an:05711224
Zbl 1202.34141
Obukhovskii, Valeri; Yao, Jen-Chih
Some existence results for fractional functional differential equations
EN
Fixed Point Theory 11, No. 1, 85-96 (2010).
00261264
2010
j
34K37 34K30 34K40 47H09 47H10
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
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.
Gis??le M. Mophou (Pointe-??-Pitre)