Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system D2y(t)+z(t)=f(t,y),∫02ϕ1(γ)Dγy(t)dγ=∫02ϕ2(γ)Dγz(t)dγ,t>0, where Dγ,γ∈[0,2]Dγ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation ∑i=1kaiDγiz=g, γi∈R,i=1,…,k in S+′ is given.