Dissipativity and Stability Analysis for Fractional Functional Differential Equations

Abstract

Abstract This paper concerns the dissipativity and stability of the Caputo nonlinear fractional functional differential equations (F-FDEs) with order 0 < α < 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.