This work consists of three sections. In the first section we state the definition and some properties of the Caputo fractional derivative. In the second section, we state the initial value problem (IVP) that we will be studying and define different sets of coupled lower and upper solutions of a Caputo fractional differential equation with initial condition where the forcing function is the sum of an increasing function and a decreasing function. Then, we develop two monotone iterative techniques, corresponding to two different sets of lower and upper solutions, in which we prove the existence of sequences that converge uniformly and monotonically to minimal and maximal solutions of the IVP. In the final section we present an example that illustrates our results.