems hold.