A solution of the fractional differential equations in the setting of $b$-metric space

Abstract

In this paper, we study the existence of solutions for the following differential equations by using a fixed point theorems $$\begin{cases} D^{\mu}_{c}w(\varsigma)\pm D^{\nu}_{c}w(\varsigma)=h(\varsigma,w(\varsigma)),& \varsigma\in J,\ \ 0<\nu<\mu<1,\\ w(0)=w_0,& \ \end{cases}$$ where $D^{\mu}$, $D^{\nu}$ is the Caputo derivative 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.