Abstract
We obtain some existence and uniqueness results for an impulsively hybrid fractional quantum Langevin (q(k)-difference) equation involving a new q(k)-shifting operator (a)Phi(qk)(m) = q(k)m +(1- qk)a and supplemented with non-separated boundary conditions containing Caputo q(k)-fractional derivatives. Our first result, relying on Banach's fixed point theorem, is concerned with the existence of a unique solution of the problem. The existence results are established by means of Leray Schauder nonlinear alternative and a fixed point theorem due to O'Regan. We construct some examples for the applicability of the obtained results. The paper concludes with interesting observations. (C) 2016 Elsevier Ltd. All rights reserved.