This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained. 1. Introduction Fractional differential equation can be extensively applied to various disciplines such as physics, mechanics, chemistry, and engineering, see [1–3]. Hence, in recent years, fractional differential equations have been of great interest and there have been many results on existence and uniqueness of the solution of FDE, see [4–8]. Especially, Diethelm and Ford [9] have gained existence, uniqueness, and structural stability of solution of the type of fractional differential equation where is a real number, denotes the Riemann-Liouville differential operator of order , and is the Taylor polynomial of order for the function at . Recently, Daftardar-Gejji and Jafari [10] have discussed the existence, uniqueness, and stability of solution of the system of nonlinear fractional differential equation where and denotes Caputo fractional derivative (see Definition 2.3). Delbosco and Rodino [11] have proved existence and uniqueness theorems for the nonlinear fractional equation where , is the Riemann-Liouville fractional derivative. Zhang [12] used the method of the upper and lower solution and cone fixed point theorem to obtain the existence and uniqueness of positive solution to (1.3). Yao [13] considered the existence of positive solution to (1.3) controlled by the power function employing Krasnosel’skii fixed point theorem of cone expansion-compression type. The existence of the local and global solution for (1.3) was obtained by Lakshmikantham and Vatsala [14] utilizing classical differential equation theorem. More recently, Zhang [15] shows the existence of positive solutions to the singular boundary value problem for fractional differential equation where is the Riemann-Liouville fractional derivative of order , . However, in the previous works, the nonlinear term has to satisfy the monotone or others control conditions. In fact, the fractional differential equations with nonmonotone function can respond better to impersonal law, so it is very important to weaken monotone condition. Considering this, in this paper, we mainly investigate the fractional differential Equation
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