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.