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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.

##### MSC:
 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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