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1

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

Helal M., Ouahab A.

Opuscula Mathematica

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2012

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Vol. 32, no. 2

249-283

EN

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.

2

On the principle of uniform boundedness for LSC convex processes in a strictly N locally convex spaces

Lahrech S., Jaddar A., Hlal J., Ouahab A., Mbarki A.

Demonstratio Mathematica

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2008

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Vol. 41, nr 1

159-163

EN

In this paper we establish the principle of uniform boundedness for LSC convex processes in some class of locally convex spaces (strictly N locally convex spaces). Thus, we generalize the same result established by S. Lahrech in [1] for sequentially continuous linear operators.

3

Existence and uniqueness of solution for initial value problem of first order differential equation involving generalized Lebesgue-Bochner spaces Lp(I,(Xv, II.II))

Lahrech S., Ouahab A., Benbrik A., Mbarki A.

Demonstratio Mathematica

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2007

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Vol. 40, nr 2

303-315

EN

We consider a bitopological vector space (X, v, II.II), where (X, v) is a topological vector space, and II.II is a norm defined on X. This paper deals with the existence and uniqueness of solution for initial value problem of first differential equation: (P)( ˙ x(t) = f(t), t is an element of]alpha, beta[ x(alpha) = x1, where the vector valued function f:]alpha,beta[-› X is assumed to be not necessarily in the classical Lebesgue-Bochner space L1(]alpha,beta[, (X, II.II). Here, by the solution of problem (P), we mean a vector valued function x acting from ]alpha,beta[ into X satisfying the conditions: 1) x is absolutely continuous with respect to the norm II.II; 2) x is almost everywhere differentiable on ]alpha,beta[ with respect to the topology v; 3) ˙ x = f(t) almost everywhere on ]alpha,beta[; 4) x(alpha) = x1. For this, we introduce a special class of integrable functions called generalized Lebesgue- Bochner space denoted L1(]alpha,beta[, (Xv, II.II)) containing (in general, strictly containing, [see the example given at the end of the paper]) the classical Lebesgue-Bochner space L1(]alpha,beta[, (X, II.II). Thus, under some conditions on the pair of topologies (v, II.II) , we prove that if f is an element of L1(]alpha,beta[, (Xv,II.II)), then the initial value problem (P) has an unique solution in the above mentioned sense. Finally, we give an example to illustrate the result given in this paper.