mathetical and computer modelling

Mathematical and Computer Modelling 49 (2009) 1260–1267 Contents lists available at ScienceDirect Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm Existence results for partial neutral functional differential equations with state-dependent delay Eduardo Hernández Morales a,∗ , Mark A McKibben b , Hernán R Henríquez c a Departamento de Matemática, ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos SP, Brazil b Department of Mathematics and Computer Science, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD 21204, USA c Departamento de Matemática, Universidade de Santiago, Casilla 307, Correo-2, Santiago, Chile article info Article history: Received 17 March 2008 Received in revised form 22 June 2008 Accepted 11 July 2008 a b s t r a c t In this paper we study the existence of mild solutions for a class of first order abstract partial neutral differential equations with state-dependent delay © 2008 Elsevier Ltd All rights reserved Keywords: Abstract Cauchy problem State-dependent delay Introduction In this paper we study the existence of mild solutions for a class of partial neutral functional differential equations with state-dependent delay described by the abstract form d (x(t ) + G(t , xt )) = Ax(t ) + F (t , xρ(t ,xt ) ), dt x0 = ϕ ∈ B , t ∈ I = [0, a], (1.1) (1.2) where A is the infinitesimal generator of a compact semigroup of bounded linear operators (T (t ))t ≥0 defined on an abstract Banach space X ; the function xs : (−∞, 0] → X , xs (θ ) = x(s + θ ), belongs to some abstract phase space B described axiomatically; and F , G : I × B → X , ρ : I × B → R are appropriate functions Functional differential equations with state-dependent delay appear frequently in applications as models of various phenomena and for this reason, the study of this type of equation has received much attention in recent years, see Chapter in [1], the papers [2–12] and the references therein We also cite [13,14,9,15] for the case of neutral differential equations with state-dependent delay The literature related to partial functional differential equations with state-dependent delay is, to our knowledge, restricted to the papers [16,17] Abstract neutral differential equations arise in many areas of applied mathematics As such, they have been largely studied during the last few decades The literature related to ordinary neutral differential equations is very extensive Hence, we refer the reader to [18], which contains a comprehensive description of such equations Similarly, for additional material concerning abstract partial neutral functional differential equations and related issues, we refer the reader to to Adimy [19], Hale [20] and Wu [21–23] for finite delay equations, and Hernández & Henriquez [24,25] and Hernández [26] for the case of unbounded delay ∗ Corresponding author E-mail addresses: lalohm@icmc.usp.br (E Hernández), mmckibben@goucher.edu (M.A McKibben), hhenriqu@lauca.usach.cl (H.R Henríquez) 0895-7177/$ – see front matter © 2008 Elsevier Ltd All rights reserved doi:10.1016/j.mcm.2008.07.011 E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 1261 Preliminaries Throughout this paper, (X , · ) is a Banach space, A : D(A) ⊂ X → X is the infinitesimal generator of a compact semigroup (T (t ))t ≥0 of bounded linear operators on X , and M > is a constant such that T (t ) ≤ M for every t ∈ [0, a] For background information related to the semigroup theory, we refer the reader to Pazy [27] In this work we will employ an axiomatic definition for the phase space B which is similar to the one used in [28] Specifically, B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm · B , which satisfies the following axioms: (A) If x : (−∞, σ + b] → X , b > 0, is such that x|[σ ,σ +b] ∈ C ([σ , σ + b] : X ) and xσ ∈ B , then for every t ∈ [σ , σ + b] the following conditions hold: (i) xt is in B , (ii) x(t ) ≤ H xt B , (iii) xt B ≤ K (t − σ ) sup{ x(s) : σ ≤ s ≤ t } + M (t − σ ) xσ B , where H > is a constant; K , M : [0, ∞) → [1, ∞), K is continuous, M is locally bounded, and H , K , M are independent of x(·) (A1) For the function x(·) in (A), the function t → xt is continuous from [σ , σ + b] into B (B) The space B is complete We now consider some examples of phase spaces Example 2.1 (The Phase Spaces Cg , Cg0 ) Let g : (−∞, 0] → [1, ∞) be a continuous, non-increasing function with g (0) = 1, g (t +θ ) which satisfies conditions (g-1) and (g-2) of [28] Briefly, this means that the function γ (t ) := sup−∞ The terminology and notations employed in this manuscript coincide with those generally used in functional analysis In particular, for Banach spaces (Z , · Z ), (W , · W ), the notation L(Z ; W ) stands for the Banach space of bounded linear operators from Z into W , and we abbreviate this notation to L(Z ) when Z = W Moreover, Br (x, Z ) denotes the closed ball with center at x and radius r > in Z and for a bounded function ξ : I → Z and ≤ t ≤ a we employ the notation ξ Z ,t for ξ (θ ) Z ,t = sup{ ξ (s) Z : s ∈ [0, t ]} (2.1) We will simply write ξ t when no confusion arises The following result, often referred to as the Leray–Schauder Alternative Theorem, plays a critical role in the proofs of some of our main results Theorem 2.1 ([29, Theorem 6.5.4]) Let D be a convex subset of a Banach space X and assume that ∈ D Let F : D → D be a completely continuous map Then, either the set {x ∈ D : x = λF (x), for some < λ < 1} is unbounded or F has a fixed point in D The remainder of the paper is divided into two sections Specifically, by using Theorem 2.1 and the classical Schauder fixed point theorem, in Section we establish the existence of a mild solution for the abstract Cauchy problem (1.1) and (1.2) Then, for illustration of the abstract theory, an application is considered in Section 1262 E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 Existence results In this section we study the existence of mild solutions for the abstract Cauchy problem (1.1) and (1.2) Throughout this section, ϕ ∈ B is a fixed function, (Y , · Y ) is a Banach space continuously included in X , and the following two conditions are assumed to hold: HY For every y ∈ Y , the function t → T (t )y is continuous from [0, ∞) into Y Moreover, T (t )(Y ) ⊂ D(A) for every t > and there exists a positive function γ ∈ L1 ([0, a]) such that AT (t ) L(Y ;X ) ≤ γ (t ), for every t ∈ I Hϕ Let R(ρ − ) = {ρ(s, ψ) : (s, ψ) ∈ I × B , ρ(s, ψ) ≤ 0} The function t → ϕt is continuous from R(ρ − ) into B and there exists a continuous and bounded function J ϕ : R(ρ − ) → (0, ∞) such that ϕt B ≤ J ϕ (t ) ϕ B for every t ∈ R(ρ − ) Remark 3.2 We point out here that condition Hϕ is often satisfied by functions that are continuous and bounded In fact, if the space of continuous and bounded functions Cb ((−∞, 0]; X ) is continuously included in B , then sup ψ(θ ) ψt B ≤L θ≤0 ψ ψ B, t ≤ 0, ψ = 0, ψ ∈ Cb ((−∞, 0]; X ) (3.1) B It is not difficult to show that Cb ((−∞, 0]; X ) is continuously included in both Cg (X ) and Cg0 (X ) Moreover, if g (·) satisfies(g- 5)–(g-7) in [28], then the space Cb ((−∞, 0], X ) is also continuously included in Cr × L2 (g ; X ) For additional details related to this matter, see Proposition 7.1.1 and Theorems 1.3.2 and 1.3.8 in [28] To establish our results, we impose the following conditions H1 The function F : I × B → X satisfies the following properties (i) For every ψ ∈ B , the function t → F (t , ψ) is strongly measurable (ii) For each t ∈ I, the function F (t , ·) : B → X is continuous (iii) There exist an integrable function m : I → [0, ∞) and a continuous non-decreasing function W : [0, ∞) → (0, ∞) such that F (t , ψ) ≤ m(t )W ( ψ B ), (t , ψ) ∈ I × B H2 The function G is Y -valued, G : [0, a] × B → Y is continuous, and there exist positive constants c1 , c2 such that G(t , ψ) Y ≤ c1 ψ B + c2 , for all (t , ψ) ∈ [0, a] × B H3 The function G is Y -valued, G : [0, a] × B → Y is continuous, and there exists LG > such that G(t , ψ1 ) − G(t , ψ2 ) Y ≤ LG ψ1 − ψ2 B, (t , ψi ) ∈ I × B H4 Let S (ϕ) be the space S (ϕ) = {x : (−∞, a] → X : x0 = 0; x|[0,a] ∈ C ([0, a]; X )} endowed with the norm of the uniform convergence topology and y : (−∞, a] → X be the function defined by y0 = ϕ on (−∞, 0] and y(t ) = T (t )ϕ(0) on [0, a] Then, for every bounded set Q such that Q ⊂ S (ϕ), the set of functions {t → G(t , xt + yt ) : x ∈ Q } is equicontinuous on [0, a] We now introduce the following concept of mild solution Definition 3.1 A function x : (−∞, a] → X is a mild solution of (1.1) and (1.2) if x0 = ϕ ; xρ(s,xs ) ∈ B for every s ∈ I; the function t → AT (t − s)G(s, xs ) is integrable on [0, t ), for every t ∈ [0, a]; and t x(t ) = T (t )(ϕ(0) + G(0, ϕ)) − G(t , xt ) − t AT (t − s)G(s, xs )ds + T (t − s)F (s, xρ(s,xs ) )ds, t ∈ I Remark 3.3 Let x(·) be a function as in axiom A Let us mention that the conditions HY , H2 , H3 are linked to the integrability of the function s → AT (t − s)G(s, xs ) In general, except for the trivial case in which A is a bounded linear operator, the operator function t → AT (t ) is not integrable over [0, a] However, if condition HY holds and G satisfies either assumption H2 or H3 , then it follows from Bochner’s criterion and the estimate AT (t − s)G(s, xs ) ≤ AT (t − s) L(Y ;X ) G(s, xs ) Y ≤ γ (t − s) sup G(s, xs ) Y , s∈[0,a] that s → AT (t − s)G(s, xs ) is integrable over [0, t ), for every t ∈ [0, a] For non-trivial examples of spaces Y for which condition HY is valid, see [30] The following lemma can be proved using the phase space axioms In the rest of this paper Ma and Ka are the constants defined by Ma = sups∈[0,a] M (s) and Ka = sups∈[0,a] K (s) Lemma 3.1 Let x : (−∞, a] → X such that x0 = ϕ and x|[0,a] ∈ C (I ; X ) Then xs ϕ B ϕ ≤ (Ma + J0 ) ϕ where J0 = supt ∈R(ρ − ) J ϕ (t ) B + Ka sup x(θ ) max{0,s} , s ∈ R(ρ − ) ∪ [0, a], E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 1263 In the rest of this paper, (S (t ))t ≥0 is the is the strongly continuous semigroup of bounded linear operators on B defined by ψ(0) ψ(t + θ ) [S (t )ψ](θ ) := for − t ≤ θ ≤ 0, for − ∞ < θ < −t We are now ready to establish the first existence result Theorem 3.1 Assume that conditions H1 and H3 are satisfied If W (ξ ) a Ka γ (s)ds) + M lim inf LG ( + m(s)ds ξ ξ →∞+ a < 1, then there exists a mild solution of (1.1) and (1.2) Proof Consider the metric space Y = {u ∈ C (I : X ) : u(0) = ϕ(0)} endowed with the norm u define the operator Γ : Y → Y by a = sups∈[0,a] u(s) , and t t T (t − s)F (s, x¯ ρ(s,¯xs ) )ds, AT (t − s)G(s, x¯ s )ds + Γ x(t ) = T (t )(ϕ(0) + G(0, ϕ)) − G(t , x¯ t ) − s ∈ I, 0 where x¯ : (−∞, a] → X is defined by the relation x¯ = ϕ and x¯ |I = x Following the discussion outlined in Remark 3.3 and using condition Hϕ , we infer that the functions s → AT (t − s)G(s, xs ) and s → T (t − s)F (s, xs ) are integrable over [0, t ), for each t ∈ [0, a], which enables us to conclude that Γ is a well-defined operator from Y into Y Let ϕ¯ : (−∞, a] → X be the extension of ϕ to (−∞, a] such that ϕ(θ ¯ ) = ϕ(0) on I We affirm that there exists r > such that Γ (Br (ϕ| ¯ I , Y )) ⊂ Br (ϕ| ¯ I , Y ) Indeed, if this property is false, then for every r > there exist xr ∈ Br (ϕ| ¯ I , Y ) and t r ∈ I such that r < Γ xr (t r ) − ϕ(0) Under these conditions, from Lemma 3.1 we find that r < Γ xr (t r ) − ϕ(0) ≤ T (t r )(ϕ(0) − G(0, ϕ)) − ϕ(0) + G(t r , S (t r )ϕ) + G(t r , (xr )t r ) − G(t r , S (t r )ϕ) tr + AT (t r − s) L(Y ;X ) G(s, (xr )s ) − G(s, S (s)ϕ) ds AT (t r − s) L(Y ;X ) G(s, S (s)ϕ) ds + M tr + tr r ≤ T (t )(ϕ(0) − G(0, ϕ)) − ϕ(0) + G(s, S (s)ϕ) a m(s)W ( xr ρ(s,(xr )s ) )ds + LG Ka xr (s) − ϕ(0) tr γ (t r − s) xr (s) − ϕ(0) s ds + G(s, S (s)ϕ) + L G Ka γ (s)ds a 0 tr ϕ m(s)W ((Ma + J0 ) ϕ +M tr a B + Ka ( xr (s) − ϕ(0) s + ϕ(0) ))ds a ≤ T (t r )(ϕ(0) − G(0, ϕ)) − ϕ(0) + LG Ka r + G(s, S (s)ϕ) a γ (s)ds + L G Ka r a + G(s, S (s)ϕ) ϕ γ (s)ds + MW ((Ma + J0 + Ka H M ) ϕ a a B + Ka r ) m(s)ds 0 and hence a ≤ Ka LG ( + γ (s)ds) + M lim inf ξ →∞+ W (ξ ) a m(s)ds , ξ which is contrary to our assumption Let r > such that Γ (Br (ϕ| ¯ I , Y )) ⊂ Br (ϕ| ¯ I , Y ) In order to prove that Γ (·) is a continuous condensing map from Br (ϕ| ¯ I, Y) into Br (ϕ| ¯ I , Y ), we introduce the decomposition Γ = Γ1 + Γ2 , where t Γ1 x(t ) = T (t )(ϕ(0) + G(0, ϕ)) − G(t , x¯ t ) − AT (t − s)G(s, x¯ s )ds, t ∈ I, t Γ2 x(t ) = T (t − s)F (s, x¯ ρ(s,¯xs ) )ds, t ∈ I From the proof of [16, Theorem 2.2], we know that Γ2 is completely continuous Moreover, from the phase space axioms and H3 we obtain a Γ1 u(t ) − Γ1 v(t ) ≤ Ka LG + γ (s)ds u−v a , t ∈ I, 1264 E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 which proves that Γ1 is a contraction on Br (ϕ| ¯ I , Y ) and so that Γ is a condensing operator on Br (ϕ| ¯ I , Y ) Consequently, from the previous remark and [31, Theorem 4.3.2] we deduce the existence of a mild solution for the system (1.1) and (1.2) The proof is complete Theorem 3.2 Assume that conditions H1 , H2 , H4 are satisfied Further, assume that ρ(t , ψ) ≤ t for every (t , ψ) ∈ I × B and a that G : [0, a] × B → X is completely continuous If µ = − Ka c1 (1 + γ (s)ds) > and µ ∞ a MKa m(s)ds < ds W (s) D ϕ where D = (Ma + J0 + MHKa ) ϕ B , + Ka C and µ a C = M G(0, ϕ) + γ (s)ds 1+ c1 ϕ ϕ B (Ma + J0 + Ka MH ) + c2 , then there exists a mild solution of (1.1) and (1.2) Proof On the space B C = {u : (−∞, a] → X ; u0 = 0, u|I ∈ C (I ; X )} endowed with the norm u we define the operator Γ : B C → B C by Γ x(t ) =  0, t ∈ (−∞, 0], t T (t )G(0, ϕ) − G(t , xt ) − a = sups∈[0,a] u(s) , t AT (t − s)G(s, xs )ds + T (t − s)F (s, xρ(s,xs ) )ds, t ∈ I, where x = y + x on (−∞, a] and y(·) is the function introduced in H4 In order to use Theorem 2.1, we will establish a priori estimates for the solutions of the integral equation z = λΓ z, λ ∈ (0, 1) Let xλ be a solution of the integral equation z = λΓ z, λ ∈ (0, 1) and α λ (s) = supθ∈[0,s] xλ (θ ) If t ∈ [0, a], from Lemma 3.1 and the fact that ρ(s, xλs ) ≤ s, s ∈ [0, a], we find that xλ (t ) ≤ T (t )G(0, ϕ) + c1 (xλ )t t B t γ (t − s)(c1 (xλ )s + c2 + B ϕ ≤ M G(0, ϕ) + c1 (Ma + J0 ) ϕ + c1 Ka MH ϕ B ϕ t + c2 + c1 Ka α λ (t ) a γ (s)ds + c1 Ka MH ϕ B a γ (s)ds + c2 B ϕ m(s)W ((Ma + J0 + MHKa ) ϕ +M B a +c1 (Ma + J0 ) ϕ m(s)W ( (xλ )s )ds + c2 )ds + M B a γ (s)ds + c1 Ka α λ (t ) γ (s)ds 0 + Ka α λ (s))ds ≤ M G(0, ϕ) + c1 ϕ B a ϕ Ma + J0 + Ka MH + (Ma + Ka MH ) a γ (s)ds + c2 + a + c1 Ka + t γ (s)ds α λ (t ) + M ϕ m(s)W ((Ma + J0 + MHKa ) ϕ γ (s)ds B + Ka α λ (s))ds Consequently, α λ (t ) ≤ C µ t M + µ m(s)W (Ma + J ϕ + MHKa ) ϕ B + Ka α λ (s) ds ϕ Using the notation ξ λ (t ) = (Ma + J0 + MHKa ) ϕ ϕ ξ λ (t ) ≤ (Ma + J0 + MHKa ) ϕ + Ka C µ + B + Ka α(t ), we obtain that t MKa µ m(s)W ξ λ (s) ds Denoting by βλ (t ) the right-hand side of the above inequality, it follows that βλ (t ) ≤ MKa µ m(t )W (βλ (t )) and hence βλ (t ) βλ (0)=D ds W (s) ≤ µ ∞ a MKa m(s)ds < D ds W (s) , which implies that the set of functions {βλ (·) : λ ∈ (0, 1)} is bounded in C (I ; R) Thus, the set of functions {xλ (·) : λ ∈ (0, 1)} is bounded on I E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 1265 To prove that Γ is completely continuous, we introduce the decomposition Γ = Γ1 + Γ2 + Γ3 where (Γi x)0 = and Γ1 x(t ) = T (t )G(0, ϕ) − G(t , xt ), t ∈ [0, a], t AT (t − s)G(s, xs )ds, t ∈ [0, a], T (t − s)F (s, xρ(s,xs ) )ds, t ∈ [0, a] Γ2 x(t ) = − t Γ3 x(t ) = The continuity of the function Γ2 is easily shown Moreover, from the proof of [16, Theorem 2.2] we know that Γ3 is completely continuous It remains to show that Γ1 is completely continuous and that Γ2 is a compact map To this end, we first prove that Γ1 is completely continuous From the assumptions it follows that Γ1 is a compact map Let (un )n∈N be a sequence in B C and u ∈ B C such that un → u From the phase space axioms we infer that the set U = [0, a] × {uns , us : s ∈ [0, a], n ∈ N} is relatively compact in [0, a] × B and that uns → us uniformly on [0, a] as n → ∞ Thus, G is uniformly continuous on U, so that G(s, uns ) → G(s, us ) uniformly on [0, a] as n → ∞, which shows that Γ1 is continuous and hence completely continuous Next, by using the Ascoli–Arzela criterion, we shall prove that Γ2 is a compact map In what follows, Br = Br (0, B C ) Step The set (Γ2 Br )(t ) = {Γ2 x(t ) : x ∈ Br } is relatively compact in X for each t ∈ I The assertion clearly holds for t = Let < < t ≤ a For u ∈ Br we see that t −ε Γ2 u(t ) = −T ( ) t AT (t − ε − s)G(s, us )ds − AT (t − s)G(s, us )ds t −ε a ∈ T ( ){x ∈ X : x ≤ (c1 Ka r + c2 ) γ (s)ds} + Br ∗ (0, X ), t where r ∗ = (c1 Ka r + c2 ) t − γ (s)ds From this, we can infer that (Γ2 Br )(t ) is totally bounded in X and hence relatively compact in X Step The set of functions Γ2 Br = {Γ2 x : x ∈ Br } is equicontinuous on I Let t ∈ (0, a) For u ∈ Br and h > such that t + h ∈ [0, a], we obtain t +h Γ2 u(t + h) − Γ2 u(t ) = (T (h) − I )Γ2 u(t ) + AT (t − s)G(s, us )ds t t +h ≤ (T (h) − I )Γ2 u(t ) + (c1 Ka r + c2 ) γ (s)ds t Since the set (Γ2 Br )(t ) is relatively compact in X and (T (t ))t ≥0 is strongly continuous, it follows that (T (h)− I )Γ2 u(t ) → as h → uniformly for u ∈ Br , which from the last inequality enables us to conclude that Γ2 Br is right equicontinuous at t ∈ (0, a) In a similar manner we can prove that Γ2 Br is right equicontinuous at zero and left equicontinuous at t ∈ (0, a] This completes the proof that Γ2 is completely continuous These remarks, in conjunction with Theorem 2.1, show that Γ has a fixed point u ∈ B C Clearly, the function x = u + y is a mild solution of (1.1) and (1.2) In the cases in which either A is bounded or X is finite dimensional (i.e., the ordinary case), our results can be simplified In fact, in such cases the function s → AT (s) is continuous on [0, a] and the results are valid for Y = X We consider these special cases in the following two results, and leave the details to the reader Proposition 3.1 Assume that condition H1 is satisfied, and that G satisfies H3 with Y = X If either A is bounded or X is finite dimensional, and Ka LG ( + e A a )+e A a lim inf W (ξ ) a m(s)ds ξ ξ →∞+ < 1, then there exists a mild solution of (1.1) and (1.2) Proposition 3.2 Assume that conditions H1 , H4 are valid and that A is bounded Further, assume that ρ(t , ψ) ≤ t on I × B and that G satisfies H2 with Y = X Suppose µ = − Ka c1 (1 + e A a ) > and e A a µ ∞ a Ka m(s)ds < D ϕ where D = (Ma + J0 + e C =e A a ds W (s) HKa ) ϕ B G(0, ϕ) + (1 + e A a A a , + Ka C µ and ) c1 ϕ B (Ma + Ka e A a H ) + c2 If either, G is completely continuous or X is finite dimensional, then there exists a mild solution of (1.1) and (1.2) 1266 E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 Examples In this section we use our abstract results to treat a concrete partial neutral differential equation We begin by introducing the technical framework Let X = L2 ([0, π]) and A be the operator Au := u with domain D(A) := {u ∈ X : u ∈ X , u(0) = u(π) = 0} It is well-known that A is the infinitesimal generator of an analytic semigroup (T (t ))t ≥0 on X Furthermore, A has a discrete spectrum with eigenvalues of the form −n2 , n ∈ N, whose corresponding (normalized) eigenfunctions are given by: zn (ξ ) := π sin(nξ ) In addition, the following properties hold: (a) {zn : n ∈ N} is an orthonormal basis for X ; ∞ ∞ −n2 t u, zn zn and Au = − n=1 n2 u, zn zn , for u ∈ D(A); (b) For u ∈ X , T (t )u = n=1 e α (c) It is possible to define the fractional power (−A) , α ∈ (0, 1), as a closed linear operator over its domain D((−A)α ) More ∞ 2α precisely, the operator (−A)α : D((−A)α ) ⊆ X → X is given by (−A)α u = u, zn zn , for all u ∈ D((−A)α ), n =1 n ∞ 2α where D((−A)α ) = {u ∈ X : n u , z z ∈ X }; n n n=1 (d) If Xα is the space D((−A)α ) endowed with the graph norm · α , then Xα is a Banach space Moreover, for < β ≤ α ≤ 1, Xα ⊂ Xβ ; the inclusion Xα → Xβ is completely continuous and there are constants Cα > such that T (t ) L(Xα ;X ) ≤ Ct αα for t > Consider the neutral partial differential equation with state-dependent delay π t d u(t , ξ ) + dt −∞ = ∂ u(t , ξ ) + ∂ξ b(t − s, η, ξ )u(s, η)dηds t a(s − t )u(s − ρ1 (t )ρ2 ( u(t ) ), ξ )ds, t ∈ [0, a], ξ ∈ [0, π], (4.1) −∞ u(t , 0) = u(t , π ) = 0, (4.2) u(τ , ξ ) = ϕ(τ , ξ ), t ∈ [0, a], τ ≤ 0, ≤ ξ ≤ π , (4.3) where ϕ ∈ C0 × L2 (g ; X ) To treat this system, we will assume that g (·) satisfies the conditions (g-5)–(g-7) in [28] We know from Theorems 1.37 and 7.1.1 in [28] that Cb ((−∞, 0]; X ) is continuously included in B Additionally, we will assume that the functions ρi : R → [0, ∞), a : R → R are continuous; LF := ∂ b(s,η,ξ ) ∂ξ (a) The functions b(s, η, ξ ), Lg := max   π  −∞ (a2 (s)) ds g (s) < ∞; and that the following condition holds: are measurable, b(s, η, π ) = b(s, η, 0) = and π 0 −∞ g (s) ∂ i b(s, η, ξ ) ∂ξ i 2 dηdsdξ   : i = 0, < ∞  By defining the maps ρ, G, F : [0, a] × B → X by ρ(t , ψ) := ρ1 (t )ρ2 ( ψ(0) ), π G(ψ)(ξ ) := −∞ b(s, ν, ξ )ψ(s, ν)dν ds, 0 F (ψ)(ξ ) := a(s)ψ(s, ξ )ds, −∞ we can transform (4.1)–(4.3) into the system (1.1) and (1.2) From these definitions, it follows that G, F are bounded linear operators with G L(X ) ≤ LG and F L(X ) ≤ LF Moreover, a straightforward estimation involving (a) enables us to prove that G is D((−A) )-valued with (−A)1/2 G ≤ LG , which implies that G is completely continuous from [0, a] × B into X since the inclusion i : X → X is completely continuous Thus, the assumptions HY , H2 , H3 all hold with Y = X The following results are consequences of Theorem 3.1 and Remark 3.2 The proofs are omitted for brevity Proposition 4.1 Assume that condition Hϕ holds and that the functions ρ1 , ρ2 are bounded If Ka LG + 2C1 then there exists a mild solution of (4.1)–(4.3) Proposition 4.2 Assume that ϕ ∈ Cb ((−∞, 0]; X ) If Ka LG + 2C1 of (4.1)–(4.3) √ a + LF a √ a + LF a < 1, < 1, then there exists a mild solution Acknowledgements The authors wish to thank the referees for their comments and suggestions The work of the third author was supported in part by FONDECYT-CONICYT, Project 1020259 and Project 7020259 E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 1267 References [1] Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther, Jianhong Wu, Functional Differential Equations with State-Dependent Delays: Theory and Applications, in: Handbook of Differential Equations: Ordinary Differential Equations, vol 3, 2006, pp 435–545 [2] Ovide Arino, Khalid Boushaba, Boussouar Khalid, Ahmed A mathematical model of the dynamics of the phytoplankton-nutrient system Spatial heterogeneity in ecological models (Alcalá de Henares, 1998), Nonlinear Anal RWA (1) (2000) 69–87 [3] Walter Aiello, H.I Freedman, Jianhong Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J Appl Math 52 (3) (1992) 855–869 [4] Maria Bartha, Periodic solutions for differential equations with state-dependent delay and positive feedback, Nonlinear Anal TMA 53 (6) (2003) 839–857 [5] Yulin Cao, Jiangping Fan, Thomas Gard, The effects of state-dependent time delay on a stage-structured population growth model, Nonlinear Analysis TMA 19 (2) (1992) 95–105 [6] Alexander Domoshnitsky, Michael Drakhlin, Elena Litsyn, On equations with delay depending on solution, Nonlinear Analysis TMA 49 (5) (2002) 689–701 [7] Ferenc Hartung, Linearized stability in periodic functional differential equations with state-dependent delays, J Comput Appl Math 174 (2) (2005) 201–211 [8] Ferenc Hartung, Parameter estimation by quasilinearization in functional differential equations with state-dependent delays: A numerical study, in: Proceedings of the Third World Congress of Nonlinear Analysis, Part (Catania, 2000), Nonlinear Analysis TMA 47 (7) (2001) 4557–4566 [9] Ferenc Hartung, Terry L Herdman, Janos Turi, Parameter identification in classes of neutral differential equations with state-dependent delays, Nonlinear Analysis TMA Ser A: Theory Methods 39 (3) (2000) 305–325 [10] Ferenc Hartung, Janos Turi, Identification of parameters in delay equations with state-dependent delays, Nonlinear Analysis TMA 29 (11) (1997) 1303–1318 [11] Y Kuang, H.L Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Analysis TMA 19 (9) (1992) 855–872 [12] Zhihui Yang, Jinde Cao, Periodic solutions for general nonlinear state-dependent delay logistic equations, Nonlinear Anal 66 (6) (2007) 1378–1387 [13] R.D Driver, A neutral system with state-dependent delay, J Differential Equations 54 (1) (1984) 73–86 [14] Ferenc Hartung, Terry L Herdman, Janos Turi, On existence, uniqueness and numerical approximation for neutral equations with state-dependent delays Volterra centennial (Tempe, AZ, 1996), Appl Numer Math 24 (2–3) (1997) 393–409 [15] Zhihui Yang, Jinde Cao, Existence of periodic solutions in neutral state-dependent delays equations and models, J Comput Appl Math 174 (1) (2005) 179–199 [16] Eduardo Hernández, Andréa Prokopczyk, Luiz Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal RWA (4) (2006) 510–519 [17] Alexander V Rezounenko, Jianhong Wu, A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors, J Comput Appl Math 190 (1–2) (2006) 99–113 [18] J.K Hale, Verduyn Lunel, Introduction to Functional-Differential Equations, in: Appl Math Sci., vol 99, Springer-Verlag, New York, 1993 [19] M Adimy, K Ezzinbi, A class of linear partial neutral functional-differential equations with nondense domain, J Differential Equations 147 (2) (1998) 285–332 [20] Jack Hale, Partial neutral functional differential equations, Rev Roumaine Math Pures Appl 39 (4) (1994) 339–344 [21] Jianhong Wu, Theory and Applications of Partial Functional-Differential Equations, in: Applied Mathematical Sciences, vol 119, Springer-Verlag, New York, 1996 [22] J Wu, H Xia, Rotating waves in neutral partial functional-differential equations, J Dynam Differential Equations 11 (2) (1999) 209–238 [23] J Wu, H Xia, Self-sustained oscillations in a ring array of coupled lossless transmission lines, J Differential Equations 124 (1) (1996) 247–278 [24] E Hernández, H Henríquez, Existence results for partial neutral functional differential equations with unbounded delay, J Math Anal Appl 221 (2) (1998) 452–475 [25] E Hernández, H Henríquez, Existence of periodic solutions of partial neutral functional differential equations with unbounded delay, J Math Anal Appl 221 (2) (1998) 499–522 [26] E Hernández, Existence results for partial neutral integrodifferential equations with unbounded delay, J Math Anal Appl 292 (1) (2004) 194–210 [27] A Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Applied Mathematical Sciences, vol 44, Springer-Verlag, New York, Berlin, 1983 [28] Yoshiyuki Hino, Satoru Murakami, Toshiki Naito, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, vol 1473, Springer-Verlag, Berlin, 1991 [29] A Granas, J Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003 [30] A Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, in: PNLDE, vol 16, Birkhauser Verlag, Basel, 1995 [31] R.H Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Robert E Krieger Publ Co., Florida, 1987  ... [24] E Hernández, H Henríquez, Existence results for partial neutral functional differential equations with unbounded delay, J Math Anal Appl 221 (2) (1998) 452–475 [25] E Hernández, H Henríquez, ... state-dependent delays equations and models, J Comput Appl Math 174 (1) (2005) 179–199 [16] Eduardo Hernández, Andréa Prokopczyk, Luiz Ladeira, A note on partial functional differential equations... + J0 ) ϕ where J0 = supt ∈R(ρ − ) J ϕ (t ) B + Ka sup x(θ ) max{0,s} , s ∈ R(ρ − ) ∪ [0, a], E Hernández et al / Mathematical and Computer Modelling 49 (2009) 1260–1267 1263 In the rest of this