Opuscula Math. 36, no. 2 (2016), 189-206
http://dx.doi.org/10.7494/OpMath.2016.36.2.189

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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• Zachary Denton
• Department of Mathematics, North Carolina A&T State University, Greensboro, NC, 27411, USA
• Communicated by Marek Galewski.
• Revised: 2015-09-26.
• Accepted: 2015-09-26.
• Published online: 2015-12-18.