This research paper focuses on investigating the solvability of the qualitative study for a coupled system of differential equation on the real half-line by applying Darboe's fixed point Theorem and the technique of the measure of noncompactness (MNC). This study has been located in space $BC(R_+)$. Furthermore, we prove the asymptotic stability of the solution of our problem, we introduce the idea of dependency of the solutions on some data. Additionally, we delve into the study of Hyers-Ulam stability. Finally, we present an example to support our findings.
The study of differential equations has received much attention over the last $30$ years or so. For papers studying such
kind of problems (see \cite{15,16,17,18}) and the references therein.\\
It is known that the nonlinear initial value problems create an important branch of nonlinear analysis and have numerous applications in describing of
miscellaneous real world problems. Such kind of these equations have been considered in numerous papers see \cite{31} and references therein.\\
The technique associated with MNC in the Banach space $BC(R_+)$ (of all bounded and continuous functions on $R_+$) have been successfully used by J. Bana\'s (see \cite{31,re178,re179}) to prove the existence of asymptotically stable solutions for some functional equation (see \cite{21,re147}).\\