TY - JOUR
T1 - Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality
AU - Abdeljawad, Thabet
AU - Al-Mdallal, Qasem M.
N1 - Funding Information:
The authors would like to express their appreciation for the valuable comments of the reviewers. The first author also would like to express his sincere appreciation to the Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17 .
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2018/9
Y1 - 2018/9
N2 - In this article, we studied the Caputo and Riemann–Liouville type discrete fractional difference initial value problems with discrete Mittag-Leffler kernels. The existence and uniqueness of the solution is proved by using Banach contraction principle. The linear type equations are used to prove new discrete fractional versions of the Gronwall's inequality. The nabla discrete Laplace transform is used to obtain solution representations. The proven Gronwall's inequality under a new defined α-Lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions. Numerical examples are discussed to demonstrate the reliability of the theoretical results.
AB - In this article, we studied the Caputo and Riemann–Liouville type discrete fractional difference initial value problems with discrete Mittag-Leffler kernels. The existence and uniqueness of the solution is proved by using Banach contraction principle. The linear type equations are used to prove new discrete fractional versions of the Gronwall's inequality. The nabla discrete Laplace transform is used to obtain solution representations. The proven Gronwall's inequality under a new defined α-Lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions. Numerical examples are discussed to demonstrate the reliability of the theoretical results.
KW - Discrete ABR and ABC fractional derivatives
KW - Discrete Laplace transform
KW - Discrete Mittag-Leffler function
KW - Discrete fractional sum
KW - Gronwall's inequality
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U2 - 10.1016/j.cam.2017.10.021
DO - 10.1016/j.cam.2017.10.021
M3 - Article
AN - SCOPUS:85040444957
SN - 0377-0427
VL - 339
SP - 218
EP - 230
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
ER -