Boundary value problems of fractional order: Analyzing iterative systems with positive solutions

This study addresses a fractional-order iterative systemHere, we consider the system where t is an integer from 1 to w, and r w+1 equals r 1 and (q) belongs to the Lebesgue space L p [0, 1] for 1 p and may exhibit singularities within (0, 1/2).The study employs Riemann-Liouville fractional derivatives of orders 1 and 2 , denoted as D 0 + .By leveraging fixed-point index theory within a Banach space cone, the existence of infinitely many positive solutions to this problem is established.Furthermore, in the specific case where w equals 1, sufficient conditions for a unique solution are derived by using Rus's theorem within a complete metric space.