Some existence results for fractional functional differential equations. (English) Zbl 1202.34141

Obukhovskii, Valeri; Yao, Jen-Chih

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)

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