Piecewise conformable fractional impulsive differential system with delay: existence, uniqueness and Ulam stability

Ideally, the state variable follows a constant motion law over time. However, due to the finiteness of motion speed, almost all systems have time delay. In this paper, we investigate a new class of piecewise conformable fractional impulsive differential system with delay under two point inhomogeneous boundary condition. In the system, the motion laws of state variable vary at different time periods, and they interact with each other through time delay "tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}" and time leading "-tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\tau $$\end{document}", so as to be more realistic. By employing the well-know fixed point theorems, the sufficient conditions for the existence and uniqueness of solutions to the system are established. Under the conditions of ensuring the existence of the system's solutions, we conclude further that the system has Ulam-Hyers stability and Ulam-Hyers-Rassias stability by means of nonlinear functional analysis method. Finally, we give a feasible example to explain our result.