The solution theory of the nonlinear q-fractional differential equations

In this paper, we study the solution theory of the nonlinear q-fractional differential equation: cD(q)(alpha)x(t)= f (t, x(t)), 0 < alpha, q < 1, with given initial value. We first give the existence theorem under the assumption that t(beta) f(t, x) is continuous where 0 <= beta < alpha. Then, by establishing a q-analogue Gronwall inequality, we prove that the solution x(t) is stable with respect to the initial value if f(t, x) satisfies the Lipschitz condition on variable x. This stability result also implies the uniqueness of solution. (C) 2020 Elsevier Ltd. All rights reserved.