Tamtam Proceedings - lamsin

120 Bouzahir1. IntroductionWe consider the following class of nonlinear partial neutral functional differentialequations with infinite delay{ ∂∂t Du t = ADu t + F (t, u t ), t ≥ 0,(1)x 0 = φ ∈ B,where A : D(A) ⊆ E → E is a linear operator on a Banach space (E, |.|), B is thephase space of functions mapping (−∞, 0] into E, which will be specified later, D is abounded linear operator from B into E defined by Dϕ = ϕ(0) − D 0 ϕ for any ϕ ∈ B,D 0 is a bounded linear operator from B into E and for each u : (−∞, b] → E, b > 0,and t ∈ [0, b], u t represents, as usual, the mapping defined from (−∞, 0] into E byu t (θ) = u(t + θ) for θ ∈ (−∞, 0] . F is an E-valued nonlinear continuous mapping onIR + × B.(B, ‖.‖ B) is a (semi)normed abstract linear space of functions mapping (−∞, 0] intoE, and satisfies the following fundamental axioms:(A) There exist a positive constant H and functions K(.), M(.) : IR + → IR + , withK continuous and M locally bounded, such that for any σ ∈ IR and a > 0, if x :(−∞, σ + a] → E, x σ ∈ B and x(.) is continuous on [σ, σ + a], then for every t in[σ, σ + a] the following conditions hold:(i) x t ∈ B,(ii) |x (t)| ≤ H ‖x t ‖ B, which is equivalent to(ii) ′ |ϕ(0)| ≤ H ||ϕ|| B , for every ϕ ∈ B ,(iii) ‖x t ‖ B≤ K(t − σ) supσ≤s≤t|x (s)| + M (t − σ) ‖x σ ‖ B.(A1) For the function x(.) in (A), t ↦→ x t is a B-valued continuous function for t in[σ, σ + a].(B) The space B is complete.Example 1. Define for a positive constant γ the following standard space{C γ := φ : (−∞, 0] → E continuous such that, limθ→−∞ eγθ φ(θ)exists in E}. (2)It is known that C γ with the norm ‖φ‖ γ= sup e γθ |φ(θ)| , φ ∈ C γ , satisfies the axiomsθ≤0(A), (A1) and (B) with H = 1, K(t) = max(1, e −γt ) and M(t) = e −γt for all t ≥ 0.Note also that, in this example, K is bounded and M(t) < 1 for all t ≥ 0.We also assume that the operator A satisfies the Hille-Yosida condition :(H1) there exist ¯M ≥ 0 and ω ∈ IN such that ]ω, +∞[ ⊂ ρ(A) andsup { (λ − ω) n ∥ ∥ (λI − A)−n ∥ ∥ : n ∈ IN, λ > ω}≤ ¯M. (3)TAMTAM –Tunis– 2005