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Zbl 1208.45004
Wang, Rong-Nian; Chen, De-Han
On a class of retarded integro-differential equations with nonlocal initial conditions.
(English)
[J] Comput. Math. Appl. 59, No. 12, 3700-3709 (2010). ISSN 0898-1221

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: $$u'(t) - \int_0^t \frac{(t-s)^{\mu -2}}{\Gamma(\mu -1)} Au(s)ds = F(t,u(t),u(\kappa(t))),\quad t\geq 0;$$ $$u(t) + H_t(u) = \phi(t),\quad-\tau \le t \le 0.$$ 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.
[Iulian Stoleriu (Iaşi)]
MSC 2000:
*45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals (real functions)
45G10 Nonsingular nonlinear integral equations

Keywords: solution operator; time delays; nonlocal initial conditions; Cauchy problem; fractional integro-differential equation; Banach space; Riemann-Liouville fractional integral; global solution

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