Positive solutions for nonlinear fractional differential equations

Abstract

We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation

$$\begin{aligned} \left\{ \begin{array}{l} ^{C}D^{\alpha }x\left( t\right) =f(t,x(t))+^{C}D^{\alpha -1}g\left( t,x\left( t\right) \right) ,\ 0<t\le T,\\ x\left( 0\right) =\theta _{1}>0,\ x^{\prime }\left( 0\right) =\theta _{2}>0, \end{array} \right. \end{aligned}$$

where $1<\alpha \le 2$. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mapping and employ Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. The results obtained here extend the work of Matar (AMUC 84(1):51–57, 2015 [7]). Finally, an example is given to illustrate our results.