TY  - JOUR
ID  - Helal2012
LB  - Helal2012
AU  - Helal, Mohmed
AU  - Ouahab, Abdelghani
TI  - Existence and solution sets of impulsive functional differential inclusions with multiple delay
ST  - Existence and solution sets of impulsive functional differential inclusions with multiple delay
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2012
VL  - 32
IS  - 2
SP  - 249
EP  - 283
DO  - http://dx.doi.org/10.7494/OpMath.2012.32.2.249
UR  - http://dx.doi.org/10.7494/OpMath.2012.32.2.249
AB  - In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: \[ \begin{cases}\frac{d}{dt}[y(t)-g(t,y_t)] \in F(t,y_t) + \sum_{i=1}^{n_*} y(t-Ti), & a.e.\, t \in J\setminus\{t_1,...,t_m\} \\ y(t_k^+)-y(t_k^-)=I_k(y(t_k^-)), & k=1,...,m, \\ y(t)=\phi(t), & t \in [-r,0],\end{cases} \] where \(J:=[0,b]\) and \(0=t_0\lt t_1 \lt ...\lt t_m\lt t_{m+1}=b\) (\(m \in \mathbb{N}^*\)), \(F\) is a set-valued map and \(g\) is single map. The functions \(I_k\) characterize the jump of the solutions at impulse points \(t_k\) (\(k=1,...,m\)). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuousdependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.
KW  - impulsive functional differential inclusions
KW  - decomposable set
KW  - parameter differential inclusions
KW  - AR-set
KW  - control theory
SN  - 1232-9274
ER  -