Abstract
The intended goal of this manuscript is to discuss the existence of the solution to the below system of tripled-fractional differential equations (TFDEs, for short):
<disp-formula> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{c} \Theta ^{\mu }\left[ k(\alpha )-\gimel (\alpha ,k(\alpha ))\right] = \Game \left( \alpha ,r(\alpha ),I^{\tau }(r(\alpha ))\right) +\Game \left( \alpha ,l(\alpha ),I^{\tau }(l(\alpha ))\right) , \\ \Theta ^{\mu }\left[ l(\alpha )-\gimel (\alpha ,l(\alpha ))\right] = \Game \left( \alpha ,k(\alpha ),I^{\tau }(k(\alpha )\right) )+\Game \left( \alpha ,r(\alpha ),I^{\tau }(r(\alpha )\right) ), \\ \Theta ^{\mu }\left[ r(\alpha )-\gimel (\alpha ,r(\alpha ))\right] = \Game \left( \alpha ,l(\alpha ),I^{\tau }(l(\alpha )\right) )+\Game \left( \alpha ,k(\alpha ),I^{\tau }(k(\alpha )\right) ), \\ k(0) = 0,\text{ }l(0) = 0,\text{ }r(0) = 0,
\end{array}
\right. a.e.\text{ }\alpha \in \Omega ,\text{ }\tau >0,\text{ }\mu \in (0,1), $\end{document} </tex-math></disp-formula>
where $ \Theta ^{\mu } $ is RL-fractional derivative of order $ \tau, \; \Omega = [0, \Lambda ], \; \Lambda > 0, $ and $ \gimel :\Omega \times \mathbb{R}
\rightarrow \mathbb{R}, $ with $ \gimel (0, 0) = 0, \; \Game :\Omega \times \mathbb{R}
\times \mathbb{R}
\rightarrow \mathbb{R}
$ are functions taken under appropriate hypotheses. The method of the proof depends on a manner of a tripled fixed point (TFP), which generalize a fixed point theorem of Burton [<xref ref-type="bibr" rid="b1">1 ] . At last, a non-trivial example to strong our results is illustrated.