## Global attractivity for some classes of Riemann-Liouville fractional differential systems.(English)Zbl 1431.34013

The authors study fractional differential equations with boundary condition given as $D^{\alpha}_{0+}x(t)= f(t,x(t)), t\in [0,+\infty), \eqno (1)$ $\lim_{t\rightarrow 0+} t^{1-\alpha}x(t)=x_{0}, \eqno (2)$ where $$\alpha\in(0,1), D^{\alpha}_{0+}$$ is the Riemann-Liouville fractional derivative, $$f: [0,+\infty)\times \mathbb{R}^{s}\rightarrow \mathbb{R}^{s}$$ satisfies a Lipschitz type condition. In order to find solutions of the problem (1) - (2) the authors construct a special function space $$C_{1-\alpha}([0,+\infty); \mathbb{R}^{s}).$$ To solve the problem (1) - (2) the authors use Banach’s theorem to prove the existence and uniqueness of a solution of the equivalent integral equation. Then the authors consider the differential equation $D^{\alpha}_{0+}x(t)= Ax(t)+Q(t)x(t)+g(t), t\in [0,+\infty), \eqno (3)$ where $$A\in\mathbb{R}^{s\times s}$$ and $$Q:[0,+\infty)\rightarrow \mathbb{R}^{s\times s}, g:[0,+\infty)\rightarrow \mathbb{R}^{s}$$ are continuous. The authors prove the globally attractivity of equation (3) in the sense of convergence to zero of solutions at infinity.

### MSC:

 34A08 Fractional ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

  F. Chen, J.J. Nieto and Y. Zhou, Global attractivity for nonlinear fractional differential systems, Nonlinear Anal. Real World Appl. 13 (2012), 287-298. · Zbl 1238.34011  N.D. Cong, T.S. Doan, S. Siegmund and H.T. Tuan, Linearized asymptotic stability for fractional differential systems, Electron. J. Qual. Theory Differ. Equ. (2016), paper no. 39. · Zbl 1363.34004  M.A. Al-Bassam, Some existence theorems on differential systems of generalized order, J. Reine Angew. Math. 218 (1965), 70-78.  D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625. · Zbl 0881.34005  K. Diethelm, The analysis of fractional differential systems: an application-oriented exposition using differential operators of Caputo type, Lecture Notes in Mathematics, 2004, Springer, Berlin, 2010. · Zbl 1215.34001  D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in $$\mathbb{R}^n$$, Fract. Calc. Appl. Anal., 14 (2011), no. 4, 538-553. · Zbl 1273.34010  N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta 45 (2006), no. 5, 765-771.  A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations, North-Holland Math. Studies 204, Elsevier, Amsterdam, 2006.  C. Kou, H. Zhou and Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential systems on the half-axis, Nonlinear Anal. 74 (2011), 5975-5986. · Zbl 1235.34022  K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993. · Zbl 0789.26002  Juan J. Nieto, Maximum principles for fractional differential systems derived from Mittag-Leffler functions, = Appl. Math. Lett., 23 (2010), 1248-1251. · Zbl 1202.34019  E. Pitcher and W.E. Sewell. Existence theorems for solutions of differential systems of non-integral order, Bull. Amer. Math. Soc. 44 (1938), 100-107. · JFM 64.0420.02  I. Podlubny, Fractional differential systems: an introduction to fractional derivatives, fractional differential systems, to methods of their solution and some of their applications, Math. Sci. Engin. 198. Academic Press, San Diego, 1999. · Zbl 0924.34008  D. Qian, C. Li, R.P. Agarwal and P.J.Y. Wong, Stability analysis of fractional differential systems with Riemann-Liouville derivative, Math. Comput. Modelling 52 (2010), 862-874. · Zbl 1202.34020  Z. Qin, R. Wu and Y. Lu, Stability analysis of fractional-order systems with the Riemann-Liouville derivative, Systems Sci. Control Engin, 2 (2014), 727-731.  S.G. Samko, A.A. Kilbas and O.I. Marichev. Integrals and derivatives of the fractional order and some of their applications, Gordon and Breach, Amsterdam (1993). · Zbl 0818.26003  H.T. Tuan, {On some special properties of Mittag-Leffler functions (2017), arXiv:1708.02277.}  T. Trif. Existence of solutions to initial value problems for nonlinear fractional differential systems on the semi-axis · Zbl 1312.34026  J.J. Trujillo and M. Rivero, An extension of Picard- Lindelöff theorem to fractional differential systems, Appl. Anal. 70 (1999), 347-361.  Y. Zhou, Attractivity for fractional differential systems in Banach space, Appl. Math. Lett. 75 (2018), 1-6. · Zbl 1380.34025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.