THE UNIQUE EXISTENCE OF SOLUTION IN THE q-INTEGRABLE SPACE FOR THE NONLINEAR q-FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

In this paper, we study the solution theory of the nonlinear [Formula: see text]-fractional differential equation of Caputo type [Formula: see text] with given initial values [Formula: see text] where [Formula: see text] is the order, [Formula: see text] and [Formula: see text] is the scale index. For [Formula: see text], by assuming that function [Formula: see text] is bounded and satisfies the Lipschitz condition on variable [Formula: see text], we prove that this problem admits a unique solution in the [Formula: see text]-integrable function space [Formula: see text] and this solution is absolutely stable in the [Formula: see text]-norm. This unique existence condition allows that [Formula: see text] is singular at [Formula: see text] and discontinuous for [Formula: see text]. Finally, a successive approximation method is presented to find out the analytic approximation solution of this problem.