In this paper, using properties of Mittag–Leffler functions, a weighted norm, and the Banach fixed point theorem, we establish a rigorous theorem on the existence and uniqueness of global solutions to delay fractional differential equations under a mild Lipschitz condition. Then, we provide a sufficient condition which guarantees these solutions to be exponentially bounded. Our theorems fill the gaps and also strengthen results in some existing papers.
Fractional differential equations delay differential equations with fractional derivatives existence and uniqueness growth and boundedness
Mathematics Subject Classification
26A33 34A08 34A12 34K12
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Bechohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008)MathSciNetCrossRefMATHGoogle Scholar
Cermak, J., Hornicek, J., Kisela, T.: Stability regions for fractional differential systems with a time delay. Commun. Nonlinear Sci. Numer. Simulat. 31, 108–123 (2016)MathSciNetCrossRefGoogle Scholar
Yang, Z., Cao, J.: Initial value problems for arbitrary order fractional equations with delay. Commun. Nonlinear Sci. Numer. Simul. 18, 2993–3005 (2013)MathSciNetCrossRefMATHGoogle Scholar
Wang, F., Chen, D., Zhang, X., Wu, Y.: The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay. Appl. Math. Lett. 53, 45–51 (2016)MathSciNetCrossRefMATHGoogle Scholar
Bhalekar, S.B.: Stability analysis of a class of fractional delay differential equations. Pramana J. Phys. 81(2), 215–224 (2013)CrossRefGoogle Scholar
Diethelm, K.: The Analysis of Fractional Differential Equations. An Application—Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)Google Scholar
Tisdell, C.: On the application of sequential and fixed-point methods to fractional differential equations of arbitrary order. J. Integral Equ. Appl. 24(2), 283–319 (2012)MathSciNetCrossRefMATHGoogle Scholar