Tamtam Proceedings - lamsin

Neutral functional differential equations 123The following result is only the combination of Lemma 3 in [2] and Proposition 11 in[1] which are proved in a general framework. Precisely, here it suffices to take h(t) :=T 0 (t)Dϕ.Proposition 1. Assume that Condition (H1) is satisfied and ‖D 0 ‖ K(0) < 1. Then,for given ϕ ∈ Y there exists a unique function u which is continuous on [0, T ) andsolves Eq. (6) on (−∞, T ). Moreover, the family of operators (T (t)) t≥0 defined on Y byT (t)ϕ = u t (., ϕ) is a C 0 -semigroup on Y.Under the hypothesis that K is bounded on [0, +∞) , the estimate (8) combined with(H3) yield the following result.Proposition 2. Assume that Conditions (H1), (H2), (H3) and (H4) are satisfied, K isbounded on [0, +∞) and ‖D 0 ‖ K(0) < 1. Suppose that M(s) < 1 for all s ∈ [0, +∞) .Then, there exists a function γ(.) ∈ L ∞ ([0, +∞) , IR + ) and ω > 0 such that‖T (t)ϕ‖ B≤ γ(t)e −ωt ‖ϕ‖ Bfor all t ∈ [0, +∞) and ϕ ∈ Y. (9)REMARK. — The above proposition is one of the keys to stability, dissipativeness andexistence of a global attractor for Eq. (1). We intend to present the extended version ofthe paper.3. References[1] M. ADIMY, H. BOUZAHIR AND K. EZZINBI, Existence and stability for some partial neutralfunctional differential equations with infinite delay, J. Math. Anal. Appl., Vol. 294, Issue 2, June15, 438-461 (2004).[2] M. ADIMY, H. BOUZAHIR AND K. EZZINBI, Local existence for a class of partial neutralfunctional differential equations with infinite delay, Differential Equations and Dynam. Systems,Vol. 12, Nos. 3-4, July-October, 353-370, (2004).[3] M. ADIMY AND K. EZZINBI, Existence and stability for a class of partial neutral functionaldifferential equations, Hiroshima Math. J., to appear.[4] J. WU, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag,(1996).TAMTAM –Tunis– 2005