Existence of solutions for multi-point non-linear $q$-differential system of fractional orders

In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional $q$-differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach s fixed point theorem, Leray-Schauder s nonlinear alternative and Leray-Schauder s degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.