GENERALIZED MONOTONE METHOD FOR CAPUTO FRACTIONAL DIFFERENTIAL SYSTEMS VIA COUPLED LOWER AND UPPER SOLUTIONS

Monotone method combined with the method of upper and lower solutions yields monotone sequences which converge uniformly and monotonically to minimal and maximal solutions of the nonlinear systems, when the forcing function is quasi monotone nondecreasing. In this paper we develop genearalized monotone method for N system of Caputo fractional differential equations when the forcing function is the sum of an increasing and decreasing functions. In generalized monotone method we use coupled upper and lower solutions and the method yields two monotone sequences which converge uniformly and monotonically to coupled minimal and maximal solutions. This method is applicable to the Lotka-Volterra equation with Caputo fractional derivative of order q when 0 < q ≤ 1. This provides an opportunity to provide better results or improve on the existing results with integer derivatives. Finally, under uniqueness condition we obtain the unique solution of the Caputo fractional differential system.