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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: \[ u'(t) - \int_0^t \frac{(t-s)^{\mu -2}}{\Gamma(\mu -1)} Au(s)ds = F(t,u(t),u(\kappa(t))),\quad t\geq 0; \]
\[ u(t) + H_t(u) = \phi(t),\quad-\tau \leq t \leq 0. \] Here, \(1< \mu < 2\), \(\tau >0\), \(A: D(A) \subset X \rightarrow X\) is a generator of a solution operator on a complex Banach space \(X\), \(\kappa: [0,\,\infty) \rightarrow [-\tau,\,\infty)\) is a function representing the delay, \(H_t: [-\tau,\,0] \times{\mathcal C}([-\tau,\,0],\,X)\rightarrow X\) 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:
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
45G10 Other nonlinear integral equations
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