Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 18, Issue 6, Pages 1399-1422Publisher
SPRINGERNATURE
DOI: 10.1515/fca-2015-0081
Keywords
fractional Halanay inequality; fractional functional differential equations; dissipativity; stability
Funding
- NSF of China [11271311, 11426178, 11301448]
- Research Foundation of Education Commission of Hunan Province of China [14A146]
- Scientific Research Program Funded by Shaanxi Provincial Education Department [2015JQ1029]
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This paper concerns the dissipativity and stability of the Caputo non-linear fractional functional differential equations (F-FDEs) with order 0 < alpha < 1. The fractional generalization of the Halanay-type inequality is proposed, which plays a central role in studies of stability and dissipativity of F-FDEs. Then the dissipativity and the absorbing set are derived under almost the same assumptions as the classical integer-order functional differential equations (FDEs). The asymptotic stability of F-FDEs are also proved under the one-sided Lipschitz conditions. Those extend the corresponding properties from integer-order FDEs to the Caputo fractional ones. The results can also be directly applied to some special cases of fractional nonlinear equations, such as the fractional delay differential equations (F-DDEs), fractional integro-differential equations (F-IDEs) and fractional delay integro-differential equations (F-DIDEs). The fractional Adams-Bashforth-Moulton algorithm is employed to simulate the F-FDEs, and several numerical examples are given to illustrate the theoretical results.
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