Opuscula Math. 36, no. 2 (2016), 189-206

Opuscula Mathematica

Monotone method for Riemann-Liouville multi-order fractional differential systems

Zachary Denton

Abstract. In this paper we develop the monotone method for nonlinear multi-order \(N\)-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders \(q_i\) where \(0 \lt q_i \lt 1\). In the development of this method we recall any needed existence results along with any necessary changes. Through the method's development we construct a generalized multi-order Mittag-Leffler function that fulfills exponential-like properties for multi-order systems. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system.

Keywords: fractional differential systems, multi-order systems, lower and upper solutions, monotone method.

Mathematics Subject Classification: 34A08, 34A34, 34A45, 34A38.

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  1. M. Caputo, Linear models of dissipation whose {Q} is almost independent, II, Geophy. J. Roy. Astronom 13 (1967), 529-539.
  2. A. Chowdhury, C.I. Christov, Memory effects for the heat conductivity of random suspensions of spheres, Proc. R. Soc. A 466 (2010), 3253-3273.
  3. Z. Denton, Monotone method for multi-order 2-systems of Riemann-Liouville fractional differential equations, Communications in Applied Analysis 19 (2015), 353-368.
  4. Z. Denton, A.S. Vatsala, Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations, Opuscula Math. 31 (2011) 3, 327-339.
  5. K. Diethelm, A.D. Freed, On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, [in:] F. Keil, W. Mackens, H. Vob, J. Werther (eds), Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, pp. 217-224 Heidelberg, Springer, 1999.
  6. M. Galewski, G. Molica Bisci, Existence results for one-dimensional fractional equations, Math. Meth. Appl. Sci., to appear.
  7. W.G. Glöckle, T.F. Nonnenmacher, A fractional calculus approach to self similar protein dynamics, Biophy. J. 68 (1995), 46-53.
  8. S. Heidarkhani, Multiple solutions for a nonlinear perturbed fractional boundary value problem, Dynam. Sys. Appl. 23 (2014), 317-332.
  9. R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing, Germany, 2000.
  10. A.A. Kilbas, H.M. Srivastava, J.J Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North Holland, 2006.
  11. V. Kiryakova, Generalized fractional calculus and applications, vol. 301, Pitman Res. Notes Math. Ser. Longman-Wiley, New York, 1994.
  12. G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman Publishing Inc., 1985.
  13. V. Lakshmikantham, S. Leela, D.J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
  14. R. Metzler, W. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phy. 103 (1995), 7180-7186.
  15. G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Letters 27 (2014), 53-58.
  16. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York - London, 2002.
  17. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • Zachary Denton
  • Department of Mathematics, North Carolina A&T State University, Greensboro, NC, 27411, USA
  • Communicated by Marek Galewski.
  • Received: 2015-04-09.
  • Revised: 2015-09-26.
  • Accepted: 2015-09-26.
  • Published online: 2015-12-18.
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Cite this article as:
Zachary Denton, Monotone method for Riemann-Liouville multi-order fractional differential systems, Opuscula Math. 36, no. 2 (2016), 189-206, http://dx.doi.org/10.7494/OpMath.2016.36.2.189

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