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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 $\Bbb X$ by means of classical fixed points methods.

34K37Functional-differential equations with fractional derivatives
34K40Neutral functional-differential equations
34K30Functional-differential equations in abstract spaces
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
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