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Some existence results for fractional functional differential equations. (English) Zbl 1202.34141
This paper deals with the local and global existence of the following Cauchy problem in a separable Banach space \(E\):
\[ D^\alpha y(t)=Ay(t)+f(t,y_t),\quad t\in [0,T];\;y(t)=\widetilde{\psi}(t),\;-\tau\leq \theta\leq 0, \]
where \(D^\alpha,\,0<\alpha<1\) stands for the Riemann-Liouville fractional derivative and \(A: D(A)\subset E\to E\) is a linear closed (not necessarily bounded) operator generating an immediately norm-continuous semigroup \(\{e^{At}\}_{t\geq 0}\), \(\widetilde{\psi}\in C=C([0,T];E)\) with \(\widetilde{\psi}(0)=0\). For \(t\in [0,T]\), the function \(y_t\in C=C([0,T];E)\) is defined as \(y_t(\theta)=y(t+\theta)\), \(-\tau\leq \theta\leq 0.\) The results are obtained by means of condensing maps theory, assuming that \(f:[0,T] \times C\to E\) is continuous such that \(f\) satisfies the Ambrosetti-Sadovskii regularity condition expressed in terms of the measures of non-compactness.

34K37 Functional-differential equations with fractional derivatives
34K30 Functional-differential equations in abstract spaces
34K40 Neutral functional-differential equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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