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

Helal M., Ouahab A.

Opuscula Mathematica

2012

Vol. 32, no. 2

249-283

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: [formula] where J : = [0; b] and 0 = t0 < t1 < ... < tm < tm+1 = b (m ∈ N*), F is a set-valued map and g is single map. The functions Ik characterize the jump of the solutions at impulse points tk (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 continuous dependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.

On Lipschitz selections of multifunctions with decomposable values

Rom C.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 2

121-127

Some conditions for existence of Lipschitz selections of multifunctions with decomposable values are given.