Positivity results on the solutions for nonlinear two-term boundary value problem involving the $\psi$-Caputo fractional derivative

In this work, we investigated a nonlinear two-term boundary value problem involving the $\psi$-Caputo fractional derivative with integral boundary conditions. By a construction of its associated Green function and application of the upper and lower solutions method together with some fixed point theorems due to Banach and Schauder, we established the existence and uniqueness of positive solutions of our considered main problem. At the end some illustrative examples are provided to validate our theoretical results.