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On a class of retarded integro-differential equations with nonlocal initial conditions. (English) Zbl 1208.45004

The paper deals with the local existence and uniqueness of a mild solution for the Cauchy problem formed by a fractional integro-differential equation with time-delay and a nonlocal initial condition:

u ' (t)- 0 t (t-s) μ-2 Γ(μ-1)Au(s)ds=F(t,u(t),u(κ(t))),t0;
u(t)+H t (u)=ϕ(t),-τt0·

Here, 1<μ<2, τ>0, A:D(A)XX is a generator of a solution operator on a complex Banach space X, κ:[0,)[-τ,) is a function representing the delay, H t :[-τ,0]×𝒞([-τ,0],X)X is an operator. The convolution integral in the equation is of the Riemann-Liouville fractional integral type. The existence of a global solution is also proven here. An illustrative example is presented at the end of the paper.

MSC:
45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
45G10Nonsingular nonlinear integral equations
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