On a class of retarded integro-differential equations with nonlocal initial conditions

Of concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial condition {u′(t)−∫0t(t−s)μ−2Γ(μ−1)Au(s)ds=F(t,u(t),u(κ(t))),t≥0,u(t)+Ht(u)=ϕ(t),−τ≤t≤0, where 1<μ<21<μ<2, τ>0τ>0, A:D(A)⊂X→XA:D(A)⊂X→X is a generator of a solution operator on a complex Banach space XX, the convolution integral in the equation is known as the Riemann–Liouville fractional integral, κ(t):[0,+∞)→[−τ,+∞)κ(t):[0,+∞)→[−τ,+∞) representing the delay property, is a function, and HtHt is an operator defined from [−τ,0]×C([−τ,0],X)[−τ,0]×C([−τ,0],X) into XX for some T>0T>0 which constitutes a nonlocal condition. The local existence and uniqueness of mild solutions for the Cauchy problem, under various criteria, are proved. Moreover, we present an existence result of the global solution. Also an example is given to illustrate the applications of the abstract results.