Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 49125, 13 pages doi:10.1155/2007/49125 Research Article Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay Hassane Bouzahir Received November 2006; Revised 16 January 2007; Accepted 19 January 2007 Recommended by Simeon Reich We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup Copyright © 2007 Hassane Bouzahir This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction Most of the existing results about functional differential equations with finite delay have been recently under verification in the case of infinite delay Our objective in this paper is to study the solution semigroup generated by the following partial functional differential equation with infinite delay: d x(t) = AT x(t) + F xt , dt x0 = φ ∈ Ꮾ, t ≥ 0, (1.1) where AT is a nondensely defined linear operator on a Banach space (E, | · |) The phase space Ꮾ can be the space Cγ , γ being a positive real constant, of all continuous functions φ : (−∞,0] → E such that limθ→−∞ eγθ φ(θ) exists in E, endowed with the norm φ γ := supθ≤0 eγθ |φ(θ)|, φ ∈ Cγ For every t ≥ 0, the function xt ∈ Ꮾ is defined by xt (θ) = x(t + θ), for θ ∈ (−∞,0] (1.2) Journal of Inequalities and Applications We assume the following (H1) F : Ꮾ → E is globally Lipschitz continuous; that is, there exists a positive constant L such that |F(ψ1 ) − F(ψ2 )| ≤ L ψ1 − ψ2 Ꮾ for all ψ1 ,ψ2 ∈ Ꮾ A typical example that can be transformed into (1.1) is the following: ∂2 ∂ w(t,ξ) = a w(t,ξ) + bw(t,ξ) + c ∂t ∂ξ + f w(t − τ,ξ) , −∞ G(θ)w(t + θ,ξ)dθ t ≥ 0, ≤ ξ ≤ π, w(t,0) = w(t,π) = 0, w(θ,ξ) = w0 (θ,ξ), (1.3) t ≥ 0, −∞ < θ ≤ 0, ≤ ξ ≤ π, where a, b, c, and τ are positive constants, f : R → R is a continuous function, G is a positive integrable function on (−∞,0], and w0 : (−∞,0] × [0,π] → R is an appropriate continuous function Effectively, in [1], an abstract treatment of (1.3) as (1.1) leads to a characterization of exponential asymptotic stability near an equilibrium of (1.3) provided that the associated linearized semigroup is exponentially stable For many quantitative studies of any problem of type (1.1) in a concrete space of functions mapping (−∞,0] into E, one should choose a space that verifies at least the fundamental axioms first introduced in [2] That is, (Ꮾ, · Ꮾ ) is a (semi)normed abstract linear space of functions mapping (−∞,0] into E, which satisfies the following (A) There is a positive constant H and functions K(·),M(·) : R+ → R+ , with K continuous and M locally bounded, such that for any σ ∈ R, a > 0, if x : (−∞,σ + a] → E, xσ ∈ Ꮾ, and x(·) is continuous on [σ,σ + a], then for every t in [σ,σ + a] the following conditions hold: (i) xt ∈ Ꮾ; (ii) |x(t)| ≤ H xt Ꮾ , which is equivalent to (ii) for each ϕ ∈ Ꮾ, |ϕ(0)| ≤ H ϕ Ꮾ ; (iii) xt Ꮾ ≤ K(t − σ)supσ ≤s≤t |x(s)| + M(t − σ) xσ Ꮾ (A1) For the function x(·) in (A), t → xt is a Ꮾ-valued continuous function for t in [σ,σ + a] (B) The space Ꮾ or the space of equivalence classes Ꮾ := Ꮾ/ · Ꮾ = {ϕ : ϕ ∈ Ꮾ} is complete However, to obtain interesting qualitative results, a concrete choice should be made on a space that verifies additional properties which are essential to investigate the equation A class of employed spaces is called uniform fading memory spaces They verify that the function K(·) is constant, limt→+∞ M(t) = 0, and the following extra property (C) If {φn }n≥0 is a Cauchy sequence in Ꮾ with respect to the (semi)norm and if φn converges compactly to φ on (−∞,0], then φ is in Ꮾ and φn − φ Ꮾ → as n → +∞ There are many examples of concrete spaces that verify the above properties In [3], it was proved, for instance, that if γ > 0, the above-defined space Cγ is a uniform fading Hassane Bouzahir memory space Another example is given by Cg := φ ∈ C (−∞,0];E : lim θ →−∞ φ(θ) =0 , g(θ) (1.4) equipped with the norm φ g := sup −∞ ω0 , n ∈ N} ≤ β, where ρ(AT ) is the resolvent set of AT and R(λ,AT ) = (λI − AT )−1 Definition 2.1 A function x : (−∞,a] → E, a > 0, is an integral solution of (1.1) in (−∞,a] if the following conditions hold: (i) x is continuous on [0, a]; t (ii) x(s)ds ∈ D(AT ), for t ∈ [0,a]; (iii) ⎧ ⎪ ⎨φ(0) + A x(t) = ⎪ ⎩φ(t), t T t x(s)ds + F s,xs ds, ≤ t ≤ a, (2.1) −∞ < t ≤ It follows from (ii) of the above definition that for an integral solution x, one has x(t) ∈ D(AT ) for all t ≥ In particular, φ(0) ∈ D(AT ) Define the part A0 of AT in D(AT ) by D A0 = x ∈ D AT : AT x ∈ D AT , A0 x = AT x, for x ∈ D A0 (2.2) Recall (cf [17]) that A0 generates a C0 -semigroup (T0 (t))t≥0 on D(AT ) It is known (see [1, 6]) that under (H1) and (H2), for φ ∈ Ꮾ such that φ(0) ∈ D(AT ), (1.1) admits a unique integral solution x(·,φ) given by the following formula: ⎧ ⎪ ⎨T (t)φ(0) + lim λ→+∞ x(t,φ) = ⎪ ⎩ φ(t), where Bλ = λR(λ,AT ) t T0 (t − s)Bλ F xs (·,φ) ds, for t ≥ 0, for t ∈ (−∞,0], (2.3) Hassane Bouzahir Set ᐄ := ϕ ∈ Ꮾ : ϕ(0) ∈ D AT (2.4) Define ᐁ(t) on ᐄ for t ≥ by ᐁ(t)ϕ = xt (·,ϕ), (2.5) where x(·,φ) is the integral solution of (1.1) The point of departure for our results is [1] where we have proved that (ᐁ(t))t≥0 is a strongly continuous semigroup satisfying the following properties: (i) (ᐁ(t))t≥0 satisfies, for t ≥ and θ ∈ (−∞,0], the following translation property ⎧ ⎨ ᐁ(t + θ)ϕ (0), ᐁ(t)ϕ (θ) = ⎩ ϕ(t + θ), if t + θ ≥ 0, if t + θ ≤ 0, (2.6) (ii) there exist two positive locally bounded functions m(·),n(·) : R+ → R+ such that, for all ϕ1 ,ϕ2 ∈ X, and t ≥ 0, ᐁ(t)ϕ1 − ᐁ(t)ϕ2 Ꮾ ≤ m(t)en(t) ϕ1 − ϕ2 Ꮾ (2.7) Moreover, if F is a bounded linear operator and Ꮾ is a subspace of C((−∞,0];E) satisfying axioms (A1), (A2), (B) and the following axiom which was introduced in [13]: (D) for a sequence (ϕn )n≥0 in Ꮾ, if ϕn Ꮾ → 0, then |ϕn (s)| → for each s ∈ (−∞,0], then Aᐁ : D(Aᐁ ) ⊆ ᐄ → ᐄ such that Aᐁ ϕ = ϕ for any ϕ ∈ D(Aᐁ ), where D Aᐁ = ϕ ∈ ᐄ : ϕ is continuously differentiable, ϕ(0) ∈ D AT , ϕ ∈ ᐄ, ϕ (0) = AT ϕ(0) + F(ϕ) , (2.8) is the infinitesimal generator of (ᐁ(t))t≥0 Main results Our first main result can be considered as an extension of the above result to the case where F is nonlinear The concrete choice of Ꮾ (LCg , LCg , or UCg ) seems to be best adapted to obtain our results Here, we suppose sufficient conditions on g The proof combines the ideas of [1, 18] or [19] Proposition 3.1 Let Ꮾ = LCg , Ꮾ = LCg (with (g1)), or Ꮾ = UCg (with (g5)) Then Conditions (H1) and (H2) imply that Aᐁ is the infinitesimal generator of (ᐁ(t))t≥0 Journal of Inequalities and Applications Proof Let ϕ ∈ ᐄ be continuously differentiable such that ϕ ∈ ᐄ, ϕ(0) ∈ D AT , ϕ (0) = AT ϕ(0) + F(ϕ) (3.1) Let x(·,ϕ) : (−∞,+∞) → E be the unique integral solution of (1.1) We have to show that limt→0+ (1/t)(ᐁ(t)ϕ − ϕ) exists in ᐄ and is equal to ϕ By definition of xt (·,ϕ) and ᐁ, ⎧ ⎨ ᐁ(t)ϕ (0), x(t,ϕ) = ⎩ ϕ(t), if t ≥ 0, if t ∈ (−∞,0], (3.2) then ⎧ ⎪1 ⎪ x(t + θ,ϕ) − ϕ(θ) (0), ⎪ ⎨ t ᐁ(t)ϕ − ϕ (θ) = ⎪ t ⎪1 ⎪ ⎩ ϕ(t + θ) − ϕ(θ) , t t + θ > 0, (3.3) t + θ ∈ (−∞,0] If t + θ ≤ 0, (1/t)(ᐁ(t)ϕ − ϕ)(θ) tends to D+ ϕ(θ) as t → 0+ , where D+ ϕ(θ) is the right derivative of ϕ in θ If t + θ > 0, we have ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) t = T0 (t + θ)ϕ(0) + lim t λ→∞ t+θ (3.4) T0 (t + θ − s)Bλ F xs ds − ϕ(θ) − ϕ (θ) Let S(t), t ≥ 0, be the integrated semigroup associated with T0 (t), t ≥ We obtain from the last equality ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) t = ϕ(0) + S(t + θ)AT ϕ(0) + lim t λ→∞ t+θ + lim λ→∞ t+θ T0 (t + θ − s)Bλ F xs − F(ϕ) ds (3.5) T0 (t + θ − s)Bλ F(ϕ)ds − ϕ(θ) − ϕ (θ) Since T0 (t)ϕ(0) = ϕ(0) + AT S(t)ϕ(0) and t+θ lim λ→∞ T0 (t + θ − s)Bλ F(ϕ)ds = lim S(t + θ)Bλ F(ϕ) λ→∞ = lim Bλ S(t + θ)F(ϕ) λ→∞ = S(t + θ)F(ϕ), (3.6) Hassane Bouzahir we deduce that ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) t = ϕ(0) + S(t + θ)ϕ (0) + lim t λ→∞ − ϕ (θ) t+θ T0 (t + θ − s)Bλ F xs − F(ϕ) ds − ϕ(θ) t+θ t+θ S(t + θ)ϕ (0) − ϕ (0)ds + lim T0 (t + θ − s)Bλ F xs − F(ϕ) ds t λ→∞ 0 1 t+θ + ϕ(0) + ϕ (0) − ϕ(θ) − ϕ (θ) t t t t+θ t+θ = T0 (s)ϕ (0) − ϕ (0) ds + lim T0 (t + θ − s)Bλ F xs − F(ϕ) ds t λ→∞ 1 + ϕ (s)ds − ϕ (θ) + ϕ (0) − ϕ (0)ds t θ t θ t+θ t+θ = T0 (s)ϕ (0) − ϕ (0) ds + lim T0 (t + θ − s)Bλ F xs − F(ϕ) ds t λ→∞ + ϕ (s) − ϕ (0) ds + ϕ (0) − ϕ (θ) t θ (3.7) = Hence 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) ≤ t t t+θ T0 (s)ϕ (0) − ϕ (0) ds t+θ T0 (t + θ − s)Bλ F xs − F(ϕ) ds lim t λ→∞ + ϕ (s) − ϕ (0) ds + ϕ (0) − ϕ (θ) t θ + (3.8) Let ε > and choose α > small enough such that if < t < α, −∞ < θ ≤ 0, and t + θ > 0, then t t+θ ε T0 (s)ϕ (0) − ϕ (0) ds < , t+θ T0 (t + θ − s)Bλ F xs − F(ϕ) ds lim t λ→∞ ϕ (s) − ϕ (0) ds + ϕ (0) − ϕ (θ) < t θ ε < , ε , (3.9) and if < t < α, −∞ < θ ≤ 0, and t + θ ≤ 0, then 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) ≤ ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) g(θ) t t ε ϕ(t + θ) − ϕ(θ) − ϕ (θ) < = t (3.10) Journal of Inequalities and Applications Consequently, if < t < α, then 1 ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) < ᐁ(t)ϕ − ϕ (θ) − ϕ (θ) < ε g(θ) t t (3.11) Hence ᐁ(t)ϕ − ϕ − ϕ t Ꮾ < ε (3.12) This proves that limt→0+ (1/t)(ᐁ(t)ϕ − ϕ) exists and is equal to ϕ Then, ϕ ∈ D(Aᐁ ) Conversely, let ϕ ∈ ᐄ such that lim t →0+ 1 ᐁ(t)ϕ − ϕ = lim xt (·,ϕ) − ϕ = ψ = Aᐁ ϕ t →0+ t t exists in ᐄ (3.13) We can easily see that axiom (D) is verified by Cγ , LCg , and UCg which implies that lim t →0+ xt (θ,ϕ) − ϕ(θ) t exists for all θ ≤ and is equal to ψ(θ) (3.14) Then, for θ ∈ (−∞,0), we have ψ(θ) = lim + t →0 ϕ(t + θ) − ϕ(θ) = D+ ϕ(θ); t (3.15) that is, D+ ϕ = ψ in (−∞,0) Since ψ is continuous, D+ ϕ is also continuous in (−∞,0) Let us recall the following result Lemma 3.2 [20] Let ϕ be continuous and differentiable on the right on [a,b) If D+ ϕ is continuous on [a,b), then ϕ is continuously differentiable on [a,b) From the above lemma, we deduce that the function ϕ is continuously differentiable in (−∞,0) and ϕ = ψ On the other hand, for θ = 0, one has limθ→0− ϕ (θ) exists and equals ψ(0) From this we infer that the function ϕ is continuously differentiable in (−∞,0] and ϕ = ψ ∈ ᐄ We also deduce that t → ᐁ(t)ϕ is continuously differentiable On the other hand, we have t x(t) = ϕ(0) + AT t x(s)ds + t F xs ds (3.16) t This implies that limt→0+ AT [((1/t) x(s)ds) + (1/t) F(xs )ds] exists and hence t t limt→0+ AT ((1/t) x(s)ds) exists From the closedness of AT and the fact that (1/t) x(s)ds t ∈ D(AT ) for t > 0, we deduce that limt→0+ (1/t) x(s)ds exists in D(AT ) and is equal to ϕ(0) Consequently, ϕ(0) ∈ D(AT ) and ϕ (0) = AT ϕ(0) + F(ϕ) This completes the proof of the proposition The next result may be considered as an extension of a similar one in [19] Our goal is to establish the Crandall and Liggett exponential formula lim n→+∞ t I − Aᐁ n −n ϕ = ᐁ(t)ϕ, ∀ϕ ∈ ᐄγ , t ≥ (3.17) Hassane Bouzahir We restrict our choice to ᐄγ := {φ ∈ Cγ : φ(0) ∈ D(AT )} with γ > Recall that this specified space is a uniform fading memory one Proposition 3.3 Let Ꮾ = Cγ with γ > Suppose that (H1) and (H2) are satisfied Then, the operator Aᐁ given by Proposition 3.1 satisfies the following Crandall Liggett conditions (a) Im(I − λAᐁ ) = ᐄγ for all λ ∈ (0,1/(L + ω0 )) (b) For all ψ1 ,ψ2 ∈ ᐄγ and λ ∈ (0,1/(L + ω0 )), I − λAᐁ −1 ψ1 − I − λAᐁ −1 ψ2 γ ≤ 1 − λ L + ω0 ψ1 − ψ γ (3.18) (c) D(Aᐁ ) is dense in ᐄγ Proof (a) It is well known that one can suppose without loss of generality that ω0 > −L and T0 (t) ≤ eω0 t To prove (a), it is clear from the definition of Aᐁ that (I − λAᐁ )(D(Aᐁ )) ⊆ ᐄγ for λ > On the other hand, for ψ ∈ ᐄγ and λ > 0, let us solve the following equation: I − λAᐁ ϕ = ψ, ϕ ∈ D Aᐁ (3.19) Recall that with γ > and λ > 0, the function W(1/λ)ψ(0) : θ → e(1/λ)θ ψ(0), θ ≤ 0, belongs to ᐄγ Also, the fact that Ꮿγ with γ > is a uniform fading memory space implies that the function Mλ ψ : θ → (1/λ) θ e(1/λ)(θ−s) ψ(s)ds, θ ≤ 0, belongs to ᐄγ (see [21, 22]) Moreover, we can see that the solution of (3.19) is ϕ(θ) = W 1 ϕ(0) (θ) + Mλ ψ (θ) = e(1/λ)θ ϕ(0) + λ λ θ e(1/λ)(θ−s) ψ(s)ds (3.20) Next, we suppose that < λω0 < From (3.19) evaluated at and the definition of Aᐁ we get ϕ(0) = I − λAT −1 ψ(0) + λF(ϕ) (3.21) Introduce the following mapping Gλ : E → E defined by ψ Gλ (x) = I − λAT ψ −1 ψ(0) + λF W x + Mλ ψ λ ∀x ∈ E (3.22) For x, y ∈ E, we have 1 x + Mλ ψ − F W y + Mλ ψ ,AT F W λ λ λ λL 1 ≤ W x−W y − λω0 λ λ γ λL ≤ sup eγθ e(1/λ)θ (x − y) − λω0 θ≤0 λL ≤ |x − y | − λω0 Gλ (x) − Gλ (y) ≤ R ψ ψ (3.23) 10 Journal of Inequalities and Applications Next, we suppose that λ ∈ (0,1/(L + ω0 )) Then, Gλ is a strict contraction and it has a ψ unique fixed point x in E Knowing that (I − λAT )−1 (E) ⊆ D(AT ), we deduce that this fixed point belongs to D(AT ) Consequently, Im(I − λAᐁ ) = ᐄγ for all λ ∈ (0,1/(L + ω0 )) (b) Let λ ∈ (0,1/(L + ω0 )) be fixed Set λ := (I − λAᐁ )−1 which is well defined from ᐄγ to D(Aᐁ ) We prove that λ is Lipschitz continuous with Lipschitz constant less than (1 − λω0 )/(1 − λ(L + ω0 )) In fact, let λ > with λ ∈ (0,1/(L + ω0 )) and λ ψ1 := ϕ1 λ ψ2 := ϕ2 for ψ1 ,ψ2 ∈ ᐄγ Given ε > 0, by definition, there exists θ ∈ (−∞,0] such that eγθ ϕ1 (θ) − ϕ2 (θ) > ϕ1 − ϕ2 γ − ε (3.24) Using (3.21) and (H2), we get eγθ ϕ1 (θ) − ϕ2 (θ) ≤ eγθ + ≤ e(1/λ)θ + ≤ ,AT λ e(1/λ)θ R λ θ e(1/λ)(θ−s) ψ1 (s) − ψ2 (s) ds 1 − λω0 (1/λ)θ e λ ψ1 (0) − ψ2 (0) + F ϕ1 − F ϕ2 λ θ ψ1 − ψ γ e(γ−1/λ)s ds e(1/λ)θ + − e(1/λ)θ − λω0 + λL ϕ1 − ϕ2 ψ1 − ψ ψ1 − ψ γ γ γ + λLe(1/λ)θ ϕ1 − ϕ2 − λω0 γ (3.25) which implies that − λω0 − λLe(1/λ)θ ϕ1 − ϕ2 − λω0 ϕ1 − ϕ2 γ γ ≤ε+ e(1/λ)θ + − e(1/λ)θ − λω0 ψ1 − ψ γ , − λω0 e(1/λ)θ + − λω0 − e(1/λ)θ + λω0 e(1/λ)θ − λω0 − λL − λω0 ≤ ψ1 − ψ γ − λ ω0 + L ≤ ψ1 − ψ γ (3.26) To prove (c), using similar arguments as in [23, the proof of Proposition 3.5], we can verify that for all λ > with λ ∈ (0,1/(L + ω0 )) and ψ ∈ ᐄγ , ψ − λ ψ γ ≤ λ λL ψ γ+ F(0) − λω − λω + ψ − ψ(0) − Mλ ψ − ψ(0) γ+ I − λAT −1 (3.27) ψ(0) − ψ(0) Since by its definition λ ψ belongs to D(Aᐁ ), assertion (c) follows from the fact that limλ→0+ |(I − λAT )−1 x − x| = for all x ∈ D(AT ) (see [24]) and the following result 0 Lemma 3.4 Set ᐄγ := {φ ∈ ᐄγ such that φ(0) = 0} Then for all λ > and ψ ∈ ᐄγ the as λ tends to zero function Mλ ψ tends to ψ in ᐄγ Hassane Bouzahir 11 Proof Set ᐄγ := {φ ∈ ᐄγ such that φ(0) = 0} We can see that the operator B0 : D(B0 ) ⊆ 0 ᐄγ → ᐄγ such that B0 φ = φ for any φ ∈ D(B0 ) where 0 D B0 = φ ∈ ᐄγ : φ is continuously differentiable and φ ∈ ᐄγ (3.28) is the infinitesimal generator of a C0 semigroup on ᐄγ Moreover, Mλ ψ = I − λB0 −1 ψ for any λ > 0, ψ ∈ ᐄγ (3.29) This implies that the assertion of the lemma is a direct consequence of a basic result that can be found, for instance, in [24, page 248] Corollary 3.5 Let ᐄ = ᐄγ with γ > Suppose that (H1) and (H2) are satisfied Then, for all ϕ ∈ ᐄγ and t ≥ 0, lim n→+∞ t I − Aᐁ n −n ϕ = ᐁ(t)ϕ (3.30) The proof of the above result is based on the following special case of the well-known Crandall and Liggett theorem (see [25]) Theorem 3.6 [25] Let (Y , · ) be a Banach space and B a nonlinear operator with dense domain D(B) in Y Suppose that there exists a positive real constant ω such that (a) Im(I − λB) = Y for all λ ∈ (0,1/ω), (b) for all y1 , y2 ∈ Y and λ ∈ (0,1/ω), (I − λB)−1 y1 − (I − λB)−1 y2 ≤ y1 − y2 − λω (3.31) Then for all y ∈ Y , the limit W0 (t)y := lim n→+∞ t I− B n −n y (3.32) exists in Y Moreover, the family of operators (W0 (t))t≥0 satisfies (i) W0 (0) = I, (ii) W0 (t1 + t2 ) = W0 (t1 )W0 (t2 ) for all t1 ,t2 ≥ 0, (iii) W0 (t)y1 − W0 (t)y2 ≤ eωt y1 − y2 for all y1 , y2 ∈ Y and t ≥ Let us now give the link between the semigroup given by the Crandall and Liggett theorem and the integral solution to (1.1) The proof will be omitted because it is very similar to the case of finite delay (see [18] or [19]) Proposition 3.7 Let ᐄ = ᐄγ with γ > Suppose that (H1) and (H2) are satisfied and the operator A : D(A) ⊆ ᐄγ → ᐄγ such that Aϕ = ϕ for any ϕ ∈ D(A), where D(A) = ϕ ∈ ᐄγ : ϕ is continuously differentiable, ϕ(0) ∈ D AT , ϕ ∈ ᐄγ , ϕ (0) = AT ϕ(0) + F(ϕ) , (3.33) 12 Journal of Inequalities and Applications satisfies the hypotheses of Crandall and Liggett theorem in ᐄγ Let (U(t))t≥0 be the nonlinear semigroup given by U(t)ϕ = lim n→+∞ t I− A n −n ϕ ∀ϕ ∈ ᐄγ , t ≥ (3.34) Then for all ϕ ∈ ᐄγ , the function y := y(·,ϕ) : (−∞,+∞) → R defined by ⎧ ⎨ U(t)ϕ (0), y(t,ϕ) = ⎩ ϕ(t), t ≥ 0, t ≤ 0, (3.35) is an integral solution of (1.1) Acknowledgments The author would like to thank Professors M Adimy and K Ezzinbi for helpful discussions This research was supported by TWAS under contract no 04-150 RG/MATHS/ AF/AC References [1] M Adimy, H Bouzahir, and K Ezzinbi, “Local existence and stability for some partial functional differential equations with infinite delay,” Nonlinear Analysis Theory, Methods & Applications Series A, vol 48, no 3, pp 323–348, 2002 [2] J K Hale and J Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj Serio Internacia, vol 21, no 1, pp 11–41, 1978 [3] Y Hino, S Murakami, and T Naito, Functional-Differential Equations with Infinite Delay, vol 1473 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991 [4] S Brendle and R Nagel, “PFDE with nonautonomous past,” Discrete and Continuous Dynamical Systems Series A, vol 8, no 4, pp 953–966, 2002 [5] J K Hale and S M Verduyn Lunel, Introduction to Functional-Differential Equations, vol 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993 [6] M Adimy, H Bouzahir, and K Ezzinbi, “Existence for a class of partial functional differential equations with infinite delay,” Nonlinear Analysis Theory, Methods & Applications Series A, vol 46, no 1, pp 91–112, 2001 [7] R Benkhalti, H Bouzahir, and K Ezzinbi, “Existence of a periodic solution for some partial functional-differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol 256, no 1, pp 257–280, 2001 [8] H Bouzahir and K Ezzinbi, “Global attractor for a class of partial functional differential equations with infinite delay,” in Topics in Functional Differential and Difference Equations (Lisbon, 1999), T Faria and P Freitas, Eds., vol 29 of Fields Inst Commun., pp 63–71, American Mathematical Society, Providence, RI, USA, 2001 [9] W M Ruess, “Linearized stability for nonlinear evolution equations,” Journal of Evolution Equations, vol 3, no 2, pp 361–373, 2003 [10] W M Ruess, “Existence of solutions to partial functional evolution equations with delay,” in ´ Functional Analysis (Trier, 1994), S Dierolf, S Dineen, and P Domanski, Eds., pp 377–387, de Gruyter, Berlin, Germany, 1996 [11] W M Ruess, “Existence and stability of solutions to partial functional-differential equations with delay,” Advances in Differential Equations, vol 4, no 6, pp 843–876, 1999 Hassane Bouzahir 13 [12] W M Ruess, “Existence of solutions to partial functional-differential equations with delay,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A G Kartsatos, Ed., vol 178 of Lecture Notes in Pure and Appl Math., pp 259–288, Dekker, New York, NY, USA, 1996 [13] W M Ruess and W H Summers, “Linearized stability for abstract differential equations with delay,” Journal of Mathematical Analysis and Applications, vol 198, no 2, pp 310–336, 1996 [14] W M Ruess and W H Summers, “Operator semigroups for functional-differential equations with delay,” Transactions of the American Mathematical Society, vol 341, no 2, pp 695–719, 1994 [15] W M Ruess and W H Summers, “Almost periodicity and stability for solutions to functionaldifferential equations with infinite delay,” Differential and Integral Equations, vol 9, no 6, pp 1225–1252, 1996 [16] A G Kartsatos and M E Parrott, “The weak solution of a functional-differential equation in a general Banach space,” Journal of Differential Equations, vol 75, no 2, pp 290–302, 1988 [17] H R Thieme, “Semiflows generated by Lipschitz perturbations of non-densely defined operators,” Differential and Integral Equations, vol 3, no 6, pp 1035–1066, 1990 [18] M Adimy, M Laklach, and K Ezzinbi, “Non-linear semigroup of a class of abstract semilinear functional differential equations with a non-dense domain,” Acta Mathematica Sinica (English Series), vol 20, no 5, pp 933–942, 2004 ` e ´ ` [19] M Laklach, “Contribution a l’´ tude des equations aux d´ riv´ es partielles a retard et de type e e neutre,” Th` se de doctorat, l’Universit´ de Pau et des Pays de l’Adour, Pau, France, 2001 e e [20] A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983 [21] T Naito, N V Minh, and J S Shin, “Spectrum and (almost) periodic solutions of functional differential equations,” Vietnam Journal of Mathematics, vol 30, supplement, pp 577–589, 2002 [22] T Naito, N V Minh, and J S Shin, “The spectrum of evolution equations with infinite delay,” unpublished [23] C C Travis and G F Webb, “Existence and stability for partial functional differential equations,” Transactions of the American Mathematical Society, vol 200, pp 395–418, 1974 [24] K Yosida, Functional Analysis, vol 123 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Berlin, Germany, 6th edition, 1980 [25] M G Crandall and T M Liggett, “Generation of semi-groups of nonlinear transformations on general Banach spaces,” American Journal of Mathematics, vol 93, pp 265–298, 1971 Hassane Bouzahir: National Engineering School of Applied Sciences (ENSA), University Ibn Zohr, P.O Box 1136, Agadir 80000, Morocco Email address: bouzahir@ensa-agadir.ac.ma ... evolution equations, ” Journal of Evolution Equations, vol 3, no 2, pp 361–373, 2003 [10] W M Ruess, “Existence of solutions to partial functional evolution equations with delay,” in ´ Functional. .. solutions to partial functional- differential equations with delay,” Advances in Differential Equations, vol 4, no 6, pp 843–876, 1999 Hassane Bouzahir 13 [12] W M Ruess, “Existence of solutions to partial. .. Ezzinbi, “Global attractor for a class of partial functional differential equations with infinite delay,” in Topics in Functional Differential and Difference Equations (Lisbon, 1999), T Faria and P