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A class of fractional evolution equations and optimal controls. (English) Zbl 1214.34010
The authors, using the techniques of fractional calculus, a singular version of Gronwall’s inequality and the Leray-Schauder fixed point theorem for compact maps, study the existence of solutions of a fractional evolution equation of the type
$D^qx(t)=-Ax(t)+f(t,x(t)),\qquad t \in J=[0,T],~q \in (0,1),$
$x(t_0)=x_0.$
They introduce a notion of $$\alpha$$-mild solution which is associated with a probability density function and a semigroup operator. Further, the existence of an optimal control for a Lagrange problem is proved. An example is given to demonstrate their results.

##### MSC:
 34A08 Fractional ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 49J27 Existence theories for problems in abstract spaces
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