Existence and Ulam stability of initial value problem for fractional perturbed functionalq-difference equations

Abstract. In this work, we discuss the existence and uniqueness of solutions to the initial value problem for perturbed functional fractional q-difference equations involving q-derivative of the Caputo sense. By applying Banach contraction principle and Burton and Kirk’s fixed point theorems. Further, we present the Ulam-Hyers and Ulam-Hyers-Rassias stabilities results by using direct analysis methods. Finally, we give two examples illustrating of the results. Mathematics Subject Classification (2010): 26A33, 34A12, 39A13, 47H10. Keywords: Initial value problem, Caputo fractional q-derivative, Burton and Kirk’s fixed point theorem, Ulam-Hyers stability, Ulam-Hyers-Rassias stability.