Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses

Abstract

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0, T ], for a class of higher‐order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.