Existence of solutions of nonlocal Cauchy problem for some fractional abstract differential equation

In this paper, by using the fixed point theory and a new methods, we study the existence and uniqueness of solutions of nonlocal Cauchy problems for the fractional differential equation Dαx(t)=f(t,x(t)),t∈[0,T],x(0)+g(x)=x0Dαx(t)=f(t,x(t)),t∈[0,T],x(0)+g(x)=x0 in a Banach space and obtain two new results.