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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:

${D}^{\alpha }x\left(t\right)=Ax\left(t\right)+f\left(t,{x}_{t},Bx\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right],$

$x\left(t\right)=\phi \left(t\right),$ $t\in \right]-\infty ,0\left[$ with $T>0$ and $0<\alpha <1$. We prove the existence (and uniqueness) of solutions, assuming that $A$ generates an $\alpha$-resolvent family $\left({S}_{\alpha }\left(t\right)\right)t⩾0$ on a complex Banach space $𝕏$ 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
##### References:
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