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Impulsively hybrid fractional quantum Langevin equation with boundary conditions involving Caputo \(q_k\)-fractional derivatives. (English) Zbl 1372.26008

Summary: 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 \(_a \Phi_{q_k}(m)=q_km+(1-q_k)a\) 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.

MSC:

26A33 Fractional derivatives and integrals
39A13 Difference equations, scaling (\(q\)-differences)
34A37 Ordinary differential equations with impulses
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