Existence Results for Partial Neutral Functional Differential Equations with Unbounded Delay

Mathematical and Computer Modelling 49 (2009) 1260–1267

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Mathematical and Computer Modelling

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Existence results for partial neutral functional differential equationswith state-dependent delayEduardo Hernández Morales a,∗, Mark A. McKibben b, Hernán R. Henríquez ca Departamento de Matemática, ICMC, Universidade de São Paulo, Caixa Postal 668, 13560-970 São Carlos SP, Brazilb Department of Mathematics and Computer Science, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD 21204, USAc Departamento de Matemática, Universidade de Santiago, Casilla 307, Correo-2, Santiago, Chile

a r t i c l e i n f o

Article history:Received 17 March 2008Received in revised form 22 June 2008Accepted 11 July 2008

Keywords:Abstract Cauchy problemState-dependent delay

a b s t r a c t

In this paperwe study the existence ofmild solutions for a class of first order abstract partialneutral differential equations with state-dependent delay.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we study the existence of mild solutions for a class of partial neutral functional differential equations withstate-dependent delay described by the abstract form

ddt(x(t)+ G(t, xt)) = Ax(t)+ F(t, xρ(t,xt )), t ∈ I = [0, a], (1.1)

x0 = ϕ ∈ B, (1.2)

where A is the infinitesimal generator of a compact semigroup of bounded linear operators (T (t))t≥0 defined on an abstractBanach space X; the function xs : (−∞, 0] → X , xs(θ) = x(s + θ), belongs to some abstract phase space B describedaxiomatically; 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 Chapter5 in [1], the papers [2–12] and the references therein. We also cite [13,14,9,15] for the case of neutral differential equationswith 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 materialconcerning 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 caseof unbounded delay.

∗ Corresponding author.E-mail addresses: [email protected] (E. Hernández), [email protected] (M.A. McKibben), [email protected] (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

2. Preliminaries

Throughout this paper, (X, ‖ · ‖) is a Banach space, A : D(A) ⊂ X → X is the infinitesimal generator of a compactsemigroup (T (t))t≥0 of bounded linear operators on X , and M̃ > 0 is a constant such that ‖T (t)‖ ≤ M̃ 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 satisfiesthe 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 inB,(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 > 0 is a constant; K ,M : [0,∞) → [1,∞), K is continuous, M is locally bounded, and H, K ,M areindependent of x(·).

(A1) For the function x(·) in (A), the function t → xt is continuous from [σ , σ + b] intoB.(B) The spaceB is complete.

We now consider some examples of phase spaces.

Example 2.1 (The Phase Spaces Cg , C0g ). Let g : (−∞, 0] → [1,∞) be a continuous, non-increasing function with g(0) = 1,which satisfies conditions (g-1) and (g-2) of [28]. Briefly, this means that the function γ (t) := sup−∞<θ≤−t

g(t+θ)g(θ) is locally

bounded on [0,∞) and that limθ→−∞ g(θ) = ∞.Let Cg(X) be the vector space consisting of the continuous functions ϕ : (−∞, 0] → X such that ϕg is bounded on

(−∞, 0], and let C0g (X) be the subspace of Cg(X) containing precisely those functionsϕ forwhichϕ(θ)

g(θ) → 0 as θ →−∞. The

spaces Cg and C0g , endowedwith the norm ‖ϕ‖g := supθ≤0‖ϕ(θ)‖

g(θ) , are both phase spaces which satisfy axioms (A), (A1), (B),see [28, Theorem 1.3.2] for details. Moreover, in this case K(t) = 1, for every t ≥ 0.

Example 2.2 (The Phase Space Cr× Lp(g;X)). Assume that g : (−∞,−r)→ R is a locally Lebesgue integrable function andthat there exists a non-negative and locally bounded function η on (−∞, 0] such that g(ξ + θ) ≤ η(ξ)g(θ), for all ξ ≤ 0and θ ∈ (−∞,−r)\Nξ , where Nξ ⊆ (−∞,−r) is a set with Lebesguemeasure zero. The space Cr × Lp(g; X), r ≥ 0, p ≥ 1,consists of all classes of functions ϕ : (−∞, 0] → X such that ϕ is continuous on [−r, 0] and g‖ϕ‖p ∈ L1((−∞,−r)). Theseminorm in Cr × Lp(g; X) is defined by

‖ϕ‖B := sup{‖ϕ(θ)‖ : −r ≤ θ ≤ 0} +(∫

−r

−∞

g(θ)‖ϕ(θ)‖pdθ)1/p

.

Under these conditions, B = Cr × Lp(g; X) satisfies axioms (A), (A1), (B). Moreover, for r = 0 and p = 2 we have that

H = 1, K(t) = 1+(∫ 0−t g(θ)dθ

)1/2andM(t) = η(−t)1/2 for t ≥ 0.

Remark 2.1. Let ϕ ∈ B and t ≤ 0. The notation ϕt represents the function defined by ϕt(θ) = ϕ(t+θ). Consequently, if thefunction x(·) in axiom (A) is such that x0 = ϕ, then xt = ϕt . We observe that ϕt is well-defined for t < 0 since the domainof ϕ is (−∞, 0]. We also note that, in general, ϕt 6∈ B; consider, for instance, a discontinuous function in Cr × Lp(g;X) forr > 0.

The terminology and notations employed in this manuscript coincide with those generally used in functional analysis. Inparticular, for Banach spaces (Z, ‖ · ‖Z ), (W , ‖ · ‖W ), the notation L(Z;W ) stands for the Banach space of bounded linearoperators from Z intoW , and we abbreviate this notation toL(Z)when Z = W . Moreover, Br(x, Z) denotes the closed ballwith center at x and radius r > 0 in Z and for a bounded function ξ : I → Z and 0 ≤ t ≤ awe 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 0 ∈ D. Let F : D → D be acompletely continuous map. Then, either the set {x ∈ D : x = λF(x), for some 0 < λ < 1} is unbounded or F has a fixed pointin D.

The remainder of the paper is divided into two sections. Specifically, by using Theorem 2.1 and the classical Schauderfixed point theorem, in Section 3 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 4.

1262 E. Hernández et al. / Mathematical and Computer Modelling 49 (2009) 1260–1267

3. Existence results

In this section we study the existence of mild solutions for the abstract Cauchy problem (1.1) and (1.2). Throughout thissection, ϕ ∈ B is a fixed function, (Y , ‖ · ‖Y ) is a Banach space continuously included in X , and the following two conditionsare 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 > 0and there exists a positive function γ ∈ L1([0, a]) such that ‖AT (t)‖L(Y ;X) ≤ γ (t), for every t ∈ I .

Hϕ LetR(ρ−) = {ρ(s, ψ) : (s, ψ) ∈ I×B, ρ(s, ψ) ≤ 0}. The function t → ϕt is continuous fromR(ρ−) intoB and thereexists a continuous and bounded function Jϕ : R(ρ−)→ (0,∞) such that ‖ϕt‖B ≤ Jϕ(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, ifthe space of continuous and bounded functions Cb((−∞, 0]; X) is continuously included inB, then

‖ψt‖B ≤ Lsupθ≤0‖ψ(θ)‖

‖ψ‖B‖ψ‖B, t ≤ 0, ψ 6= 0, ψ ∈ Cb((−∞, 0]; X). (3.1)

It is not difficult to show that Cb((−∞, 0]; X) is continuously included in both Cg(X) and C0g (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 relatedto 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 functionm : I → [0,∞) and a continuous non-decreasing functionW : [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 > 0 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 uniformconvergence 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 } isequicontinuous 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; thefunction t → AT (t − s)G(s, xs) is integrable on [0, t), for every t ∈ [0, a]; and

x(t) = T (t)(ϕ(0)+ G(0, ϕ))− G(t, xt)−∫ t

0AT (t − s)G(s, xs)ds+

∫ t

0T (t − s)F(s, xρ(s,xs))ds, t ∈ I.

Remark 3.3. Let x(·) be a function as in axiomA. Let usmention that the conditionsHY,H2,H3 are linked to the integrabilityof the function s → AT (t − s)G(s, xs). In general, except for the trivial case in which A is a bounded linear operator, theoperator function t → AT (t) is not integrable over [0, a]. However, if condition HY holds and G satisfies either assumptionH2 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) sups∈[0,a]

‖G(s, xs)‖Y ,

that s 7→ AT (t − s)G(s, xs) is integrable over [0, t), for every t ∈ [0, a]. For non-trivial examples of spaces Y for whichcondition 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 constantsdefined byMa = 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‖B ≤ (Ma + Jϕ

0 )‖ϕ‖B + Ka sup ‖x(θ)‖max{0,s}, s ∈ R(ρ−) ∪ [0, a],

where Jϕ0 = supt∈R(ρ−) Jϕ(t).

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 onB defined by

[S(t)ψ](θ) :={ψ(0) for − t ≤ θ ≤ 0,ψ(t + θ) 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

Ka

(LG(1+

∫ a

0γ (s)ds)+ M̃ lim inf

ξ→∞+

W (ξ)ξ

∫ a

0m(s)ds

)< 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‖a = sups∈[0,a] ‖u(s)‖, anddefine the operator Γ : Y → Y by

Γ x(t) = T (t)(ϕ(0)+ G(0, ϕ))− G(t, x̄t)−∫ t

0AT (t − s)G(s, x̄s)ds+

∫ t

0T (t − s)F(s, x̄ρ(s,x̄s))ds, s ∈ I,

where x̄ : (−∞, a] → X is defined by the relation x̄0 = ϕ and x̄ |I = x. Following the discussion outlined in Remark 3.3 andusing condition Hϕ , we infer that the functions s 7→ AT (t − s)G(s, xs) and s 7→ T (t − s)F(s, xs) are integrable over [0, t), foreach t ∈ [0, a], which enables us to conclude that Γ is a well-defined operator from Y into Y .Let ϕ̄ : (−∞, a] → X be the extension of ϕ to (−∞, a] such that ϕ̄(θ) = ϕ(0) on I . We affirm that there exists r > 0

such that Γ (Br(ϕ̄|I , Y )) ⊂ Br(ϕ̄|I , Y ). Indeed, if this property is false, then for every r > 0 there exist xr ∈ Br(ϕ̄|I , Y ) andt 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)tr )− G(t r , S(t r)ϕ)‖

+

∫ tr

0‖AT (t r − s)‖L(Y ;X)‖G(s, (xr)s)− G(s, S(s)ϕ)‖ds

+

∫ tr

0‖AT (t r − s)‖L(Y ;X)‖G(s, S(s)ϕ)‖ds+ M̃

∫ tr

0m(s)W (‖xrρ(s,(xr )s)‖)ds

≤ ‖T (t r)(ϕ(0)− G(0, ϕ))− ϕ(0)‖ + ‖G(s, S(s)ϕ)‖a + LGKa‖xr(s)− ϕ(0)‖tr

+ LGKa

∫ tr

0γ (t r − s)‖xr(s)− ϕ(0)‖sds+ ‖G(s, S(s)ϕ)‖a

∫ a

0γ (s)ds

+ M̃∫ tr

0m(s)W ((Ma + J

ϕ

0 )‖ϕ‖B + Ka(‖xr(s)− ϕ(0)‖s + ‖ϕ(0)‖))ds

≤ ‖T (t r)(ϕ(0)− G(0, ϕ))− ϕ(0)‖ + LGKar + ‖G(s, S(s)ϕ)‖a + LGKar∫ a

0γ (s)ds

+‖G(s, S(s)ϕ)‖a

∫ a

0γ (s)ds+ M̃W ((Ma + J

ϕ

0 + KaHM̃)‖ϕ‖B + Kar)∫ a

0m(s)ds

and hence

1 ≤ Ka

(LG(1+

∫ a

0γ (s)ds)+ M̃ lim inf

ξ→∞+

W (ξ)ξ

∫ a

0m(s)ds

),

which is contrary to our assumption.Let r > 0 such thatΓ (Br(ϕ̄|I , Y )) ⊂ Br(ϕ̄|I , Y ). In order to prove thatΓ (·) is a continuous condensingmap fromBr(ϕ̄|I , Y )

into Br(ϕ̄|I , Y ), we introduce the decomposition Γ = Γ1 + Γ2, where

Γ1x(t) = T (t)(ϕ(0)+ G(0, ϕ))− G(t, x̄t)−∫ t

0AT (t − s)G(s, x̄s)ds, t ∈ I,

Γ2x(t) =∫ t

0T (t − s)F(s, x̄ρ(s,x̄s))ds, t ∈ I.

From the proof of [16, Theorem 2.2], we know that Γ2 is completely continuous. Moreover, from the phase space axiomsand H3 we obtain

‖Γ1u(t)− Γ1v(t)‖ ≤ KaLG

(1+

∫ a

0γ (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(ϕ̄|I , Y ) and so that Γ is a condensing operator on Br(ϕ̄|I , Y ). Consequently, fromthe previous remark and [31, Theorem 4.3.2] we deduce the existence of a mild solution for the system (1.1) and (1.2). Theproof 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 andthat G : [0, a] ×B → X is completely continuous. If µ = 1− Kac1(1+

∫ a0 γ (s)ds) > 0 and

M̃Kaµ

∫ a

0m(s)ds <

∫∞

D

dsW (s)

,

where D = (Ma + Jϕ

0 + M̃HKa)‖ϕ‖B +KaCµand

C = M̃‖G(0, ϕ)‖ +(1+

∫ a

0γ (s)ds

) (c1‖ϕ‖B(Ma + J

ϕ

0 + KaM̃H)+ c2),

then there exists a mild solution of (1.1) and (1.2).

Proof. On the space BC = {u : (−∞, a] → X; u0 = 0, u|I ∈ C(I; X)} endowed with the norm ‖u‖a = sups∈[0,a] ‖u(s)‖,we define the operator Γ : BC → BC by

Γ x(t) =

0, t ∈ (−∞, 0],

T (t)G(0, ϕ)− G(t, xt)−∫ t

0AT (t − s)G(s, xs)ds+

∫ t

0T (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 apriori estimates for the solutions of the integral equation z = λΓ z, λ ∈ (0, 1). Let xλ be a solution of the integral equationz = λΓ z, λ ∈ (0, 1) and αλ(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‖B + c2 +∫ t

0γ (t − s)(c1‖(xλ)s‖B + c2)ds+ M̃

∫ t

0m(s)W (‖(xλ)s‖)ds

≤ M̃‖G(0, ϕ)‖ + c1(Ma + Jϕ

0 )‖ϕ‖B + c1KaM̃H‖ϕ‖B + c2 + c1Ka‖αλ(t)‖

+c1(Ma + Jϕ

0 )‖ϕ‖B

∫ a

0γ (s)ds+ c1KaM̃H‖ϕ‖B

∫ a

0γ (s)ds+ c2

∫ a

0γ (s)ds+ c1Kaαλ(t)

∫ a

0γ (s)ds

+ M̃∫ t

0m(s)W ((Ma + J

ϕ

0 + M̃HKa)‖ϕ‖B + Kaαλ(s))ds

≤ M̃‖G(0, ϕ)‖ + c1‖ϕ‖B

(Ma + J

ϕ

0 + KaM̃H + (Ma + KaM̃H)∫ a

0γ (s)ds

)+ c2

(1+

∫ a

0γ (s)ds

)+ c1Ka

(1+

∫ a

0γ (s)ds

)αλ(t)+ M̃

∫ t

0m(s)W ((Ma + J

ϕ

0 + M̃HKa)‖ϕ‖B + Kaαλ(s))ds.

Consequently,

‖αλ(t)‖ ≤Cµ+M̃µ

∫ t

0m(s)W

((Ma + Jϕ + M̃HKa)‖ϕ‖B + Kaαλ(s)

)ds.

Using the notation ξλ(t) = (Ma + Jϕ

0 + M̃HKa)‖ϕ‖B + Kaα(t), we obtain that

ξλ(t) ≤ (Ma + Jϕ

0 + M̃HKa)‖ϕ‖ +KaCµ+M̃Kaµ

∫ t

0m(s)W

(ξλ(s)

)ds.

Denoting by βλ(t) the right-hand side of the above inequality, it follows that

β ′λ(t) ≤M̃Kaµm(t)W (βλ(t))

and hence∫ βλ(t)

βλ(0)=D

dsW (s)

≤M̃Kaµ

∫ a

0m(s)ds <

∫∞

D

dsW (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 (Γix)0 = 0 and

Γ1x(t) = T (t)G(0, ϕ)− G(t, xt), t ∈ [0, a],

Γ2x(t) = −∫ t

0AT (t − s)G(s, xs)ds, t ∈ [0, a],

Γ3x(t) =∫ t

0T (t − s)F(s, xρ(s,xs))ds, t ∈ [0, a].

The continuity of the function Γ2 is easily shown. Moreover, from the proof of [16, Theorem 2.2] we know that Γ3is completely continuous. It remains to show that Γ1 is completely continuous and that Γ2 is a compact map. To thisend, 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 BC and u ∈ BC such that un → u. From the phase space axioms we infer that the setU = [0, a] × {uns , us : s ∈ [0, a], n ∈ N} is relatively compact in [0, a] × B and that uns → us uniformly on [0, a] asn→∞. Thus, G is uniformly continuous on U , so that G(s, uns )→ G(s, us) uniformly on [0, a] as n→∞, which shows thatΓ1 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,BC).

Step 1. The set (Γ2Br)(t) = {Γ2x(t) : x ∈ Br} is relatively compact in X for each t ∈ I.The assertion clearly holds for t = 0. Let 0 < ε < t ≤ a. For u ∈ Br we see that

Γ2u(t) = −T (ε)∫ t−ε

0AT (t − ε − s)G(s, us)ds−

∫ t

t−εAT (t − s)G(s, us)ds

∈ T (ε){x ∈ X : ‖x‖ ≤ (c1Kar + c2)∫ a

0γ (s)ds} + Br∗(0, X),

where r∗ = (c1Kar + c2)∫ tt−ε γ (s)ds. From this, we can infer that (Γ2Br)(t) is totally bounded in X and hence relatively

compact in X .Step 2. The set of functions Γ2Br = {Γ2x : x ∈ Br} is equicontinuous on I .Let t ∈ (0, a). For u ∈ Br and h > 0 such that t + h ∈ [0, a], we obtain

‖Γ2u(t + h)− Γ2u(t)‖ =∥∥∥∥(T (h)− I)Γ2u(t)+ ∫ t+h

tAT (t − s)G(s, us)ds

∥∥∥∥≤ ‖(T (h)− I)Γ2u(t)‖ + (c1Kar + c2)

∫ t+h

tγ (s)ds.

Since the set (Γ2Br)(t) is relatively compact in X and (T (t))t≥0 is strongly continuous, it follows that ‖(T (h)−I)Γ2u(t)‖ → 0as h → 0 uniformly for u ∈ Br , which from the last inequality enables us to conclude that Γ2Br is right equicontinuous att ∈ (0, a). In a similar manner we can prove that Γ2Br 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 ∈ BC . 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 thesespecial 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 finitedimensional, and

Ka

(LG(1+ e‖A‖a)+ e‖A‖a lim inf

ξ→∞+

W (ξ)ξ

∫ a

0m(s)ds

)< 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 × Band that G satisfies H2 with Y = X. Suppose µ = 1− Kac1(1+ e‖A‖a) > 0 and

e‖A‖aKaµ

∫ a

0m(s)ds <

∫∞

D

dsW (s)

,

where D = (Ma + Jϕ

0 + e‖A‖aHKa)‖ϕ‖B + KaC

µand

C = e‖A‖a‖G(0, ϕ)‖ + (1+ e‖A‖a)(c1‖ϕ‖B(Ma + Kae‖A‖aH)+ 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

4. Examples

In this sectionwe use our abstract results to treat a concrete partial neutral differential equation.We begin by introducingthe 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 hasa discrete spectrum with eigenvalues of the form−n2, n ∈ N, whose corresponding (normalized) eigenfunctions are given

by: zn(ξ) :=√2πsin(nξ). In addition, the following properties hold:

(a) {zn : n ∈ N} is an orthonormal basis for X;(b) For u ∈ X , T (t)u =

∑∞

n=1 e−n2t〈u, zn〉zn and Au = −

∑∞

n=1 n2〈u, zn〉zn, for u ∈ D(A);

(c) It is possible to define the fractional power (−A)α ,α ∈ (0, 1), as a closed linear operator over its domainD((−A)α). Moreprecisely, the operator (−A)α : D((−A)α) ⊆ X → X is given by (−A)αu =

∑∞

n=1 n2α〈u, zn〉zn, for all u ∈ D((−A)α),

where D((−A)α) = {u ∈ X :∑∞

n=1 n2α〈u, zn〉zn ∈ X};

(d) If Xα is the spaceD((−A)α) endowedwith the graph norm ‖·‖α , then Xα is a Banach space.Moreover, for 0 < β ≤ α ≤ 1,Xα ⊂ Xβ ; the inclusion Xα → Xβ is completely continuous and there are constants Cα > 0 such that ‖T (t)‖L(Xα ;X) ≤

Cαtα

for t > 0.

Consider the neutral partial differential equation with state-dependent delayddt

[u(t, ξ)+

∫ t

−∞

∫ π

0b(t − s, η, ξ)u(s, η)dηds

]=∂2u(t, ξ)∂ξ 2

+

∫ 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, t ∈ [0, a], (4.2)u(τ , ξ) = ϕ(τ , ξ), τ ≤ 0, 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 :=(∫ 0−∞

(a2(s))g(s) ds

) 12<∞; and that the following condition holds:

(a) The functions b(s, η, ξ), ∂b(s,η,ξ)∂ξ

are measurable, b(s, η, π) = b(s, η, 0) = 0 and

Lg := max

(∫ π

0

∫ 0

−∞

∫ π

0

1g(s)

(∂ ib(s, η, ξ)

∂ξ i

)2dηdsdξ

) 12

: i = 0, 1

<∞.

By defining the maps ρ,G, F : [0, a] ×B → X byρ(t, ψ) := ρ1(t)ρ2(‖ψ(0)‖),

G(ψ)(ξ) :=∫ 0

−∞

∫ π

0b(s, ν, ξ)ψ(s, ν)dνds,

F(ψ)(ξ) :=∫ 0

−∞

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 linearoperators with ‖G‖L(X) ≤ LG and ‖F‖L(X) ≤ LF . Moreover, a straightforward estimation involving (a) enables us to provethat G is D((−A)

12 )-valued with ‖(−A)1/2G‖ ≤ LG, which implies that G is completely continuous from [0, a] × B into X

since the inclusion i : X 12→ X is completely continuous. Thus, the assumptions HY,H2,H3 all hold with Y = X 1

2.

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

√a+ LFa

)< 1,

then there exists a mild solution of (4.1)–(4.3).

Proposition 4.2. Assume that ϕ ∈ Cb((−∞, 0]; X). If Ka(LG + 2C1

√a+ LFa

)< 1, then there exists a mild solution

of (4.1)–(4.3).

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

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