Opuscula Math. 32, no. 2 (2012), 249-283
http://dx.doi.org/10.7494/OpMath.2012.32.2.249

Opuscula Mathematica

# Existence and solution sets of impulsive functional differential inclusions with multiple delay

Mohmed Helal
Abdelghani Ouahab

Abstract. 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.

Keywords: impulsive functional differential inclusions, decomposable set, parameter differential inclusions, AR-set, control theory.

Mathematics Subject Classification: 34K45, 34A60, 54C55, 54C60.

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• Mohmed Helal
• Sidi-Bel-Abbes University, Department of Mathematics, P.B. 89, 22000. Sidi-Bel-Abbes, Algeria
• Abdelghani Ouahab
• Sidi-Bel-Abbes University, Department of Mathematics, P.B. 89, 22000. Sidi-Bel-Abbes, Algeria
• Revised: 2011-05-04.
• Accepted: 2011-05-08.