Existence-uniqueness result for a nonlinear n-term fractional equation

We prove the existence-uniqueness of the solution to the nonlinear n-term time-fractional differential equation with constant coefficients in the Banach space C([0,T]),(1)b0Dβ0u(t)+∑i=1m−1biDβiu(t)+∑i=mn−1biDαiu(t)+bnDαnu(t)=f(t,u(t)),t∈(0,T),x∈R,u(0)=f(0),ut(0)=g(0),0<β1<β2<⋯<βm−1<1<αm<αm+1<⋯<αn<2(respectively 0<β1<⋯<βm−1<αm<αm+1<⋯<αn<1,1<β1<⋯<βm−1<αm<αm+1<⋯<αn<2), f(t,u(t))∈C([0,T)×C([0,T))) is a given function meant to be composed with a real valued function u and satisfies Assumption 1.