This paper extend recent works on \(\mathbb{BVP}\) s regarding the existence of solutions for nonlinear fractional differential equations and inclusions involving fractional \(\mathtt{q}\) -derivative of order in \((1,2)\) with Atangana-Baleanu type based on the fixed point theorem of Schauder. We specify a relation between the fractional \(\mathtt{q}\) -differential equations, \(\mathtt{q}\) -differential inclusions and the corresponding fractional integral equation in Banach spaces under infinite dimension, and characterize the solution function form them. Also, some generalized achievements about the existence and uniqueness of the solution for \(\mathtt{q}\) -differential equations and \(\mathtt{q}\) -differential inclusions are given. Further, when we consider fractional \(\mathtt{q}\) -differential equations and \(\mathtt{q}\) -differential inclusions under impulsive effects, we establish sufficiency conditions for the existence of the solution and also demonstrate the existence of anti-periodic solution for the fractional \(\mathtt{q}\) -differential equations and \(\mathtt{q}\) -differential inclusions. At the end, before conclusion, some examples illustrate to verify our all theoretical achievements.
