Nonlocal controllability of fractional measure evolution equation

In this paper, we consider the following kind of fractional evolution equation driven by measure with nonlocal conditions: {D0+αCx(t)=Ax(t)dt+(f(t,x(t))+Bu(t))dg(t),t∈(0,b],x(0)+p(x)=x0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} {}^{C} D_{0+}^{\alpha }x(t)=Ax(t)\,dt+ (f(t,x(t))+Bu(t) )\,dg(t), \quad t\in (0, b],\\ x(0)+p(x)=x_{0}. \end{cases} $$\end{document} The regulated proposition of fractional equation is obtained for the first time. By noncompact measure method and fixed point theorems, we obtain some sufficient conditions to ensure the existence and nonlocal controllability of mild solutions. Finally, an illustrative example is given to show practical usefulness of the analytical results.