Analysis of Riesz–Caputo Type Fractional Thermostat Equation

We analyze a nonlinear fractional boundary value problem utilizing the symmetric Riesz–Caputo derivative, which captures global interval dependencies, coupled with nonlocal boundary conditions. Analysis of the derived Green’s function reveals a singularity complicating standard positivity proofs. General solution existence is established via Schaefer’s fixed-point theorem. Then, we develop conditions guaranteeing positive solutions using adapted cone-theoretic methods (Guo–Krasnoselskii). The Hyers–Ulam stability of the problem is also demonstrated. This theoretical analysis of fractional systems with symmetric operators and complex boundary effects is supported by numerical and visual illustrations.