We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order , where . We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute. 1. Introduction The study of fractional differential equations has acquired popularity in the last few decades due to its multiple applications, see [1–5] for more information. However, it was not until recently that a study on the existence of solutions by using upper and lower solutions, which is well established for ordinary differential equations in [6], has been done for fractional differential equations. See [3, 7–16] for recent work. In this paper we recall a comparison theorem from [3] for a Caputo fractional differential equation of order , , with initial condition. We will use coupled lower and upper solutions combined with a generalized monotone method of initial value problems to prove the existence of coupled minimal and maximal periodic solutions. The results developed provide natural sequences and intertwined sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. Instead of the usual approach as in [6, 17] where the iterates are solutions of linear periodic boundary value problems we have used a generalized monotone method of initial value problems. This idea was presented in [18] for integrodifferential equations. The advantage of this method, compared to what was developed in [3, 15], is that it avoids computing the solution of the linear periodic boundary value problem using the Mittag-Leffler function at every step of the iterates. We also modify the comparison theorem which does not require the H？lder continuity condition as in [3]. 2. Preliminary Definitions and Comparison Results In this section we state some definitions and recall some results for a Caputo initial value problem which we need in our main results. Consider the initial value problem of the form: Here, is the Caputo derivative of order for , which is defined in [1–3, 5] as In this paper we will denote . The relation between the Caputo fractional derivative and the Riemann-Liouville fractional derivative, , is given by Throughout this paper we consider the Caputo derivative of order , for and . We start by showing some comparison results relative to
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