This paper, we investigat the existence and uniqueness of solutions for a initial value problem for impulsive fractional q-difference equations involving the Caputo fractional q-difference derivative. The results are obtained using the Banach contraction principle and Krasnoselskii’s fixed point theorems. Also, we discuss the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of its solutions. Finally, an example is given to illustrate the effectiveness of our results. AMS (MOS) Subject Classifications : 26A33, 39A13, 47H10.

The authors investigate the existence of solutions to a class of boundary value problems for fractional q-difference equations in a Banach space that involves a q-derivative of the Caputo type and nonlinear integral boundary conditions. Their result is based on Mönch’s fixed point theorem and the technique of measures of noncompactness. This approach has proved to be an interesting and useful approach to studying such problems. Some basic concepts from the fractional q-calculus are introduced, including q-derivatives and q-integrals. An example of the main result is included as well as some suggestions for future research.