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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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