Impulsively hybrid fractional quantum Langevin equation with boundary conditions involving Caputo qk-fractional derivatives

We obtain some existence and uniqueness results for an impulsively hybrid fractional quantum Langevin (qk-difference) equation involving a new qk-shifting operator aΦqk(m)=qkm+(1−qk)a and supplemented with non-separated boundary conditions containing Caputo qk-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.