In this paper we study the existence of solutions for the following differential equations by using a fixed point theorems \begin{align*} \left\{\begin{array}{ll} D^{\mu}_{c}w(\varsigma)\pm D^{\nu}_{c}w(\varsigma)=h(\varsigma,w(\varsigma)),\quad\varsigma\in J,~~0<\nu<\mu<1,\\ w(0)=w_0,~~~~~~~~~~~~~~~~ \end{array} \right. \end{align*} where $D^{\mu}$, $D^{\nu}$ are the Caputo derivatives of order $\mu$, $\nu$ (respectively)\\ and $h:J\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous. The results are well demonstrated with the aid of exciting examples.
