Received: 4 March 2018 DOI: 10.1002/mma.5292 RESEARCH ARTICLE Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions Kamal Shah 1 Arshad Ali 1 Samia Bushnaq 2 1 Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan 2 Department of Basic Sciences, Princess Sumaya University for Technology, Amman, Jordan Correspondence Samia Bushnaq, Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan. Email: s.bushnaq@psut.edu.jo Communicated by: D. Baleanu MSC Classification: 26A33; 34A08; 35B40 In this article, we deal with the existence and Hyers-Ulam stability of solution to a class of implicit fractional differential equations (FDEs), having some initial and impulsive conditions. Some adequate conditions for the required results are obtained by utilizing fixed point theory and nonlinear functional analysis. At the end, we provide an illustrative example to demonstrate the applications of our obtained results. KEYWORDS Caputo derivative, existence theory, Hyers-Ulam stability, impulsive condition 1 INTRODUCTION Fractional calculus is the extension of classical calculus. This concept is not a new one. In 1695, Leibnitz and Hospital's discussed it in a well-known correspondence through a letter. Later on, Lacroix (1819), Abel (1823), Fourier, Liouville (1832), Heaviside (1892), and Riemann (1953), etc, formalized this theory. For detail study, see previous studies. 1-3 Frac- tional calculus have some interesting characteristics over the classical calculus. For example, fractional-order derivative is a global operator, in which we have many choices of derivative to be considered, while the integer-order derivative is a local operator, in which we have just one option of derivative that can be considered. Moreover, fractional-order sys- tems are more stable as compare with integer-order systems. We give an example of two dynamical systems with initial condition v(0) as in the following example taken from Li et al. 4 d dt v(t)= t -1 ,  ∈(0, 1), (1) C 0   t v(t)= t -1 , 0 << 1,  ∈(0, 1). (2) Now, the exact solutions of (1) and (2) are t  + v(0) and Γ()t +-1 Γ(+) + v(0), respectively. We see that for any  ∈(0, 1), the integer-order system (1) is unstable. However, the system presented by (2) is stable for 0 < ≤ 1 - . Certain real-world problems can be described more accurately via fractional differential equations (FDEs). These equations have applications in modeling of video tape problems, blood alcohol level problems, world population growth problems, and in problems concerning fluid mechanics, fluids dynamics, electrodynamics, aerodynamics, signal processing, image processing, control systems and computer network systems, etc. See, for example, other studies. 5-10 Due to their multidimensional applications, FDEs have gained much consideration from the researchers. Fixed point theory for various classes of FDEs have been studied countably. See, for instance, literature. 11-17 Math Meth Appl Sci. 2018;1–15. wileyonlinelibrary.com/journal/mma © 2018 John Wiley & Sons, Ltd. 1