منیره نصرتی

‎In this study‎, using some new contractions‎, ‎we obtain an existence and uniqueness conclusion for a fractional differential equation with Atangana-Baleanu derivative as follows‎: ‎\begin{align}\label{202}‎ ‎&(_0^{ABC}D^{\xi}\delta)(s)=h(s,\delta(s)),~~~~~~~~~~~~~~~~‎~ ‎0\leq s,\xi\leq 1,~~~~~~~~~~~~~~~~\\‎ ‎\nonumber &\delta(0)=\delta_0,~~~~~~~~~~~~~~~~‎~ ‎\end{align}‎ ‎where $^{ABC}D^{\xi}$ is the Atangana-Baleanu derivative of order $\xi$ and‎ ‎$f$ is continuous with $f(0,\hslash(0))=0$‎.