No abstract

a b s t r a c tOf concern is the following Cauchy problem for fractional integro-differential equations with time delay and nonlocal initial conditiongenerator of a solution operator on a complex Banach space X , the convolution integral in the equation is known as the Riemann-Liouville fractional integral, κ(t) : [0, +∞) → [−τ , +∞) representing the delay property, is a function, and H t is an operator defined from [−τ , 0] × C([−τ , 0], X ) into X for some T > 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.

In the present paper, we deal with the Cauchy problems of abstract fractional integro-differential equations involving nonlocal initial conditions in a-norm, where the operator A in the linear part is the generator of a compact analytic semigroup. New criterions, ensuring the existence of mild solutions, are established. The results are obtained by using the theory of operator families associated with the function of Wright type and the semigroup generated by A, Krasnoselkii's fixed point theorem and Schauder's fixed point theorem. An application to a fractional partial integrodifferential equation with nonlocal initial condition is also considered. Mathematics subject classification (2000) 26A33, 34G10, 34G20

We study the local and global existence of mild solutions to a class of semilinear fractional Cauchy problems in the α-norm assuming that the operator in the linear part is the generator of a compact analytic C0-semigroup. A suitable notion of mild solution for this class of problems is also introduced. The results obtained are a generalization and continuation of some recent results on this issue.