Analysis on a nonlinear fractional differential equations in a bounded domain [1,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[1,\mathcal {T}]$$\end{document}

In this manuscript, based on the most widespread fixed point theories in literature. The existence of solutions to the system of nonlinear fractional differential equations with Caputo Hadmard fractional operator in a bounded domain is verified by using Mönoch’s fixed point theorem, The stability of the coupled system is also investigated via Ulam-Hyer technique. Finally, an applied numerical example is presented to illustrate the theoretical results obtained.