On integrable and approximate solutions for Hadamard fractional quadratic integral equations

This article addressed the integrable and approximate solutions of Hadamard-type fractional Gripenberg's equation in Lebesgue spaces $L_1[1,e]$. It is well known that the Gripenberg's equation has significant applications in mathematical biology. By utilizing the fixed point (FPT) approach and the measure of noncompactness (MNC), we demonstrated the presence of monotonic integrable solutions as well as the uniqueness of the solution for the studied equation in spaces that are not Banach algebras. Moreover, the method of successive approximations was successfully applied and, as a result, we obtained the approximate solutions for these integral equations. To validate the obtained results, we provided several numerical examples.