Journal of Integral Equations and Applications

Global attractivity for some classes of Riemann-Liouville fractional differential systems

H.T. Tuan, Adam Czornik, Juan J. Nieto, and Michał Niezabitowski

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Abstract

We present results for existence of global solutions and attractivity for multidimensional fractional differential equations involving Riemann-Liouville derivative. First, by using a Bielecki type norm and the Banach-fixed point theorem, we prove a Picard-Lindelof-type theorem on the existence and uniqueness of solutions. Then, applying the properties of Mittag-Leffler functions, we describe the attractivity of solutions to some classes of Riemann-Liouville linear fractional differential systems.

Article information

Source
J. Integral Equations Applications, Volume 31, Number 2 (2019), 265-282.

Dates
First available in Project Euclid: 23 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1569225676

Digital Object Identifier
doi:10.1216/JIE-2019-31-2-265

Mathematical Reviews number (MathSciNet)
MR4010587

Subjects
Primary: 34A08: Fractional differential equations
Secondary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 34A30: Linear equations and systems, general 34D05: Asymptotic properties

Keywords
Fractional differential equation Riemann-Liouville derivative asymptotic behaviour of solutions existence and uniqueness

Citation

Tuan, H.T.; Czornik, Adam; Nieto, Juan J.; Niezabitowski, Michał. Global attractivity for some classes of Riemann-Liouville fractional differential systems. J. Integral Equations Applications 31 (2019), no. 2, 265--282. doi:10.1216/JIE-2019-31-2-265. https://projecteuclid.org/euclid.jiea/1569225676


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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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • K.S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
  • Juan J. Nieto, Maximum principles for fractional differential systems derived from Mittag-Leffler functions, = Appl. Math. Lett., 23 (2010), 1248–1251.
  • 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.
  • 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.
  • 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.
  • 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).
  • 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}, JOURNAL = Fract. Calc. Appl. Anal. 16 (2013), no. 3, 595–612.
  • 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.