Soumia Bourchia,b, Fahd Jaradc, Yassine Adjabia,d, Thabet Abdeljawade, Ibrahim Mahariqf
a Department of Mathematics, Faculty of Sciences, University of M’Hamed Bougara, Boumerdes, Algeria.
b Dynamic of Engines and Vibroacoustic Laboratory, University of M’Hamed Bougara of Boumerdes.
c Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, Ankara, 06790, Turkey.
d Dynamic Systems Laboratory, Faculty of Mathematics, U.S.T.H.B., Algeria.
e Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, Riyadh, 11586, KSA.
f College of Engineering and Technology, American University of the Middle East, Kuwait.
In this paper, we consider a class of abstract Cauchy problems in the framework of a generalized Caputo-type fractional derivative. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators, and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results.
Keywords: Caputo-type generalized fractional operators, abstract Cauchy problem, existence, uniqueness, continuous dependence on parameters, stability, uniformly continuous semigroups, fixed point theorems, Mittag-Leffler type function.
Bourchi, S., Jarad, F., Adjabi, Y., Abdeljawad, T., & Mahariq, I. (2023). On abstract Cauchy problems in the frame of a generalized Caputo type derivative. Advances in the Theory of Nonlinear Analysis and Its Application, 7(1), 1–28.