On Erdélyi–Kober fractional Urysohn–Volterra quadratic integral equations

A very general nonlinear singular integral equation is introduced, namely u(τ)=f1(τ,u(τ))+βf2(τ,u(τ))Γ(γ)∫0τsβ−1k(τ,s,(Au)(s))(τβ−sβ)1−γds,0≤τ≤1,β > 0 and 0 < γ < 1. The above equation is called Erdélyi–Kober fractional Urysohn–Volterra quadratic integral equation. The main goal is to show that the above equation has solutions in C[0, 1] and these solutions are nonnegative and nondecreasing on [0, 1]. By means of a measure of noncompactness and Darbo fixed point theorem we prove our main results. In the end of the paper, we give an example to show that our assumptions of our abstract results are rather easy to verify.