Existence of Positive Solutions for a Nonlinear Iterative System of Boundary Value Problems with Tempered Fractional Order Derivative

This paper investigates the existence of positive solutions for an iterative system of nonlinear two-point tempered fractional boundary value problem. Utilizing Krasnoselskii’s fixed point theorem in a cone, we establish criteria for the existence of positive solutions. The proofs involve transforming the problem into an equivalent Fredholm integral equation of the second kind. We further explore solution uniqueness using Rus’s theorem and examine Hyers–Ulam stability, particularly for the case when only a single fractional differential equation is considered, m=1. Our study represents a significant departure from previous works by including Riemann–Liouville tempered fractional derivative operators and an iterative equation. This research sheds light on the diverse applications of iterative functional differential equations, extending beyond noniterative counterparts. Throughout the paper, presumptive conditions are applied, and the results are validated through illustrative examples.