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Stability for Caputo Fractional Differential Systems

DOI: 10.1155/2014/631419

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We introduce the notion of h-stability for fractional differential systems. Then we investigate the boundedness and h-stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give examples to illustrate our results. 1. Introductions and Preliminaries Lakshmikantham et al. [1–5] investigated the basic theory of initial value problems for fractional differential equations involving Riemann-Liouville differential operators of order . They followed the classical approach of the theory of differential equations of integer order in order to compare and contrast the differences as well as the intricacies that might result in development [6, Vol. I]. Li et al. [7] obtained some results about stability of solutions for fractional-order dynamic systems using fractional Lyapunov direct method and fractional comparison principle. Choi and Koo [8] improved on the monotone property of Lemma?? in [5] for the case with a nonnegative real number . Choi et al. [9] also investigated Mittag-Leffler stability of solutions of fractional differential equations by using the fractional comparison principle. In this paper we introduce the notion of -stability for fractional differential equations. Then, we investigate the boundedness and -stability of solutions of Caputo fractional differential systems by using fractional comparison principle and fractional Lyapunov direct method. Furthermore, we give some examples to illustrate our results. For the basic notions and theorems about fractional calculus, we mainly refer to some books [5, 10, 11]. We recall the notions of Mittag-Leffler functions which were originally introduced by Mittag-Leffler in 1903 [12]. Similar to the exponential function frequently used in the solutions of integer-order systems, a function frequently used in the solutions of fractional order systems is the Mittag-Leffler function, defined as where and is the Gamma function [11]. The Mittag-Leffler function with two parameters has the following form: where and . For , we have . Also, . Note that the exponential function possesses the semigroup property (i.e., for all ), but the Mittag-Leffler function does not satisfy the semigroup property unless or [13]. We recall briefly the notions and basic properties about fractional integral operators and fractional derivatives of functions [5, 10]. Let . Definition 1 (see [5]). The Riemann-Liouville fractional integral of order of a function is defined as where (provided that the integral exists in the Lebesgue

References

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