Existence Theory for a Langevin Fractional q-Difference Equations in Banach Space

In this work, we establish an existence theorem of solutions for a new class of nonlinear Langevin fractional q-difference equation involving Caputo q-derivative in Banach space. Indeed, we will introduce the notion of kuratowski measure of noncompactness and the Mo<spacing diaeresis>nch's fixed-point theorem, on which; our analysis of the problem will essentially be based. An example is provided to show the applicability of the main result.