Obukhovskii, Valeri; Yao, Jen-Chih Some existence results for fractional functional differential equations. (English) Zbl 1202.34141 Fixed Point Theory 11, No. 1, 85-96 (2010). 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. Reviewer: Gisèle M. Mophou (Pointe-à-Pitre) Cited in 11 Documents 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 Keywords:fractional derivative; fractional differential equation; functional differential equation; neutral functional differential equation; mild solution; Cauchy problem; existence theorem; measure of noncompactness; fixed point; condensing map PDF BibTeX XML Cite \textit{V. Obukhovskii} and \textit{J.-C. Yao}, Fixed Point Theory 11, No. 1, 85--96 (2010; Zbl 1202.34141) Full Text: Link