This article considers the existence and the uniqueness of monotonic solutions of a delay functional integral equation of fractional order in the weighted Lebesgue space $L_1^N({\mathbb{R}}^{+})$. Our analysis uses a suitable measure of noncompactness, a modified version of Darbo's fixed point theorem, and fractional calculus in the mentioned space. An illustrated example to show the applicability and significance of our outcomes is included.

We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $L_{\phi }$, namely when the generating $N$-functions fulfill ${\mathrm{\Delta }}^{\prime }$ and ${\mathrm{\Delta }}_3$-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.