This paper aims to investigate the Caputo fractional differential equation involving ρ(τ) Laplacian operator and nonlocal multi-point of Riemann-Lioville’s fractional integral. We also prove the uniqueness of positive solutions for Boyd and Wong’s non-linear contraction via the Guo-Krasnoselskii fixed point theorems in Banach spaces. Finally, we illustrate the theoretical results and show that by solving the nonlocal problems is possible to obtain accurate approximations of the solutions.