In this investigation, by applying definition of the fractional $q$-derivative of the Caputo type and the fractional $q$-integral of the Riemann–Liouville type, we study the existence and uniqueness of solutions for a multi-term nonlinear fractional $q$-integro-differential equations under some boundary conditions ${}^cD_q^{\alpha} x(t) = w \big( t, x(t), (\varphi_1 x)(t), (\varphi_2 x)(t), {}^cD_q^{ \beta_1} x(t), {}^cD_q^{\beta_2} x(t), \ldots, {}^cD_q^{ \beta_n}x(t) \big)$. Our results are based on some classical fixed point techniques, as Schauder's fixed point theorem and Banach contraction mapping principle. Besides, some instances are exhibited to illustrate our results and are reported all algorithms required along with numerical result obtained.
