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Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. (English) Zbl 1191.34098
Summary: We deal with the mild solution for fractional semilinear differential equations with infinite delay:
$D^\alpha x(t) = Ax(t) +f(t,x_t , Bx(t)),\quad t \in [0,T],$
$$x(t)=\phi (t),$$ $$t \in ]-\infty ,0[$$ with $$T>0$$ and $$0<\alpha <1$$. We prove the existence (and uniqueness) of solutions, assuming that $$A$$ generates an $$\alpha$$-resolvent family $$(S_\alpha (t)){t\geqslant 0}$$ on a complex Banach space $$\mathbb X$$ by means of classical fixed points methods.

##### MSC:
 34K37 Functional-differential equations with fractional derivatives 34K40 Neutral functional-differential equations 34K30 Functional-differential equations in abstract spaces 47N20 Applications of operator theory to differential and integral equations
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