Analysis of positive solutions for the fractional derivative with delay and integral boundary conditions

Abstract

The goal of the manuscript is to investigate the existence of positive solutions of the nonlinear fractional differential equation with delay and integral boundary conditions. The system is governed by the Riemann-Liouville fractional derivative of order 2 < α ≤ 3 and includes a nonlinear term influenced by both the current state and a solution-dependent delay. To establish the existence of a positive solution, we transform the fractional differential equation into an equivalent integral equation and apply the Guo-Krasnoselskii fixed-point theorem in a cone. Additionally, a practical example is provided to validate the theoretical results and demonstrate their real-world relevance.