Nondecreasing solutions of fractional quadratic integral equations involving Erdélyi-Kober singular kernels

Abstract

In this paper, we firstly present the existence of nondecreasing solutions of a fractional quadratic integral equations involving Erdélyi-Kober singular kernels for three provided parameters $\alpha\in ({1}/{2},1)$, $\beta\in (0,1]$ and $\gamma\in [\beta(1-\alpha)-1,\infty)$. Moreover, we prove this restriction on $\alpha$ and $\beta$ can not be improved. Secondly, we obtain the uniqueness and nonuniqueness of the monotonic solutions by utilizing a weakly singular integral inequality and putting $\gamma\in [{1}/{2}-\alpha,\infty)$. Finally, two numerical examples are given to illustrate the obtained results.