**Shuman Meng <sup>1</sup> and Yujun Cui 2,\***


Received: 19 January 2019; Accepted: 14 February 2019; Published: 16 February 2019

**Abstract:** In this article, by using the monotone iterative technique coupled with the method of upper and lower solution, we obtain the existence of extremal iteration solutions to conformable fractional differential equations involving Riemann-Stieltjes integral boundary conditions. At the same time, the comparison principle of solving such problems is investigated. Finally, an example is given to illustrate our main results. It should be noted that the conformal fractional derivative is essentially a modified version of the first-order derivative. Our results show that such known results can be translated and stated in the setting of the so-called conformal fractional derivative.

**Keywords:** fractional differential equations; Riemann-stieltjes integral; monotone iterative method; upper and lower solutions
